Posts | James Colliander

Digital public goods for Earth system management: U.S. Greenhouse Gas Center launches

Tue, 05 Dec 2023 17:01:48 +0000

(This post first appeared on the 2i2c blog.)

Abstract

The International Interactive Computing Collaboration (2i2c.org), working with NASA VEDA, Development Seed and other partners, operates an interactive computing platform for The U.S. Greenhouse Gas Center. The U.S. GHG Center, announced yesterday at the 28th annual United Nations Climate Conference (COP-28) in Dubai, is an interagency collaboration of the Environmental Protection Agency (EPA), the National Aeronautics and Space Administration (NASA), the National Institute of Standards and Technology (NIST), and the National Ocean and Atmospheric Administration (NOAA). This note places the launch of the U.S. GHG Center in a scientific, international, and national context and argues that similar digital public goods are needed for humanity to understand and manage the Earth system.

Scientific Context

It was controversial in 1827 when Joseph Fourier (the discoverer of the law of heat conduction) argued ¹ that the atmosphere keeps the Earth warm, like a puffy down comforter, but it’s not now. Gases in the atmosphere trap heat near Earth. How much heat is trapped depends on the gas mixture. Putting more heat-trapping gases in is like putting a wool blanket on top of the down comforter. Human activity since industrialization is injecting lots more heat-trapping gas into the atmosphere and changing the Earth’s climate.

The science is clear. The up-to-date consensus view of the global scientific community is expressed in the Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC):

International Context

The international community officially recognized human-influenced climate change at the World Climate Conference (WCC-1) ² in 1979. The 1979 declarationis remarkably prescient and detailed. A complex and interconnected collection of scientific and diplomatic activities were catalyzed by WCC-1. Some important milestones from this history are captured in the chart and numbered list below.

gantt
dateFormat YYYY-MM-DD
title International Climate Change Milestones
section Study
WCC-1 (1979) :crit, done, admin0, 1979-02-12, 1979-02-23
WCP :crit, adminA, 1979-06-01, 2025-12-31
CMIP1: crit, done, adminT, 1995-01-01,1995-12-31
CMIP2: crit, done, adminR, 1997-01-01, 1998-12-31
CMIP2+: crit,done,adminS, 2000-05-09, 2001-12-31
CMIP3 :crit, done, adminP, 2004-10-01, 2006-12-31
CMIP5 :crit, done, adminO, 2008-09-01, 2013-03-15
CMIP6 :crit, done, adminQ, 2014-02-01, 2024-12-31
IPCC :crit, admin1, 1988-12-06, 2025-12-31
IPCC-AR1 :crit, done, adminH, 1990-08-01, 1992-06-30
IPCC-AR2 :crit, done, adminI, 1995-01-01,1995-12-31
IPCC-AR3 :crit,done, adminJ, 2001-01-01, 2001-12-31
IPCC-AR4 :crit, done,adminK, 2007-01-01,2007-12-31
IPCC awarded Nobel Prize :crit, done, adminN, 2007-10-12, 2007-11-12
IPCC-AR5 :crit,done,adminL, 2014-01-01,2014-12-31
IPCC-AR6 :crit,done,adminM,2023-01-01,2023-12-31
section Treaties
Rio Earth Summit (1992) :crit, done, adminC, 1992-06-03, 1992-06-14
UNFCC :crit, admin2, 1994-03-21, 2025-12-31
Berlin (COP-1) :crit, done, adminE, 1995-03-28, 1995-04-07
Byrd-Hagel Resolution :crit, done, adminX, 1997-07-25, 1997-07-30
Kyoto (COP-3) :crit, done, adminD, 1997-12-01, 1997-12-10
Kyoto Protocol :crit, done, admin3, 1997-12-11, 2020-12-31
Paris (COP-21) :crit, done, adminF, 2015-11-30, 2015-12-12
Paris Agreement :crit, adminG, 2016-11-04, 2025-12-31
Glasgow (COP-26) :crit, done, adminV, 2021-10-31, 2021-11-12
Dubai (COP-28) :crit, adminW, 2023-11-20, 2023-12-12
ETF :crit, adminU, 2024-01-01, 2025-12-31

The table above describes a subset (for a more systematic review see ³, ⁴) of key milestones in global efforts to understand and address climate change. A glossary of acronyms and additional background:

The First World Climate Conference (WCC-1) ² was held in 1979.
The World Climate Programme (WCP), an activity overseen by the World Meteorological Organization was established after WCC-1. WCP, in partnership with other organizations, operates programs (e.g. the World Climate Research Program (WCRP)) that organize and integrate international scientific efforts to understand the climate. The WMO also operates the Integrated Global Greenhouse Gas Information System (IG3IS), a natural partner for the emerging work described below.
WCRP manages the Common Model Intercomparison Project (CMIP). CMIP serves as a kind of league for intercomparing models of the Earth’s climate system developed by teams who approach the problems with different methods and designs. Intercomparison, an approach that enables finding the consensus views of teams with divergent approaches to problems, is used in other modeling scenarios.
Research papers on the climate are rapidly produced by scholars from essentially all knowledge disciplines. This overwhelming stream of content, like snowflakes in a blizzard, is coalesced into coherent and carefully scrutinized IPCC Assessment Reports by the Intergovernmental Panel on Climate Change (IPCC).
The Earth Summit held in Rio de Janeiro in 1992 led to the United Nations Framework Convention on Climate Change (UNFCC). The UNFCCC is an international treaty that recognizes the dangers to the climate system caused by human activity, calls for ongoing study, and establishes recurring Conference of the Parties (COP) meetings.
COP-3 (Kyoto) led to the Kyoto Protocol Treaty.
The Byrd-Hagel Resolution was a unanimous United States Senate Resolution that stipulated the United States would not sign treaties promising greenhouse gas reductions by developed countries that did not mandate similar reductions from developing countries. This killed USA participation in the Kyoto Protocol Treaty.
The Paris Agreement, established at COP-21 (Paris), effectively replaces the Kyoto Protocol, includes specifications by participant countries on greenhouse gas reductions called National Determined Contributions (NDCs). The United States entered the Paris Agreement under President Obama, exited under President Trump and rejoined under President Biden.
COP-26 (Glasgow)established an accountability system for the Paris Agreement called the Enhanced Transparency Framework (ETF). Participant countries to the Paris Agreement will submit their first Biennial Transparency Reports (BTR1) under the ETF in 2024.

🇺🇸 U.S.A. Context

The United States Environmental Protection Agency (EPA) annually releases the Inventory of U.S. Greenhouse Gas Emissions and Sinks reports. These reports are submitted to the United Nations in accordance with the UNFCCC. The EPA openly shares software, tools, data, and builds capacity to assist other nations to assemble their own greenhouse gas inventories.

An Interagency Working Group (IWG) on the Social Cost of Carbon was set up by the Obama Administration in 2010. The IWG, renamed as the Interagency Working Group on Social Cost of Greenhouse Gases in 2016, synthesizes research on integrated assessment modelling to quantify the dollar costs associated to damage caused by an incremental increase in GHG emissions in a given year. Quantifying the impacts of GHG emissions in monetary terms is vital to effective rulemaking across the Federal Government. This EPA fact sheet on the social costs of carbon provides further background. A 2017 consensus report ⁵ of the National Academies of Science Engineering and Medicine (NASEM) offered recommendations for ongoing research to improve the assignment of social costs to GHG emissions.

Other federal agencies have developed expertise, data, and analyses that give insights into GHG emissions that compliment and potentially extend the Inventory reports developed annually by the EPA. How should the United States integrate federal agency efforts to monitor and measure greenhouse gas emissions? A 2022 NASEM consensus report ⁶ investigated this question. In January of this year, the Biden Administration released guidance⁷ for federal agencies on incorporating GHG emissions information in policies and reports. Shortly thereafter, a draft federal strategy to advance an integrated greenhouse gas monitoring system was released by NASA with a request for public input.

Some important insights from the NASEM consensus report, the draft federal strategy, and the IWG reports:

Data streams on greenhouse gas emissions can be sorted into two broad categories:
- Activity-based: Activity-based data, sometimes called “bottom-up” data, quantify GHG emissions by measuring activities that generate GHG emissions. Economic or business activity data (gallons of diesel sold in a county on Tuesday; miles flown by a 747 in October) can be converted into quantified GHG emissions information.
- Atmospheric-based: Atmosphere-based data, sometimes called “top-down” data, quantify GHG emissions by performing atmospheric measurements. For example, the Orbiting Carbon Observatory (OCO2) remotely senses $CO_2$ from space.
- A hybrid approach that blends activity-based and atmospheric-based GHG data has the potential to provide new insights.
Interoperable and easily accessed tools and data products for analyzing GHG emissions information and assigning costs should be made available across the Federal government.
GHG emissions information is needed in scenarios outside the international context of participating nations reporting for UNFCCC and Paris Agreement ETF compliance:
- Subnational governments – cities, states, provinces, counties – want GHG data products to measure their progress on emission reduction programs.
- Facilities – harbours, toll roads, power plants, factories, universities – similarly want GHG data products.
- Companies – airlines, trucking, construction – want GHG data products.
Reliable and transparent GHG emissions information is required to enable effective environmental-social-governance (ESG) investment without “greenwashing”.
Accurate and improving quantifications of the social costs associated to GHG emissions require ongoing research.

The U.S. Greenhouse Gas Center

The launch of U.S. GHG Center, an interagency collaboration of the Environmental Protection Agency (EPA), the National Aeronautics and Space Administration (NASA), the National Institute of Standards and Technology (NIST), and the National Ocean and Atmospheric Administration (NOAA), was announced on 2023-12-04 at COP-28 (Dubai)

How much GHG emission is generated through oil and gas production? How much GHG emission is generated by urban centers? Do landfills contribute significant GHG emissions? How do human-generated GHG emissions compare to natural sources of GHG emissions? The U.S. GHG Center is designed to assemble the data, tools, and people to scientifically address these kinds of questions.

The U.S. Greenhouse Gas Center builds on EPA’s leadership to openly share the data and tools for the Inventory and the 2023 Year of Open Science. Instead of building a walled garden with proprietary technology from a vendor selected through RFP, the Center launched a prototype platform using curated open source tools integrated with public federal data. This open toolchain approach amplifies the open data efforts developed over the past two decades.

The U.S. GHG Center’s interactive computing platform is open source science infrastructure. The platform is:

transparently operated by 2i2c on a cloud data center under a right to replicate that ensures zero vendor lock-in with data integrations and visualizations built with using software from a vibrant open source ecosystem by Development Seed, NASA VEDA, and collaborators;
proximate to and optimized ⁸ for analyzing geospatial data (e.g. NASA, NOAA);
designed to be a digital public good.

Research, data and recommendations developed by scientists and engineers that influence policies set by democratic governments should be accessible by voters. No entity should own the ways humans communicate and learn about the Earth system. The U.S. Greenhouse Gas Center’s generous approach to digital infrastructure – an open toolchain adjacent to open data – is vital for democracy and should be replicated across government agencies worldwide.

References

Mémoire sur les températures du globe terrestre et des espaces planétaires. in Oeuvres de Fourier: Publiées par les soins de Gaston Darboux (eds. Fourier, J. B. J. & Darboux, J. G.) vol. 2 95–126 (Cambridge University Press, 2013). ↩︎
World Climate Conference. World Climate Conference - Declaration and supporting documents. https://library.wmo.int/records/item/54699-world-climate-conference-declaration-and-supporting-documents (1979). ↩︎
Gupta, J. A history of international climate change policy. WIREs Climate Change 1, 636–653 (2010). ↩︎
Zillman, J. A history of climate activities. WMO Bulletin 58, (2009). ↩︎
Valuing Climate Changes: Updating Estimation of the Social Cost of Carbon Dioxide. (National Academies Press, 2017). doi:10.17226/24651. ↩︎
Greenhouse Gas Emissions Information for Decision Making: A Framework Going Forward. (National Academies Press, 2022). doi:10.17226/26641. ↩︎
CEQguidance. National Environmental Policy Act Guidance on Consideration of Greenhouse Gas Emissions and Climate Change. Federal Register https://www.federalregister.gov/documents/2023/01/09/2023-00158/national-environmental-policy-act-guidance-on-consideration-of-greenhouse-gas-emissions-and-climate (2023). ↩︎
Abernathey, R. P. et al. Cloud-Native Repositories for Big Scientific Data. Computing in Science & Engineering 23, 26–35 (2021). ↩︎

Democracy’s Duty to Open Science

Thu, 16 Nov 2023 19:33:06 +0000

In my role as a NASA TOPS Community Panelist, I participated in a meeting hosted by the Office of Science and Technology Policy (OSTP) at the White House (EEOB) yesterday. I thank Lisa-Joy Zgorski of the National Science Foundation (NSF) for encouraging me to write down a comment I made during our meeting. I also thank my colleagues at 2i2c and NASA TOPS.

Democracies have long recognized that citizens need to be able to read and write. Literacy is a requirement for citizens to vote and participate in governance. Citizens in a democracy have a right to literacy; democratic governments have a duty to literacy.

Citizens in democracies today need more than reading and writing skills. Effective participation in the democratic process requires citizens to have skills to learn and understand themselves, society, and the universe. Citizens need access and tools and skills to use information. Citizens in a democracy should have a right to participate in science — the collective human enterprise to understand. Democratic governments have a duty to open science.

2022 NASA SMD Data and Computing Architecture Study

Mon, 21 Nov 2022 21:50:43 -0800

NASA Science Mission Directorate (SMD) Data and Computing Architecture Study

2022 Jack Eddy Symposium

Mon, 21 Nov 2022 21:45:28 -0800

3rd Jack Eddy Cross Disciplinary Symposium

Why did UBC spend $70M for 3 acres in Surrey?

Wed, 03 Nov 2021 21:22:21 +0000

Of all the ways to spend $70M, the University of British Columbia announced yesterday that it had purchased three acres of real estate in Surrey, nowhere near the Point Grey, Okanagan or Robson Square campuses. UBC did not announce a major investment in data science the way peer institutions (UofT, Berkeley) have done. The University did not double down on innovation with a leveraged investment into the Digital Technology Supercluster which is headquartered at the Robson Square campus. No major investment toward decarbonizing the economy, additional classroom space, or a new research facility was made. Nope! UBC bought the Grace Hanin Community Church property.

I found this curious so I looked into it.

Where did UBC buy?

The Grace Hanin Community Church property looks like this today:

The property is conveniently located near King George station and sits strategically between Surrey Memorial Hospital and the Surrey campus of Simon Fraser University (SFU).

What’s the business case?

The mission-aligned purpose of this real estate investment is not conveyed well in yesterday’s announcement. UBC’s press release includes “artistic renderings” with people wearing white lab coats in front of a shiny new building. How will the new building be financed? What units of the university will occupy the new building? President Ono’s quote in the press release namechecks Fraser Health and First Nations Health Authority so perhaps there are plans to use this space for medical research and training programs. Are there plans for a public-private venture to build on the purchased site? Who are the anticipated co-investors and property developers?

Is it reasonable to pay $70M for three acres in Surrey? How is commercial real estate priced? I don’t have expertise on these questions but would like to know. While this purchase may make sense as a pure play real estate investment, it would be interesting to collectively understand the business logic and the larger scope of UBC’s strategic use of land and capital managed by UBC Properties Trust.

Is this linked to the Broadway Subway Project?

A research-driven transformation of the BC economy was championed by former UBC president Arvind Gupta in an address to the Vancouver Board of Trade in 2014. Politics intervened for a while but the vision eventually took root when the Broadway Subway project was funded, albeit only to Arbutus.

With the purchase of the Grace Hanin Community Church property, UBC has a place to develop facilities near the terminus of the Expo Line.

Is this connected to the new medical school?

SFU has been working for decades to establish a medical school. On first glance, this sounds like a great idea: more doctors; new facilities; better care for everyone. However, medical schools are costly and a new operation at SFU will likely cut into funds that would otherwise be allocated to UBC’s medical school. A shiny new school at SFU will also emerge as a competitor for federal CIHR funds. UBC successfully opposed the creation of a new medical school and maintained its Lower Mainland monopoly during the Gordon Campbell and Christy Clark led era for the Province. Then things changed. The BC NDP promised a new medical school on SFU’s Surrey campus during the last provincial election. SFU provided details on plans last month in a medical school information session.

The property purchased by UBC is a short walk from Surrey Memorial Hospital.

What about quantum?

The Province made another $17M investment for Surrey with the creation of the Quantum Algorithms Institute (QAI) in 2019. QAI is incorporated as an institute for British Columbia. That said, the Government’s document announcing QAI says the Institute “will be housed on the Simon Fraser University (SFU) Surrey campus and include partners at post-secondary institutes throughout B.C.” The QAI launch event in 2019 took place at SFU and with no representation from other universities. While UBC, together with the University of Victoria and SFU, is represented on the Board, the location of QAI could make SFU a more viable recipient of the $360M in quantum research funding promised by the federal government in Budget 2021.

From the new site in Surrey, UBC has the opportunity to establish quantum research programs aligned with and nearby the Quantum Algorithms Institute.

I look forward to learning more about why UBC bought three acres between SFU’s Surrey campus and the Surrey Memorial Hospital. A strategic unification of the higher education and health systems, newly linked by SkyTrain and improved infrastructure, has the potential to massively improve the Lower Mainland.

NSERC Open Data Exploration

Sat, 24 Nov 2018 00:00:00 +0000

NSERC Open Data Exploration

The 2018 federal budget of Canada 🇨🇦 announced significant new investments in research. The Canada Research Coordinating Committee is defining a work plan to promote “international and risky research; support for Canada’s next generation of researchers; equity, diversity and inclusion; and indigenous research and capacity building”.

This site explores the NSERC Awards Data with the goal of helping Canada to achieve the best outcomes for research and training of future researchers. Open analysis of research investments also helps advance the goals outlined in the Mandate Letter to the Minister of Science.

Canada’s Open Government Portal includes NSERC Awards Data from 1995 through 2016. The awards data (in .csv format) were copied to an Amazon Web Services S3 bucket and analyzed using computing resources from PIMS on cloud infrastructure from Compute Canada and Cybera.

You are invited to collaborate on the analysis of investments made by NSERC. Access to the data and source code are shared in a public repository on GitHub.

TODO Set up similar analyses for SSHRC

TODO Set up similar analysis for CIHR

Overview

Interactive Notebook: Exploring NSERC

NSERC Budget

Year	Total	Not Discovery	Discovery
1995	451159747	267113329	184046418
1996	434116337	247533171	186583166
1997	417146970	227810152	189336818
1998	477141539	266731883	210409656
1999	526133116	299956302	226176814
2000	537934451	294508093	243426358
2001	554698568	316537927	238160641
2002	615320903	357333281	257987622
2003	696997644	423803197	273194447
2004	765643760	488699482	276944278
2005	820640167	519567082	301073085
2006	854567999	545233090	309334909
2007	968713281	655321429	313391852
2008	980889930	664642572	316247358
2009	1003483757	680426976	323056781
2010	1026366682	702101715	324264967
2011	1035205511	705872398	329333113
2012	1021894152	685991741	335902411
2013	1018139165	681486392	336652773
2014	1038149414	698064881	340084533
2015	1068044894	727279990	340764904
2016	1142066427	791496311	350570116

Accumulated Program Investments [1995 - 2016] Top 25

ProgramNameEN	Reported Dollars
Discovery Grants Program - Individual	5949789135
Canada Research Chairs	1590512911
Collaborative Research and Development Grants	948122955
Strategic Projects - Group	908959760
Networks of Centres of Excellence	762559105
Research Tools and Instruments - Category 1 (<$150,000)	565583275
Postgraduate Scholarships	435601321
Industrial Research Chairs	398835692
Research Networks	382614308
Alexander Graham Bell Canada Graduate Scholarships - Doctoral	352888065
Postgraduate Scholarships - Doctoral	340062867
Postdoctoral Fellowships	283101969
Subatomic Physics Envelope - Project	267988202
University Undergraduate Student Research Awards	258863102
Major Resources Support Program - Infrastructure	201128769
Centres of Excellence for Commercialization and Research - Group	186214812
Engage Grants Program	185938402
Alexander Graham Bell Canada Graduate Scholarships - Master’s	175254486
Cooperative Activities	163228194
College and Community Innovation Program	161161262
Major Facilities Access Grants	159523772
Collaborative Research and Training Experience	147907095
Canada Excellence Research Chairs	137314666
Discovery Grants Program - Accelerator Supplements	121369119
Postgraduate Scholarships - Master’s	114880771

Accumulated Principal Investigator Investments [1995-2016] Top 25

Name	Reported Dollars
Hormes , Josef	99836800
Gupta , Arvind	97587570
McWalter , Ian	75503588
Fortier , Louis	72542772
Frise , Peter	63874923
Barge , Brian	50400000
Thomlinson , William	46610800
Fedosejevs , Robert	40720798
McDonald , Arthur	36781300
Orr , Robert	32043881
Smith , Donald	31227906
Mufti , Aftab	28761712
Wallace , Douglas	28433833
Servos , Mark	28130452
Rogers , Harry	27590000
Salama , Andre	26296562
Burton , Howard	25000000
Yada , Rickey	24982617
Clowes , Ronald	24072844
Pelton , Robert	23985763
Svensson , Carl	22782441
Lamb , Robert	21349800
Chrisman , Nicholas	21286250
Larter , Stephen	20024564
Bourbonnais , Normand	17753965

TODO NSERC’s Open Data does not report gender of PIs. Find ways to measure equity, diversity, inclusion metrics over past investments.
TODO How can we connect the reported dollar investments to measurements of research output?

Discovery

Interactive Notebook: Evaluation Groups Analysis

Evaluation Groups

Award Amounts

TODO Why do 1501 and 1502 receive so much more investment than other EGs?

Number of Awards

TODO Why are there so many more awards in 1501 and 1502 than in other EGs?

Bin Frequency

Example: 1504 (Chemistry); FiscalYear 2013

TODO Find a way to infer bin levels based on awards data amounts

Engage

Interactive Notebook: Engage Grants

TODO Find a way to connect Engage Grant investments to economic impact metrics.

National Scale Interactive Computing

Wed, 22 Aug 2018 19:32:16 +0000

(Invited post on Jupyter blog: https://blog.jupyter.org/national-scale-interactive-computing-2c104455e062)

This is an invited post from Jim Colliander, Professor of Mathematics at UBC and Director of the Pacific Institute for the Mathematical Sciences.¹

In 2017, the Pacific Institute for the Mathematical Sciences (PIMS), in partnership with Compute Canada and Cybera, launched Syzygy, a cloud-hosted interactive computing platform that delivers JupyterHub deployments for universities across Canada.

Syzygy has been used by over 16,000 students at 20 universities. The main results of the Syzygy experiment so far are:

Demand for interactive computing is ubiquitous² and growing strongly at universities.
The Jupyter ecosystem is an effective way to deliver interactive computing.
A scalable, sustainable, and cost-effective interactive computing service for universities is needed as soon as possible.

Demand for interactive computing

Both research and teaching at universities are adapting to major societal changes driven by explosions in data and computational tools. New educational programs that prepare students to think computationally are emerging, while research strategies are changing in ways that are more open, reproducible, collaborative, and interdisciplinary. These transformations are inextricably linked and are accelerating demand for interactive computing. The Syzygy experiment has shown that using Jupyter in educational programs drives interest in using Jupyter for research (and vice versa). For example, students in mathematics, statistics, and computer science collaborated with a researcher from St. Paul’s Hospital in Vancouver using Syzygy to identify new pathways to prevent death from sepsis. Research communities typically need access to deeper computational resources and often span multiple universities, but the common thread is the need to expand access to interactive computing.

A map of JupyterHub deployments deployed by Syzygy.

Technical milestone achieved

The Syzygy project has demonstrated that it’s possible to deploy tools for interactive computation at a national scale rapidly and efficiently using an entirely open source technology stack. Students, faculty and staff across Canada use Syzygy to access Jupyter through their browsers with their university single-sign-on credentials. The JupyterHubs range from a “standard” configuration to bespoke environments with specially curated tools and data integrations. This richness is possible because of the architecture of the Syzygy and Jupyter projects and the flexibility of the underlying cloud resources. As a case-study, Syzygy demonstrates that the Jupyter community has achieved a significant technical milestone: interactive computing can be delivered at national scale using cloud technologies.

Service level requirements

The validation that interactive computing can be technically delivered at national scale prompts universities to ask a variety of questions. Can interactive computing service be delivered robustly? How will users be supported? What are the uptime expectations? What is the data security policy? How is privacy protected? Can the robustness of the service be clarified in a service level agreement? Syzygy, as an experimental service offered to universities at no charge and without a service level agreement, does not properly address these questions. To advance on their education-research-service mission and address growing demand, universities need a reliable interactive computing service with a service level agreement.

What’s next?

How should universities address their needs for interactive computing over the next five years? Right now, universities are following two primary approaches:

🙏 Ad hoc: faculty figure out how to meet their own needs for interactive computing; IT staff deploys JupyterHub on local or commercial cloud servers; this approach gives universities control over their deployments and hardware, though requires time and expertise that many may not have.
🎩 Use a cloud provider’s service: Google Colab, Amazon Sagemaker, Microsoft Azure Notebooks, IBM Watson Studio; this approach allows universities to quickly launch interactive computing services, with a loss of flexibility and some risks by becoming reliant upon a particular vendor’s closed-source and proprietary software.

These approaches are not sustainable over the long term. If universities all deploy their own JupyterHub services, many will need technical expertise they do not currently have and will involve a significant duplication of effort. If universities rely on hosted cloud notebook services, the reliance on proprietary technology will impair their ability to switch between different cloud vendors, change hardware, customize software, etc. Vendor lock-in will limit the ability of universities to respond to changes in price for the service. Universities will lose agility in responding to changes in faculty, staff, and student computing needs.

There is a third option that addresses the issues with these two approaches and generates other benefits for universities:

🤔 Form an interactive computing consortium: universities collaborate to build an interactive computing service provider aligned with their missions to better serve their students, facilitate research, and avoid risks associated with vendor lock-in.

To retain control over their interactive computing stacks, avoid dependence⁴ on cloud providers, and accelerate the emergence of new programs, universities should work together to deploy interactive computing environments in a vendor-agnostic manner. This might take the form of a consortium — an organization dedicated to serving the needs of universities through customized shared infrastructure for interactive computing. The consortium would also ensure that universities will continue to play a leadership role in the development of the interactive computing tools used for education and research.

The Syzygy experiment confirmed that growing demand for interactive computation within universities can be supplied with the available technologies advanced by the Jupyter open source community. In the coming year, we aim to build upon the success of the Syzygy experiment and seed an initial node of a consortium in Canada with the intention of fostering a global network of people invested in advanced interactive computing. If you are interesting in partnering, please get in touch! See this Jupyter Community Forum post to continue the discussion.

The author gratefully acknowledges feedback on this piece from Ian Allison, Lindsey Heagy, Chris Holdgraf, Fernando Perez, and Lindsay Sill.
Interactive computing needs have been identified in agriculture, applied mathematics, astronomy, chemistry, climate science, computer science, data science, digital humanities, ecology, economics, engineering, genomics, geoscience, health sciences, K-12 education, neuroscience, political science, physics, pure mathematics, statistics, and sociology.
Relying on commercial cloud vendors to provide the interactive computing service for universities risks recreating the problems associated with scientific publishing that emerged with the internet.

Notes on Academic Freedom Dialogue hosted by UBCFA

Tue, 24 Oct 2017 00:00:00 +0000

Robert Lee Auditorium in the Alumni Center, UBCFA Fall General Meeting:

My central takeaway from the discussion below and the preceding actions by the Sr. Advisor on Academic Freedom is that faculty at UBC should be concerned. Guppy’s Historical Note on academic freedom at UBC references the Smith report but presents no analysis of the issues surrounding the Berdahl Case. Why? When asked about the lack of consequences to senior and honorific administrators who collectively violated academic freedom at UBC, Professor Guppy confirms that the history shows that there are no consequences. His position is not equipped with any authority other than to provide guidance. When pressed, Professor Guppy pointed out that his role is to advise the Provosts and that he was not appointed by the President. However, the position that Guppy presently occupies was created by M. Piper. The terms of reference in the announcement of the position promises that UBC will hire a “specialist who will proactively work with faculty, staff, and governors to ensure that academic freedom is safeguarded and preserved at UBC.” Guppy has previously reported that he does not have access to the full Smith report. Why? And why wouldn’t he demand access to that document to carry out his charge? Between the time of Piper’s promises and today, it seems that the important role of the Sr. Advisor on Academic Freedom has been directed to avoid interacting with any aspects of the Berdahl/Gordon/Helsley/Montalbano case that launched the position.

UBCFA Fall General Meeting, Tues. Oct. 24

Posted on October 10, 2017 at 1:00 AM. Technical delays led to the meeting actually starting at 1:13pm.

Join us for a Dialogue on Academic Freedom, presented by Dr. Neil Guppy, Professor of Sociology & Senior Advisor to the Provost, Academic Freedom The UBCFA Constitution requires that a general meeting be held every Fall term. This year, the first order of business will be to appoint Hedden Chong LLP as auditors for the 2017 fiscal year. Following this, Professor Neil Guppy will lead a Dialogue on Academic Freedom.

Date and Time: Tuesday, October 24th starting at 1:00 p.m. Locations:

Robert H. Lee Family Boardroom, Alumni Centre (6163 University Blvd) UBC Okanagan (Video Conference) – Sci 331 We hope you can join us!

Faculty Association Executive

Nancy Langton brings the meeting together at 1:12p. The agenda is approved. The auditors are approved. The business actions are carried out very efficiently.

Neil Guppy is introduced as a Professor and as the Sr. Advisor to the Provost on Academic Freedom. Neil gave a discussion of academic freedom to the Board of Governors. They were perhaps not as enthusiastic as this audience will be to hear this presentation.
The creation of the Sr. Advisor on Academic Freedom postion was announced by Interim President M. Piper on 2015-10-15. In the same announcement, Piper reported that “UBC accepts and will be guided by the findings” of the Smith Report.
Neil Guppy was appointed Sr. Advisor on Academic Freedom on 2016-05-25. The announcement of this appointment was made by Provost (pro tem) Angela Redish and Provost Cynthia Matheson.
Neil Guppy wrote Academic Freedom at UBC: Historical Notes.
Another page on Academic Freedom at UBC.

Neil Guppy

This is (essentially) identical to the presentation I gave to the Board of Governors. He highlights the intial tasks:

lead a dialogue
education program
online tool
formalized module

This presentation was delivered many times across campus. The one thing that has changed up about this….I did not know that much about academic freedom when I stepped forward for this role. Many things come out of the blue and I’ve found the topic to be expansive and surprising. Does academic freedom apply to staff? I’d say yes at this time.

UBC Senate statement (1976) on academic freedom

What is academic freedom?

(a) freedom to pursue knoledge claims in whatever directions they take
(b) tenure to protect knowledge-seeking activity of all legitimate kinds
(c) collegial governance and scholarly critique of the admininstration
(d) ability of the scholar to participatge freely in public life

At what point should a Dean be able to criticize a strategic plan for the university? Is there a line where the critique of the administration should not be allowed? cites the Potter case at McGill. M. Piper, wrt this line, would not be able to speak her personal opinion since she would be speaking on behalf of the university.

My own view is that UBC is good on a, b and not so good on c, d.

Smith report remarks

Gupta….Montalbano/Berdahl…..slide changed too fast for me to keep up.

The public version of the Smith report is up online. Key points

Who has academic freedom?

individuals
universities

Who controls academic freedom?

Professions/Disciplines
- contexts/units in which we develop and disseminate knowledge – what is “academic”?
- questioning accepted wisdome key but within set boundaries – questioning, boundary setting
- duty to uphold standards and monitor disciplinary ranks

Restrictions and Responsibilities

Are there restrictions on academic freedom?
- Yes
How does academic freedom coexist with academic responsibilities?

Is academic freedom more than “freedom of expression”?

If yes, how does it differ?
- quality control – academic modifies the freedom
Is it less than “fredom of expression”?
Example: Why is the theory of evoution taught but not creationism?

Specific Issues of Academic Freedom

Teaching
Scholarly work
Governance
Donor relations

Questions

How do we handle the collision between the university and an individual?

What are the consequences for the failure to defend of academic freedom? When Justice Smith finds that Mr. Montalbano, Chancellor Gordon, and the leadership at Sauder contributed to a collective failure of academic freedom at UBC, why are there no consequences? OK, Mr. Montalbano resigned by the Chancellor, a position that is meant to be honorific, was reappointed to the Board. What is the impact of violating academic freedom?

“It appears that there are no consequences.”
“I am not able to provide further input on any specific case.”
Why not? “I was appointed as a special advisor to the Provost, not by the President. You seem to know more about this issue than me…”

What would you do now if a case like Berdahl/Montalbano arose again?

Based on the principles, I’d provide guidance. I’m not in a position in which I have disciplinary or punitive power.
The conclusion I see here is that the creation of this position has no authoritity to provide defense or intervention.
Based on what I see today, I wonder what UBC considers by the term “positive obligation” to protect academic freedom?

Cases

Buckingham case at USask

Media attention on the Buckingham case led to his reinstatement.
Quick resolution led to a legal non-disclosure agreement. There should have been a complete analysis.

Issue with grade changes case

Who owns the content created for online courses?

The meeting closed at 1:59pm.

Canadians Land on Jupyter

Tue, 11 Jul 2017 17:00:00 +0000

The proliferation of mobile devices, social networks and sensor networks, the massive data streams they generate, and increasing computational power generate research challenges and provoke widespread interest in mathematical sciences expertise and insight. Democratizing access to this expertise and insight catalyzes meaningful change. Higher education institutions are launching interdisciplinary programs in digital humanities, data science, scientific computation, mathematical modeling, bioinformatics and epigentics to address these challenges. To achieve success, these programs require easy access to state-of-the-art computing environments to support research, teaching and training activities.

The Pacific Institute for the Mathematical Sciences (PIMS), in partnership with Compute Canada and Cybera, launched a cloud-hosted scientific computing and data science platform for Canada. The service, syzygy.ca, delivers Jupyter to faculty, staff and students at Canada’s universities using single-sign-on (SSO) via their university user account. By eliminating the requirement to install customized software on personal computers, syzygy.ca makes it easier for research teams to collaborate using the right tools for their investigations. The platform delivers an interactive coding environment for literate programming in Python 2, Python 3, R (and sometimes Julia, Octav, Sage and other languages).

The data8 program at Berkeley, the open source quantitative economics course, and computational fluid dynamics course are inspirational examples showcasing the potential for Jupyter. PIMS is leveraging syzygy.ca and other tools to develop expertise in scientific computing, data science, machine intelligence, optimization, etc.

Jupyter service is available today at several universites (UBC, SFU, UofT, Waterloo, Queen’s, Victoria, Saskatchewan, Manitoba, Calgary, Lethbridge). A few hubs have been deployed that can be accessed using other identity authentication providers (Google OAuth; GitHub; GitHub Enterprise). Colleges, universities, and other prospective partners can request Jupyter service via syzygy.ca by clicking here.

PIMS developed some support resources: e-book introduction; Discourse forum.

The platform has been used for seminars on Python and Git, machine learning with SciKit Learn, neural networks and deep learning, undergraduate courses on mathematical computing, computer science, and statistics, and graduate courses on mathematical modeling for industry, seismic inverse problems, and computational finance.

Academy-industry partnerships are forming to investigate data science challenges arising in business through a workshop built atop syzygy.ca.

The syzygy.ca platform democratizes access to digital research infrastructure. PIMS and our partners advance Canada’s research capacity by connecting human talent to curated tools from the mathematical sciences, diverse data sources, excellent documentation and training programs.

Budget 2017, Naylor’s review, and the Mathematical Sciences in Canada

Fri, 21 Apr 2017 19:29:59 +0000

James Colliander, Director of the Pacific Institute for the Mathematical Sciences (PIMS)
Nassif Ghoussoub, Director of the Banff International Research Station (BIRS)
Ian Hambleton, Director of the Fields Institute for Research in Mathematical Sciences (Fields)
Luc Vinet, Directeur du Centre de Recherches Mathématiques (CRM)

The direct funding of research initiatives on artificial intelligence (AI) and quantum computing via Budget 2017, and the release of the report of Canada’s Fundamental Science Review present an opportunity to reflect on the role of mathematical sciences within Canada’s scientific heritage and future, but also on our country’s ways of funding research.

The importance of the mathematical sciences (mathematics, statistics and computer science) is deepening in almost all areas of knowledge. Mathematical sciences provide a conceptual infrastructure underpinning advances in biology, engineering, humanities, medicine, social sciences and beyond. Progress in our understanding in all these fields depends upon advanced research and high-level training in the mathematical sciences.

Canadian Mathematician John Charles Fields, the creator of the Fields medal (often dubbed the Nobel prize for mathematics), also played a key role in the founding of the National Research Council in 1916. Today, Canada is served by a collaborative network of mathematical sciences research institutes: the Centre de Recherches Mathématiques (CRM) in Quebec, the Fields Institute for Research in Mathematical Sciences in Ontario, and the Pacific Institute for the Mathematical Sciences (PIMS) in Western Canada.

The institutes amplify Canada’s capacity for discovery and invention through partnerships that intertwine our nation’s universities with academic and industrial researchers from across the globe. Together, the institutes created Mitacs in 1999, which, under the leadership of Arvind Gupta, became a cornerstone of the government’s effort to link our graduate students (in all disciplines) with industry. This accomplishment was amply recognized in Budget 2017. In 2003, they collaborated with Berkeley’s Mathematical Sciences Research Institute (MSRI) to found the Banff International Research Station (BIRS), a unique North-American research infrastructure on Canadian soil, that provides an environment for creative, synergetic, intense and prolonged interactions between mathematical scientists and investigators in other areas of research.

In 2002, the three institutes committed to provide long-term funding to the Atlantic Association for Research in the Mathematical Sciences (AARMS), a network that plays an important role in the mathematical sciences research activities of the Atlantic region.

In 2003, PIMS collaborated with Berkeley’s Mathematical Sciences Research Institute (MSRI) to found the Banff International Research Station (BIRS), a unique North-American research infrastructure on Canadian soil, that provides an environment for creative, synergetic, intense and prolonged interactions between mathematical scientists and investigators in other areas of research.

The need for leadership to advance the mathematical, computational and statistical understanding of information, the development of data science, and the advent of machine learning, prompted the institutes in 2012 to use their own resources to invest in the creation of CANSSI, the Canadian Statistical Sciences Institute. That NSERC did not have the resources to do so at that time sheds some light on the community’s reaction to how Budget 2017 continues to shut out the Tri-Council.

AI rests on mathematical sciences. Indeed, some of this field’s prominent leaders pursued their foundational research within BIRS, CRM, Fields and PIMS. AI needs further advances in the mathematical sciences to thrive. We celebrate strong support of AI research but the disjointed approach used by government for its substantial investment in the area of deep learning totally missed the opportunity to include and exploit the national resource that BIRS, CRM, Fields, PIMS and CANSSI represent.

This example is but one of many examples that illustrate the importance of some of the excellent recommendations of the report of the Science Review Panel. Indeed, Canada developed world leading expertise and operations in science policy during the era of Pierre Elliot Trudeau. The work of the Senate Special Committee on Science Policy, chaired by Senator Maurice Lamontagne, identified principles to guide Canada’s future governments. The consultation of the scientific community overseen by the panel chaired by David Naylor aligns with these best practices. Direct investments disbursed through political channels, instead of through the Tri-Council’s scientific peer review process, undermine the transparency of research funding programs, and can miss opportunities such as the one we described. Funding allocations to support research should follow consistent and rigorous evaluation processes incorporating independent scientific peer review.

Fortier's Attack on Academic Freedom

Tue, 28 Mar 2017 19:28:49 +0000

McGill University’s Mission Statement and Statement on Academic Freedom define the values and parameters for how the University should react when the Premier of Quebec lashes out at an article in Maclean’s authored by a scholar at McGill. Before analyzing the situation at McGill, consider two fictional scenarios involving Institute Directors.

The UK Home Secretary Amber Rudd wishes to eliminate end-to-end encryption for WhatsApp and other social messaging platforms following the recent Westminster terrorist attack. I can envision circumstances in Canada, painful as they would be, where our government might also request the power to intercept private communications between citizens. I can also imagine the Executive Director of Ryerson’s Privacy and Big Data Institute staking out scholarly, and potentially controversial, positions defending the public’s right to private conversation through encrypted channels. Should a Director who defends the public’s right to privacy following a terrorist attack in Canada be forced to resign?

I can imagine the Fields Institute Director convening a scholarly workshop exploring quantum computing skepticisim. I can also imagine that such a workshop might offend Minister Navdeep Bains. Should the Fields Institute Director be forced to resign if Minister Bains expressed disappointment in such a workshop?

We turn our attention to the recent case at McGill. Here are three examples where Principal Fortier’s comments in The Globe and Mail on 2017-03-26 corrode academic freedom.

1. Requirement to convene a nonpartisan environment

“The institute has to maintain [an] environment which is not partisan. It is anybody’s judgment if after an article like that, politicians would be happy to come to an event,” Dr. Fortier said. “That’s not pressure, that’s just reality.”

Fortier’s “reality” focuses pressure (from Premier Couillard?) on any future Director of the McGill Institute for the Study of Canada (MISC) to “maintain an environment which is not partisan”. This is an imposition of “political orthodoxy” upon a member of McGill’s scholary community. No such requirement appears in the MISC Mission Statement. By demanding an environmental nonpartisan critereon on the operation of the Institute, Fortier weakens the capacity of MISC to carry out its mission to study Canada (a constitutional monarchy with government convened through a partisan election process).

2. Fortier’s Maxim: Never utter a controversial statement as Director

“It was an unfortunate article,” Dr. Fortier said. “It was perhaps a moment not remembering what his new role was and falling back on a previous role,” she said. “If he had written this article as Andrew Potter [period], nothing would have happened. He wrote it as director of the McGill Institute for the Study of Canada.” “You have to remember what responsibilities come with new positions. When you are an academic administrator, there are things you must be more prudent about doing,” Dr. Fortier said.

Fortier asserts that “nothing would have happened” if Maclean’s had not identified Potter as Director of MISC. Following this logic, we derive Fortier’s Maxim for Institute Directors: Never utter a controversial statement as Director, a statement strikingly similar to Academic Freedom Destroyed: Never utter a controversial statement. Fortier’s Maxim commands Institute Directors to filter their comments based on controversy and then choose to speak with their little, personal voice on controversial topics and their big, prudent, Institute voice on vanilla topics. Fortier’s guidance weakens the academic leadership capacity of Institutes and their Directors.

3. Derision of scholarship

“I think he would be the first to admit that it is not a good piece of scholarship, which is important when you are director of an institute.”

After the Premier of Quebec criticizes a member of the scholarly community of McGill University, the central duty of the Principal is to intervene and protect academic freedom. Dr. Fortier, a protein crystallographer, may have various opinions on Canada, Quebec and the quality of work by McGill’s scholars. That said, Fortier’s scholarship is not focused on the study of Canada. Instead of carrying out the Principal’s central duty to protect academic freedom, Fortier chose to characterize Potter’s view of his work as “not a good piece of scholarship”. Fortier, speaking as Principal of the University, disparaged the scholarship of a member of McGill’s academic community. Fortier’s actions permit “infringement and undue external influence” from Premier Couillard and others to weaken McGill’s “institutional autonomy”.

Here is McGill’s Statement on Academic Freedom:

Academic freedom is central to McGill University’s mission of advancing learning through teaching, scholarship and service to society.

The scholarly members of the university have the freedom to pursue research and artistic creation and to disseminate their results, without being constrained by political or disciplinary orthodoxies, monetary incentives or punitive measures as a result of their academic pursuits. They may exercise this freedom in the service of both the university and the wider society. When scholarly members of the university participate in public forums and debates, they should represent their views as their own.

The exercise of academic freedom requires collegial governance with the full participation of scholarly members. They retain the right of free expression, including the freedom to criticize one another, university policies and administration.

The university and its officers have a duty to protect the academic freedom of its scholarly community, both individually and collectively, from infringement and undue external influence as well as to maintain the university’s institutional autonomy.

The scope of protection in the statement is wide: “scholarly members of the university have the freedom”. The protection is not restricted to faculty. The statement does not reference tenure as a requirement for academic freedom protection. The statement does not exclude administrators, directors, or others from the “scholarly community”. The statement does not reference prudence or impose any additional requirements upon directors or administrators for exercising their academic freedom. The character of academic speech, whether it is partisan or impartial is not referenced in the statement.

Although “the university and its officers have a duty to protect” academic freedom, the officers of McGill University, and especially Principal and Vice Chancellor Suzanne Fortier, failed to defend academic freedom of Andrew Potter as Director of the McGill Institute for the Study of Canada. Fortier’s public relations platitudes and distinctions are inconsistent with McGill’s mission statement. Fortier’s actions harm McGill and weaken Canada’s intellectual capacity.

Toward Global Science Excellence

Sun, 28 Aug 2016 19:27:19 +0000

Prime Minister Justin Trudeau’s ministerial mandate letters for Science and Innovation, Science and Economic Development (ISED) outline an agenda to investigate and improve the nation’s innovation ecosystem. Roundtable discussions on various themes are taking place across the nation as part of Canada’s Innovation Agenda, an initiative driven by Minister Navdeep Bains (ISED) and Minister Kirsty Duncan (Science). Minister Duncan has also launched Canada’s Fundamental Science Review and empowered an eminent panel chaired by Dr. David Naylor with a mandate to review and make recommendations to improve Canada’s science policy.

I recently participated in a roundtable discussion on the theme Global Science Excellence. The event was convened by Dr. Maurice Moloney, CEO of the Global Institute for Food Security, and took place at the University of Saskatchewan. The roundtable panel included representatives from biology, chemistry, computer science, environmental science, geoscience, mathematics, pharmacy, physics and leaders from provincial and federal governments.

After reflecting on the panel discussion, here are four points that stand out for me:

Transparent sharing of government data through resources like the Open Government Portal should continue. Government decisions will be better when based upon insights informed by publicly accessible data. (Some panelists requested access to my open Jupyter notebooks on NSERC investments during 1995-2014 and 2012-2016.)
Decisions on government investments in scientific research should be decided scientifically through vigorous peer review and not politically through direct line items in the federal budget.
Ideas from mathematical sciences (computer science, statistics, mathematics) are influencing almost all areas of inquiry. The measured impact of mathematical sciences on the economy far exceeds expectations typically held by policy makers. Systematic underfunding of mathematics, statistics and computer science obstructs Canada’s potential for global science excellence.
Researchers from various disciplines request greater coherence across the programs offered by government. Advancing human knowledge through basic research, applying breakthroughs to solve problems in society or industry, developing and validating applications and prototypes, and scaling innovations through commercialization or other social processes are all endeavours that merit government investment. These activities unfold over differing time scales, involve differing sources of inspiration, and are driven by differing incentive systems and require a coherent collection of programs designed around these differences.

The Government of Canada is reviewing its approach to supporting research, development and commercialization. I encourage everyone to share their views.

Artificial Intelligence as a Service: Text Analysis

Fri, 03 Jun 2016 19:11:25 +0000

The computing infrastructure as a service offered by Amazon Web Services (AWS) may have originally been conceived as a resource to support their own electronic commerce business but other striking applications emerged. Netflix toppled Blockbuster and transformed the way we consume video atop the AWS infrastructure. Dropbox changed the way we store and share digital resources by building on AWS. My company Crowdmark leverages AWS to store and serve images of student work for evaluation by graders and further analysis. Recently, the Google Cloud Vision API, the IBM Watson Developer Cloud and the Receptiviti.ai API started offering artificial intelligence as a service available for purchase like a utility. These resources may form the foundation for a new era of technological metamorphosis.

Experimenting with Text Analysis

I wondered whether text analysis might generate useful insights for Crowdmark or other applications so I performed some experiments. I ran the texts from inaugural addresses by four presidents of the United States through some text analysis tools I found online. The IBM Watson Tone Analyzer uses “cognitive linguistic analysis methods” to measure the emotional tone in text. The Linguistic Inventory Word Count (LIWC) is a computer text analysis tool developed and psychometrically validated by James Pennebaker. Receptiviti.ai is a Toronto-based startup that offers text analysis as a service based on LIWC and other technology.

The texts

The tools

Screen captures of the results appear below.

Conclusion

The IBM Watson Tone Analyzer results are exposed within an intuitive and interactive user interface. The results do not correspond well with my own emotional response reviewing the videos or reading these speeches. Based on these experiments, I am not convinced that Tone Analyzer will generate useful insights into the emotional characteristics of text. Based on what I observed, there appear to be too few dimensions of emotional tone generated by Tone Analyzer for it to drive improvements to the dialogue between instructors and students. There may be a rich superset of output measurements not exposed in this free demonstration.

The LIWC results exposed through this free sample analysis are a small collection of the many dimensions measured by LIWC. It is not easy to glean insights into the speaker’s personality or their emotional tone based on the reports externalized in this free demo.

Short text descriptions of personality traits of the speaker emerged in the results from Receptiviti.ai. I found the text descriptions interesting but with limited precision in describing the speakers. I’m curious to know whether the personality decription accuracy increases with larger text samples from the same speaker.

Files of various types (text, images) may now be sent to increasingly sophisticated online analysis engines poised and ready to extract data and return insights to the sender. What will your robot assistant read for you tomorrow?

Experiments with IBM Watson Tone Analyzer

IBM Watson Tone Analyzer

Reagan Experiment

As a first experiment, I copied the text from the January 20, 1981 Inaugural Address of President Ronald Reagan (video) and pasted it into the IBM Watson Tone Analyzer. Here are the overview results:

Moving the mouse over portions of the text reveals the results of tonal analysis of the highlighted paragraph.

Tonal dimensions can be highlighted and sentences can be ranked based on tonal strength. For example, here is the sentence ranked highest for Disgust.

Clinton Experiment

As a second experiment, I processed the text from the January 20, 1993 Inaugural Address of President Bill Clinton (video) with Tone Analyzer. Here are the overview results:

Tone Analyzer reports that Clinton’s speech is most dominated by Fear.

Kennedy Experiment

For a third experiment, I chose the text from January 20, 1961 Inaugural Address of President John Kennedy (video):

Eisenhower Experiment

For the final experiment, I processed the text from January 21, 1957 Inaugural Address of President Dwight Eisenhower (video):

Overview Comparisons using IBM Watson Tone Analyzer

Reagan 1980

Clinton 1993

Kennedy 1961

Eisenhower 1957

Experiments with LIWC

LIWC

Reagan via LIWC
Clinton via LIWC
Kennedy via LIWC
Eisenhower via LIWC

Experiments with Receptiviti Personality Insights

Receptiviti.ai

Reagan via Receptiviti.ai
Clinton via Receptiviti.ai
Kennedy via Receptiviti.ai
Eisenhower via Receptiviti.ai

The Lost Opportunity

Wed, 16 Mar 2016 19:25:29 +0000

I will vote no confidence in the UBC Board of Governors. My main reason is that this Board’s governance failure led to massive losses in opportunity. By forcing President Gupta to resign, John Montalbano, Lindsay Gordon and Greg Peet did great harm to UBC, the Province of British Columbia, Canada and our international partners. The consequences of their actions disqualify Mr. Gordon and Mr. Peet from playing any role in selecting the next President.

Hiring Gupta as President was no Fluke

Those who believe the whispers that Arvind Gupta lacked the “administrative experience” to be UBC President are invited to kiss the back side of a high velocity shovel. After achieving Full Professor with tenure at SFU, Dr. Gupta served as CEO and Scientific Director of Mitacs, a non-profit company that drives research-based innovation through partnerships intertwining government, industry and academia. Gupta led Mitacs on a tear:

Revenue grew at an average annual rate of 23% over 14 years.
More than $300 million of new money flowed into the Canadian higher education system.
Canada’s R&D ecosystem was enriched through 15,000 Mitacs academic-industrial internships.

Academic leaders noticed Gupta’s “Mitacs touch”: 60 (out of 95 nationwide) of Canada’s universities have become Mitacs partners. Government leaders noticed the Mitacs track record so Gupta was invited to serve on the Science, Technology and Innovation Council (STIC) and the Expert Panel that reviewed Federal support for R&D. The Expert Panel reviewed every program in Canada’s research and development ecosystem giving Gupta enormous insight. Industrial partners (like The Boeing Company) also noticed and partnered with Mitacs. The opportunities created by Gupta and his team were also noticed outside Canada: Brazil, France, India, Germany, Mexico, Tunisia, Turkey and Vietnam have all formed partnerships with Mitacs.

In Arvind Gupta, UBC had a leader who built partnerships with the Chrétien, Martin and Harper governments, with mutiple governments in each of the provinces, with thousands of companies, and with many foreign countries. UBC’s appointment received national media attention. Gupta, in what appears to have been a first for a university president, was named on the Maclean’s Power List of top 50 most important people in Canada. Gupta was also invited, another first for a university president, to the federal finance minister’s policy retreat. He may have lacked the experience of being a Dean but Arvind Gupta’s vision, connections and accomplishments qualified him to serve as President of UBC.

Lost Serendipity: Federal Election

2015 was a year of magical opportunity for the higher education system. Canada’s top three universities had new leadership. Former President of NSERC, Dr. Suzanne Fortier was entering her second year as Principal of McGill. Dr. Meric Gertler was entering his second year as President of the University of Toronto. Working together in advance of the federal election, Fortier, Gertler and Gupta had the chance to transform the government’s role in higher education. With Fortier’s Tri-Council background, Gertler’s expertise on cities, and Gupta’s record with Mitacs, these three university leaders had a unique opportunity to advocate for specific investments from the new government.

The Trudeau Government will soon announce details for the 2016 budget. If they’d had the chance, could Fortier, Gertler and Gupta have enlarged the Tri-Council budget by an additional $3.5 billion? Could the anticipated infrastructure investments have been redirected to support the knowledge economy? With Gertler and Fortier adding support to the innovative arguments presented to the Vancouver Board of Trade, could President Gupta have secured billions in federal infrastructure investment for the Broadway transportation corridor? These serendipitous opportunities were lost when the UBC Board of Governors broke the synergy of this trio by forcing the resignation of President Gupta.

Lost Serindipity: Provincial Election

Gupta’s vision for UBC, as outlined in the leaked document from January 30, 2015, included a province-wide strategy with an expanded role for UBC Okanagan and a renewed focus on research excellence. The document also revealed plans for sustainable need-based and merit-based bursaries to support students. These plans have emerged as the Centennial Scholars Program, a legacy milestone during Gupta’s abridged presidency. Equipped with mathematical models, government relations expertise, and new mechansims to fund students, Gupta was poised to transform the rigid 2% formula for funding higher education that has been in place since 2005. With Ontario’s recent move to make tuition free for thousands of students and the upcoming provincial election, UBC had a special opportunity to influence the platforms of the political parties. The election is scheduled for May 9, 2017 so the platform planning activities will be taking place this summer and fall. UBC lost its opportunity to have an established president in place, with a team and strategy ready to influence government during this period. The Board took actions that left UBC tactically disadvantaged to leverage the 41st General Election of British Columbia.

Lost Money, Lost Leadership, Lost Momentum

The presidential search, inaugurations, transition costs and legal fees will cost UBC millions of dollars. In the absence of a clear strategic vision, a mishmash of administrators from the Toope, Gupta and interim Piper presidencies direct their units as best they can. The next president will certainly make personnel changes which will incur costs. Thirty-nine days before President Gupta was forced to resign, UBC welcomed his administration’s new Vice President Finance Andrew Simpson. Mr. Simpson relocated to Vancouver from New Zealand. Actions by the current UBC Board created a situation in which Arvind Gupta continues to be paid presidential salary while UBC also pays the salary of Interim President Piper. Instead of investing in scholarships, research facilities, or faculty renewal, this Board wasted millions of taxpayer dollars on a presidential transition. Their reasons for taking these actions remain unexplained.

No Confidence

UBC lost a great opportunity in Arvind Gupta as president. Rather than a different president, what UBC needed was a successful presidency, which requires support of the Board of Governors. Actions by the Board have created initial conditions that make a successful presidency extremely unlikely. The leaked documents reveal that Chancellor Lindsay Gordon, Vice Chair Greg Peet and former Chair John Montalbano acted as a secret committee to force the resignation of President Gupta, without rightful involvement of other governors, and without the formal performance review promised in his contract. The unexplained termination of the previous president caused a loss of momentum and reputational damage to an institution that will be dealing with a prolonged period of unstable, incoherent, and unfocused management. Mr. Gordon and Mr. Peet are leading the search for the next president. Why, and on what basis and authority, were these men allowed to abort a presidency? With no new governance review or new best practices in place, what will prevent the formation of a future secret committee to control, interfere with or terminate the next presidency? The circumstances are suspicious. Can some members of the Board, acting secretly, wield excessive coercion over the next UBC president? The actions of this Board of Governors have done great harm to UBC, to British Columbia and to Canada. I stand with hundreds of my faculty colleagues to publicly indicate No Confidence in the UBC Board of Governors.

Calculating MRR over variable term subscriptions using Pandas

Thu, 10 Mar 2016 20:20:39 +0000

Mathematical Scientist's Guide to Innovation Funding Sources in Canada

Sun, 06 Mar 2016 20:14:04 +0000

The NSERC investment in the Pacific Institute for the Mathematical Sciences (PIMS), the Fields Institute, and the Centre de Recherches Mathématiques (CRM) was last renewed in 2014. These institutes contributed funds from this investment to support the launch of the Canadian Statistical Sciences Insitute (CANSSI) and continued to provide support for the Atlantic Association for the Research in the Mathematical Sciences (AARMS). The Institutes received additional seed funds, not sourced through the math/stats Discovery envelope, to support the launch of the Insitutes Innovation Platform (IIP), a program aimed at fostering “partnerships between mathematics and statistics researchers and non-academic partners in the public and private sectors.” This post provides guidance to mathematical scientists who may wish to seek funding to support collaborations with non-academic partners. So far, I’ve explored programs offered by NSERC and Mitacs and hope to supplement this guide with programs offered by the provinces sometime in the future.

NSERC Innovation Programs

NSERC Connect

Mathematical scientists can explore the idea to collaborate with industry using funds offered through the NSERC Connect program. NSERC Connect is ramified into three levels with funding support up to CAD5K, CAD10K and CAD25K. The CAD5K grant can be used to cover travel expenses related to the development of a new academy-industry collaboration. Regional programs aimed at promoting new academy-industry research collaborations or student-industry training programs may be supported at the CAD10K level. National programs targeting “substantial new research collaborations” between post-secondary researchers and non-academic partners may be supported at the CAD25K level.

NSERC Engage

The NSERC Engage program offers CAD25K to support academic researchers working to solve problems for an industry partner. The funds are directed to the academic researcher. All intellectual property developed in the collaboration are owned by the collaborating industry partner. Engage grants can be extended with Engage Plus. Industrial internships for graduate student and postdoctoral researchers can be supplemented with Mitacs Accelerate (see below). The application process is quite simple. Pawel Pralat of Ryerson has secured seven Engage grants. Some further notes (in the form of a GitHub Gist) on NSERC Engage including the application template are available here.

NSERC supports deeper investments in academic-industrial research collaboration through the Strategic Partnership Grants and Collaborative Research and Development Grants (CRD) programs. NSERC also offers funding support for talented undergraduates to collaborate with industry through the Industrial Undergraduate Student Research Awards (IUSRA) program.

Mitacs Innovation Programs

Mitacs Accelerate

The Mitacs Accelerate program generates CAD15K to support four month long industrial internships for graduate student and postdoctoral researchers. The intern receives at least CAD10K as a stipend during the collaboration. Mitacs provides CAD7.5K and the industry partner provides the matching CAD7.5K. NSERC Engage grants can be combined with Mitacs Accelerate internships to fund a longer term interaction between university researchers and industry. The Mitacs Accelerate Cluster program is designed to support the formation of teams focused on advancing on a single cohesive research plan. Relative to a collection of single Accelerate internships, there are significant financial benefits generated through the Mitacs Accelerate Cluster program. Some further notes (in the form of a GitHub Gist) including the application template and “two-pagers” for industry partners and universities on Mitacs Accelerate are available here.

Mitacs Elevate

The Mitacs Elevate program provides a two-year training program to support postdoctoral fellows who lead an industrially relevant research project. The program supplements the academic activities of a postdoctoral fellowship with specialized training in communication, leadership and other skills crucial to success in business. Elevate Fellows receive a salary of at least CAD50K per year.

Further Assistance

Researchers with some interest in collaborating with non-academic partners are encouraged to reach out to staff at the Institutes, Mitacs, NSERC or their institution’s research services office. If I can be of any assistance, please feel free to contact me.

I am Joining PIMS and UBC Math

Mon, 28 Apr 2014 19:10:03 +0000

I’m excited to be joining the Pacific Institute of Mathematical Sciences as Deputy Director and the Department of Mathematics at the University of British Columbia.

Here is a link to the announcement: http://www.pims.math.ca/news/james-colliander-appointed-deputy-director-pims

(I have moved my blog and web presence over here: http://colliand.com/)

Ubiquity of Mathematics: Ingrid Daubechies

Sun, 12 May 2013 19:09:11 +0000

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University of Toronto Math Department Colloquium Videos Winter 2013

Wed, 08 May 2013 19:08:10 +0000

The Department of Mathematics at the University of Toronto has recently deployed a new granular content management system called KBasix for sharing files and web content. We’ve deployed and enriched a KBasix system to create a video upload-transcode-stream service for the videos of seminars and colloquia that take place in our department. I am grateful to Pamela Brittain, Marco de la Cruz-Heredia, Emile LeBlanc and Habiba Mohtadi for their efforts at making it possible for the world to tune into our colloquium stream!

Wednesday April 24, 2013

“Linearization of Lie groupoids – Rui Loja Fernandes“

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Thursday March 28, 2013

“What is quantum probability? - Greg Kuperberg“

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Wednesday March 20, 2013

“Cherednik algebras and torus knots - Pavel Etingof”

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Wednesday March 13, 2013

“Thin Matrix Groups and Diophantine Analysis - Peter Sarnak”

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Wednesday March 06, 2013

“Trees and Wheels and Balloons and Hoops and Why I Care – Dror Bar-Natan”

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Wednesday February 06, 2013

“Universal spaces for birational invariants - Yuri Tschinkel”

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Wednesday January 30, 2013

“Operator limits of random matrices - Balint Virag”

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Wednesday January 23, 2013

“Unexpected applications of polynomials in combinatorics - Larry Guth”

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Ubiquity of Mathematics: Luis Seco

Tue, 07 May 2013 19:07:02 +0000

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Luis Seco Interivew - Transcript

Ubiquity of Mathematics: Spyros Alexakis

Mon, 06 May 2013 19:06:12 +0000

Your browser does not support the <video> tag. Spryridon Alexakis - Transcript

Ubiquity of Mathematics: Adrian Nachman

Fri, 03 May 2013 19:04:57 +0000

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Adrian Nachman Interview - Transcript

Ubiquity of Mathematics: Charles Fefferman

Thu, 02 May 2013 19:03:43 +0000

This is Episode 1 of a 5 part video series entitled Ubiquity of Mathematics. I am grateful to the Department of Mathematics, the Fields Institute, Richard Cerezo, and Andrea MacLeod for making this project happen. I also thankful the mathematicians interviewed for this series:

The other episodes will appear shortly. Here is the transcript and here is the video of the interview with Charles Fefferman:

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Zaher Hani Seminar: A Continuum Large Box Limit for the Cubic Nonlinear Schrödinger Equation

Thu, 21 Mar 2013 19:02:33 +0000

Analysis & Applied Math Seminar 2013-03-21

Speaker: Zaher Hani

Institution: New York University

Abstract: Inspired by the general paradigm of weak turbulence theory, we consider the 2D cubic nonlinear Schrödinger equation on a box of size L with periodic boundary conditions. In an appropriate “large box regime” (L very large), we derive a continuum equation on ℝ2 that governs the dynamics of the discrete frequency modes over nonlinear time scales. This equation turns out to satisfy many surprising symmetries and conservation laws, as well as several families of explicit solutions. (This is joint work with Erwan Faou (INRIA, France) and Pierre Germain (Courant Institute, NYU)).

#Introduction#

Setup

2d cubic defocsusing or focusing NLS on a box of size $L$. Energy and mass conservation. NLS is GWP for small data in $H^s$ for $ s \geq 1$. We are not concerned with the existence issue. We are working in the setting of global-in-time solutions.

Physical and mathematical setup: weak nonlinearity.

Aim: understand out-of-equilibrium dnamics of small solutions. e.g. CKSTT
Take small data….nonlinear time scale is $\epsilon^{-2}$ where we imagine $ u \thicksim \epsilon v$ and we study $v$ with an $\epsilon^2$ coefficient and the data is of the size 1.
Fourier ansatz, transfer dynamics onto the coefficients. Comments on the $L^2$ norm dependence upon the box size parameter $L$.
Express the dynamics in terms of the $a_k (t)$.
Define the 4-frequency convolution hypersurface. He calls that $S_K$.
Interaction Representation. Conjugate by the fast linear dynamic….remove the linear dynamics. The new Fourier variable is called $\tilde{a}_k (t).$

All that has happened was a change of variables enabling us to view the dynamics on the Fourier side.

Weak turbulence “paradigm”

Aim: statistical description of out-of-equilibrium dynamics of small solutions (Zakharov 60s, Kolmogorov 50s)
RPA rand phase and amplitude.
$n(K,t) = {\mathbb{E}} |a_k (t)|^2$ is the wave spectrum or mass density.
propagation of chaos assumption. True at $t=0$, but not propagated.
Roughly, we have three main steps:
1. Statistical and time averaging.
2. large-box limit $L \rightarrow \infty$.
3. weak nonlinearity limit $\epsilon \rightarrow 0$ to arrive at a continuum equation for $n(K), ~ K \in {\mathbb{R}}^2.$

The Kolmogorov-Zakharov kinetic equation. Long convolution equation localized on the convolution hypersurvface and further localized on the resonant set.

Admits explicit stationary solutions called KZ spectra. These solutions are thought to offer some explanation to some cascade phenomena.
Non-rigorous, KZ spectra are not integrable, negliects some finite-size effects, some numerical discrepancies, the appearance of some coherent structures called “quasi-solitons” even in defocusing problems.

A New Limiting equation

Statistical averaging was causing problems. Let’s dispense with that but still take the large box and weak nonlinearity limits.

Resonant cutoff/normal forms transformation. He goes to the board and describes the separation. On the non-resonant portion, he makes a stationary phase type integration by parts, and then makes a crude estimate using the equation. This shows the non-resonant portion contributes at size $\epsilon^4 L^2$. Therefore, we concentrate our attention on the resonant terms.

(slide 12/37 and we are 20 minutes into the talk….)

He analyzes the convolution + resonance condition and identifies orthogonality properties based on the pythagorean relationship among frequencies.

Parametrization of rectangles in $\mathbb{Z}^2 / L$.

A lattice point $ J \in \mathbb{Z}^2 / L$ is called visible if $z = (p,q)/L$ with $gcd (|p|, |q|) =1. $ These points can be connected by a straight to the origin without hitting another point in the lattice.

Some new coordinates involving an $\alpha$ and $\beta$.

Co-prime equidistribution:

You can, in certain circumstances, replace sums by corresponding integrals with bounds. A classical number theory result establishes the density of visible lattice points in $\mathbb{Z}^2 / M$ is $\frac{6}{\pi^2}$. This lets you translate equidistribution into a co-prime equidistribution statement enabling us to replace sums by corresponding integrals with bounds.

Q:…interesting, I wonder to what extent similar ideas can be used on the sums we have omitted earlier in the argument. Perhaps those sums can also be represented as integrals with appropriate bounds?

Continuum limit.

Following these formal arguments leads to an integral equation resembling the KZ equation. Q: What are the differences/similarities with the KZ equation? One difference is that it preserves the Hamiltonian structure and has a positive definite Hamiltonian. He calls this equation $*$.

Symmetries lead to conserved quantities.

He writes the trilinear term in the equation as $\mathcal{T}(f,g,h)$.

Hamiltonian
Mass
Momentum
Position
Second momentum
Kinetic energy
Angular momentum

A scaling property.

Invariance under Fourier transform.

If $g$ solves $$ then $\hat{g}$ also solves $$.

Properties of the continuum equation

Boudedness properties and well-posedness. He reports on LWP and GWP properties of the equation $*$.

Gaussian family is a family of explicit stationary solutions. Gaussians are the unique maxima of the Hamiltonian functional.

Heavy tailed solutions.

Are there more?

Invariance of Harmonic oscillator eigenspaces. Hermite polynomials. The associated linear spans are invariant under the nonlinear flow $*$. The Hamiltonian of the harmonic oscillator is an integral of motion so the two flows commute and you get this easily.

Question: Is this equation $*$ completely integrable?

Rigorous Approximate Results

In analogy to the CKSTT cascade result, there is a reduction to an equation related to NLS. Can we transfer information from $*$ back to learn something about NLS?

Three difficulties:

Pass to the resonant sum.
Obtain good discrete to continuum error estimates.
Trilinear estimates on resonant sums.

….discussion of these issues…. identifies the small nonlinearity regime characterized by the condition $$ \epsilon^4 L^2 \ll \frac{\epsilon^2 \log L}{L^2}.$$

Möbius inversion formula.

Convergence Theorem: ….long statement. He gets a convergence statement on an interval that is longer than the nonlinear time scale by a factor $ M \leq \log \log L$.

Further Questions

Numerical study comparing NLS and $*$.
Other explicit solutions of $*$? Cascading solutions? Videos.
Is $*$ completely integrable?
Similar continuum limit for other equations?

Larry Guth Colloquium: Unexpected Applications of Polynomials in Combinatorics

Mon, 11 Feb 2013 20:01:01 +0000

My former colleague Larry Guth (now at MIT) visited us recently and gave a beautiful colloquium talk. The Department has recently deployed a video streaming service so we are able to share Larry’s talk with the world. We look forward to sharing other videos in the future.

Here is the video:

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Unexpected applications of polynomials in combinatorics

by Larry Guth | MIT Time: 16:10 (Wednesday, Jan. 23, 2013) Location: BA6183, Bahen Center, 40 St George St Abstract: In the last five years, several hard problems in combinatorics have been solved by using polynomials in an unexpected way. In some cases, the proofs are very short, and I will present a complete proof in the lecture. One of the problems is the joints problem. Given a set of lines in $R^3$, a joint is a point that lies in three non-coplanar lines. Given $L$ lines in $R^3$, how many joints can there be? Another problem is the distinct distance problem in the plane. If P is a set of points in the plane, the distance set of $P$ is the set of all distances from one point of $P$ to another. If $P$ is a set of $N$ points in the plane, how small can the distance set of $P$ be? The proofs involve studying a set of points in a vector space by finding a polynomial of controlled degree that vanishes at the points, and then using the geometry of the zero-set to understand the combinatorial properties of the points. The goal for the talk is to give an overview of this new method.

Canada is Retreating from Investment in Science and Engineering

Tue, 11 Sep 2012 18:59:42 +0000

(The following is a slightly edited version of an invited post appearing on The Inside Agenda Blog on TVO’s web space.)

Canada retreats from Science

Canada is retreating from investment in science and engineering. Public letters (by 10 prominent physicists, 336 mathematicians, 49 leading researchers) have signaled alarms at changes to the NSERC Discovery Grants Program and the elimination of the Major Resources Support (MRS) and Research Tools and Instruments (RTI) programs. Investments in the training of the next generation of researchers through the Postdoctoral Fellowships Program have been slashed. Without funds to operate laboratories, without funds for new tools, and without funds for young researchers, Canada’s science and engineering research enterprise faces disaster.

Mission Drift

The program cuts are not driven by a decrease in the budget to NSERC. The program cuts are instead the result of a transfer of funds away from people and discovery into new programs giving money to businesses, a transformation characterized by the recent report of the federal R&D panel as “mission drift.” The Engage Program, described by NSERC President Suzanne Fortier as spawning business-academy “first dates” provides an illustration. Consider the details of the program:

NSERC provides $25K of taxpayer funds to pay for a six-month research and development project between a university researcher and a company already involved in research and development.
The company is not required to invest any money on the project.
Any intellectual property developed by the project is owned by the company.
There is no direct return back to taxpayers, to the university researcher, or to the university on the investment.

The program description reports that “these grants are intended to foster the development of new research partnerships between an academic researcher and a company that have never collaborated together before.” However, the Engage Program does not appear to be producing robust collaborative partnerships. There have been cool anecdotes about ski goggles and fiber optic guitar pickups but insufficient reporting on the program as a whole. Recently, in response to an inquiry from the official opposition regarding the conversion rate of Engage grants into the more substantial Collaborative Research and Development (CRD) grants, NSERC reported:

“348 distinct researchers have received both Engage grants and Collaborative Research and Development grants since these programs have operated, and this without regard to the years or the order in time. This number represents 10.62% of the total number of grantees for these two programs.”

This is a confusing statement and does not accurately reveal how many Engage Grants matriculated to become CRD projects. NSERC President Fortier has written that

“Nearly a thousand Canadian companies have benefited from the Engage experience to date.”

This represents an investment of $25,000,000. From a program level perspective, and not just anecdotally, what was achieved?

Despite announcements to the contrary by NSERC and Minister of State (Science and Technology) Gary Goodyear, the evidence shows that NSERC and the NRC (now described as a “business concierge”) are transfering funds away from “blue sky” basic research programs to Canadian businesses through programs like Engage.

Inadequate Consultation

Major changes in NSERC funding have often involved the research community through a long range plan (LRP) consultation. Long range plans for Subatomic Physics and Astronomy were recently completed; the LRP for Mathematics/Statistics is close to completion. The LRP consultation process activates a nationwide discussion by a community of researchers, contributes scientific input to the federal research investment strategy, and, in some cases, identifies opportunities for cost savings. The broad consultation of the LRP process respectfully empowers researchers to contribute to the policy discussions affecting them and, ultimately, all of Canada.

In contrast, there was no broad consultation in advance of the recent decisions to eliminate the Major Resource Support (MRS) and Research Tools and Instruments (RTI) programs. Shortly after University of Ottawa Chemistry Professor David Bryce’s letter and related public messages appeared, Minister Goodyear announced that these actions would only be a moratorium for one year as the government “seeks counsel” from the scientific community. Minister Goodyear’s remarks were reassuring but the terms of the RTI consultation have turned out to be much more narrow in scope. Instead of seeking creative input from the Canadian scientific community on how best to consolidate the “plethora of programs” and to “simplify the application process,” the consultation asks scientists and engineers to choose between Option 1 (rock) and Option 2 (hard place).

Canada was and can be a spectacular place for scientific and engineering studies. Canada had a research investment strategy that was once the “envy of the world.” Rapid policy changes with inadequate participation by the research community in the decision process threaten Canada’s long-term prosperity.

On Today's NSERC Contact Newsletter Item Regarding Postdocs

Fri, 07 Sep 2012 18:58:36 +0000

I received today the September 2012 Contact Newsletter (volume 36, number 4) from NSERC via email. The fourth item in the newsletter reads:

Postdoctoral Fellowships - no change to number of awards
Over the last ten years, the volume of applications to the NSERC PDF Program has doubled to about 1,300, impacting the workload of volunteer selection committee members. A change to the eligibility rules for the Postdoctoral Fellowships (PDF) Program was made to ensure that applicants' and reviewers' time was used productively.

The eligibility rules were changed to allow students to apply only once during the eligibility window. Please note that this change does not affect the budget for the PDF Program or the number of awards.

More information about the new policy is outlined in the Program Guide for Students and Fellows.

Over the last ten years, the faculty at Canada’s Universities has expanded by the addition of 2000 Canada Research Chairs and other strategic recruitment. This “brain gain” has had the desired effect: more highly qualified personnel are being produced by the system. Over the period 1999-2009, there has been an expansion in enrollment in degree granting programs (data extracted from 2010-2011 Tables):

Bachelor's enrollment expanded 34% (Table 44)
Master's enrollment expanded by 56% (Table 45)
Doctoral enrollment expanded by 70% (Table 46)

It does involve a lot of work for volunteers to assess postdoctoral fellowship applications. Instead of punishing the next generation of scientists by restricting the number of competitions they can enter to one, an alternate solution to the workload problem would be to correspondingly expand the size of the volunteer review committee. I volunteer to help with those assessments in mathematics. Other Canadian scientists and engineers could indicate their willingness to help with the assessments by politely contacting their program officers.

The Contact Newsletter item also reports that the budget and number of awards will not be changed. Note the substantial changes that have already occurred between 2010 and 2012.

Canada Restricts Athlete Participation to One Olympic Games per Lifetime

Thu, 16 Aug 2012 18:57:08 +0000

The (false) headline conveys the sporting analog of NSERC’s new policy on Postdoctoral Fellowship Competitions:

Effective as of the 2013 competition, you can only apply once to the NSERC Postdoctoral Fellowships (PDF) Program; however, applicants whose first PDF application was submitted prior to the 2013 competition may submit a second application provided they are within the eligibility window.

What's going on? Why would Canada choose to limit the pool of participants competing for advanced training opportunities in science and engineering? A recent letter to the Canadian Association of Postdoctoral Scholars by NSERC's Director (Scholarships and Fellowships Division) Serge Villemure gives the following reasons:

In recent years, NSERC has seen a growing disparity between the number of applications submitted to the Postdoctoral Fellowships (PDF) program and the number of awards available. As a result, NSERC has decided to reduce the maximum number of applications an individual may submit in a lifetime to its PDF program from two to one.
This change to the eligibility rules will contribute to a better alignment between both the number of applications submitted and the awards available, thereby streamlining the application and review processes. Limiting the number of applications an individual may submit to the program will not impact the the current budget projections or the number of anticipated awards available.

The success rate for the postdoctoral fellowships competition in 2011 was 9.3% and in 2012 the rate was 7.8%. (The tables and visualizations are appended below.) Another strategy to confront the “growing disparity” is to invest more money into Canadian human capacity for research and development by expanding the number of awards. However, changes in NSERC policy over the past decade have transferred investment away from its mission supporting discovery and the training of highly qualified personnel into many new programs aimed at commercialization of research. Restricting Canada’s young scientists to one postdoctoral fellowship competition per lifetime has “better alignment” with the transfer of funds toward commercialization, but it is a bad policy change.

Funding support for graduate programs from Ontario (and likely from other provinces too?) is frequently limited to four years. This means that faculty and departments are under pressure to have their graduate students complete their PhD in four years. Unfortunately, many students do not meet this timeline. Funds to pay for extensions of PhD studies into a fifth and sometimes a sixth year must come from other sources and are often uncertain, conditional upon adequate progress, and may involve expanded teaching responsibilities. Graduate students know all this.

Consider the point of view of a graduate student. Suppose the key advances for the student’s thesis are completed during the summer between the third and fourth year of studies and the student starts writing the thesis during the Fall of the fourth year. The funding uncertainty for the fifth year motivates the student to want to finish the PhD in the fourth year. To maintain a career in science, the student needs to also spend that Fall preparing job and fellowship applications, a process that can take up a lot of time and mental energy. The student’s application materials (research statement, letters of recommendation, thesis abstract) will be not as strong as they would be if the thesis were entirely nailed down. Nevertheless, the funding uncertainty for the fifth motivates the student to submit postdoc applications in the fourth year.

What happens next? In this situation, students sometimes get a postdoc but more frequently don’t. When they don’t, they stay on for another year and often make substantial advances. Their science comes together during the fourth year and the summer thereafter. They have a working draft of their thesis at the start of the fifth year and can concentrate on applications. Instead of merely talking about the student’s potential, the letters of recommendation can reference accomplishments. Students who fail to land a postdoc offer in their fourth year often emerge as extremely strong candidates in the next year.

PhD students will soon be asking graduate advisers for advice: should I apply for an NSERC postdoc now or should I wait until next year? The right answer was both. Under the new policy, the answer is not clear. A certain outcome: some excellent candidates will be forbidden to enter the competition because they applied the year before.

NSERC’s new one-postdoc-competition-per-lifetime rule combined with the funding uncertainties around fifth and sixth year support are a lethal combination. The victim is Canada’s scientific research capacity.

I’ve set up a “hive” on BuzzData (an open social media platform for discussions around data) focused on NSERC. My view is that there is a need for respectful discussion about Canada’s research and development policy driven by transparent data. So far, there are four public Datarooms devoted to the following topics:

Others are welcome to join the hive.

NSERC Funded PDFs data (Thanks David Kent.)

Awards/Applicants (Year)
250 / 1169 (08)
254 / 1220 (09)
286 / 1341 (10)
133 / 1431 (11)
98 / 1254 (12)

(Extracted from NSERC’s Scholarships and Fellowships Competition Results.)

The acronyms appearing in the tables are defined as follows:

CGS M/PGS M (one-year scholarship for the first or second year of graduate studies);
CGS D2/PGS D2 (two-year scholarship tenable during the first five years of doctoral studies);
CGS D3/PGS D3 (three-year scholarship tenable during the first five years of doctoral studies); and
PDF (two-year postdoctoral fellowship).

Visualizations by Brent Pym:

2012-08-21 Addendum (New visualizations by Brent Pym.)

GWP of Gross-Pitaevskii Equation on R4

Mon, 28 May 2012 18:55:42 +0000

Last week, I had a chance to visit Edinburgh in part to serve as the external examiner on the PhD Thesis (papers) of Tim Candy. Tim is now Dr. Timothy Candy and has an exciting research program to develop as a postdoc at Imperial.

It turned out I had lucky timing since my visit overlapped with a visit by Oana Pocovnicu. I had a chance to hear her speak about her recent work on the Gross-Pitaevskii equation. I took some notes during Oana’s talk and they appear below.

(joint work with Rowan Killip, Tadahiro Oh, and Monica Visan)

Edinburgh talk. 2012-05-21

Dynamics becomes more interesting with a nonvanishing condition at infinity.
This is the so-called energy critical case.

$$ i \partial_t u + \Delta u = (|u|^2 - 1)u, u(0) = u_0 $$

The modulus will tend to 1 as $ |x| \rightarrow 1$.

Literature

$R$
- Zhidkov 1987: introduced Zhidkov spaces.
- Gall 2004. gGWP in $X^1 (R)$
$R^2, R^3$
- Bethuel-Saut 1999 in $1+ H^1$.
- Gourbet 2007
- Gallo 2008
- Gerard 2006 in the energy space.
$R^4$
- Gerard 2006, small energy data such that $\nabla u \in L^2_t L^4_x.$

Remark: energy critical in $R^4$.

Gerard 2006 considered the energy space:

$$ E_{GP} = [ u = \alpha + v: |\alpha | =1, v \in \dot{H}^1, |v|^2 + 2 \Re (\overline{\alpha}v) \in L^2 (R^d)]. $$

Finite energy data do not have winding at spatial infinity. Therefore, to treat the finite energy case, it suffices to reduce the study to the setting where $u = 1 + v$ and $v$ satisfies…. She reduces the study to finite energy data so the set up excludes vortices right away.

Theorem (K-O-P-V): GP is GWP in the energy space $E_{GP} (R^4)$.

Two ingredients:

GWP of energy-critical defocusing NLS on $R^4$.
Perturbation theory: We will treat the equation as a perturbation off the cubic NLS.

Scaling Invariance

Dilation invariance of solutions for cubic NLS is described.
Dependence of $\dot{H}^s$ in terms of the scaling parameter $\lambda$.
critical, subcritical, supercritical.
Cubic NLS on $R^4$ is critical in $\dot{H}^1$. Quintic NLS on $R^3$ is also critical in $\dot{H}^1$.

Strichartz Estimates

Dispersive decay estimate
Strichartz Norm; supremum over the admissible pairs.
$N(I \times R^d)$ is the dual space of the Strichartz space $S(I\times R^d)$.
Homogeneous Strichartz estimate
Inhomogeneous Strichartz estimate
Admissible pairs on $R^4: (\infty, 2), (2,4), (6, \frac{12}{5})$.
By Sobolev embedding, we have some nice Strichartz containments.

Energy Critical NLS

LWP. Cazenave-Weissler 1989
GWP for small data. She then describes this by passing through Strichartz and identifies:
- If $\| \nabla e^{it \Delta } w_0 \|{L^6_t L^{12/5}_x}$ is small, we can close the argument.
- The smallness of this expression can be insured by shrinking $T$, but this depends upon the profile properties not just upon the norm of the data.
- GWP for small data follows.
Explains the blowup critereon showing that the spacetime $L^6$ norm controls the GWP+Scattering theory.

Main Results on defocusing energy-critical NLS

Bourgain 1999: GWP + Scattering, quintic NLS on $R^3$ with radial data.
- induction on energy
- localized Morawetz estimate
Grillakis 2000: global regularity for quintic NLS on $R^3$ with radial data.
CKSTT 2003: removed the radial assumption on $R^3$.
Ryckman-Visan 2007: GWP and scattering for cubic NLS on $R^4$.
Visan 2010: Simpler method for GWP+Scattering for cubic NLS on $R^4$, building on work of Dodson.
Kenig-Merle 2006: focusing energy-critical NLS on $R^3, R^4, R^4, R^5$. GWP+ Scattering for radial data with energy and kinetic energy smaller than those of the stationary solution.

Goal: prove existence of a global solution with control on the spacetime $L^6$.

Contradiction strategy.
Minimal blowup solution must exist.
Minimal blowup solutions mut be almost periodic. They are localized in physical and Fourier space.
Frequency localized Morawetz inequality. (only true for the minimal blowup solution). This is obtained by localizing in frequency the interaction Morawetz estimate.
This show that we have a smallness property on the spacetime $L^3$ norm on the high frequencies.
With some interpolation, we can then prove that the spacetime $L^6$ is bounded, contradicting the hypothesis.

Cubic NLS on $R^4$ (Visan)

(Original proof due to Ryckman-Visan but Visan recently simplified that following some ideas of Dodson.)

By contradiction and using concentration-compactness we have a minimal blowup solution.
There are only two scenarios. Rapid frequency cascade scenario; quasi-soliton scenario.

These are excluded using the long-time Strichartz estimates in the spirit of Dodson. The quasisoliton case is excluded using Morawetz.

Perturbation theory

Recalls the perturbation lemma from CKSTT, adapted to this problem.

She nicely describes the reduction to proving a local result on a time interval controlled by the energy. Once we have this type of local theory, we essentially convert the critical problem into one that behaves like the subcritical problem so GWP will follow.

Remarks on Proof

Subcritical quadratic terms in the Duhamel-Strichartz analysis on local intervals have a time factor. If this time factor is small enough, these subcritical terms can be absorbed. Oh, now I understand! The point here is that GP can be viewed as the energy-critical NLS plus some quadratic terms which don’t destroy energy conservation. This perspective guides the KOPV analysis. They show that the GP equation can be treated as a perturbation off the dilation invariant energy critical case.

Cubic-Quintic NLS with non-vanishing BC on $R^3$

They write $u=1+v$ and observe that $v$ satisfies energy critical NLS with subcritical lower order terms. The Hamiltonian is not sign definite so does not provide coercive control over the kinetic energy term. This is compensated for by using a lower order term $M(v)$, the $L^2$ norm of the real part of $v$. This quantity is not conserved. They show that it satisfies a Gronwall type estimate and that turns out to suffice.

Scattering for the GP equation in the case of large data

GP equation has traveling wave solutions that do NOT scatter.
Formation of traveling waves require a minimal energy in $R^d, d \geq 3$. Bethuel-Gravejat-Saut 2009, de Laire 2009.
Solutions with sufficiently small energy scatter. (Gustafson-Nakanish-Tsai 2006)
Can one prove scattering up to the minimal energy of a traveling wave?

Our goal is to fill in the gap. But, this problem does not seem too easy to attack, so we tried to apply these ideas on a simpler problem.

Killip-Oh-Pocovnicu-Visan

For a Cubic-Quintic NLS with zero boundary conditions (which has conserved mass and energy and has soliton solutions) the are working to show that if $v_0 \in H^1 (R^3)$ then scattering holds true if the mass is smaller than the mass of any soliton OR if it has positive energy, smaller than the enrgy of any solution.

(Final statement is a work in progress.)

Business Earmarks or Merit Competition: Which is the Better Federal Research Strategy?

Fri, 13 Apr 2012 18:54:34 +0000

Investments by governments to support research and development are crucial to economic prosperity, job creation, scientific advancement, and improvements to the future to be inherited by children. How should these investments be selected?

Merit review is a competitive process leveraging the expertise of a specially qualified panel to direct investments in research and development.
Earmarks are appropriations given to specific recipients or targeted areas, without competition, to satisfy the intent of government.

When viewed over the time scale on which benefits of research occur (decades), the merit review process is the better strategy. When viewed over the time scale of election cycles (years), governments often consider earmarks to be the better strategy. Visionary governments who followed the advice of scientists like Francis Bacon and Vannevar Bush built research systems that spawned the industrial revolution, the space and electronics industries, the internet, …. Canadian governments of the past who listened to John Charles Fields and Maurice Lamontaigne launched Canadian industries through the NRC and NSERC.

“The wealth of a nation once depended on its natural resources or the sheer size of its land, or its potential labour force; but now it is coming to depend more on its reservoirs of knowledge and its ability to organize and utilize them than on the older criteria.” –A Science Policy for Canada, Lamontagne Report, v. 3, (1976)
(Concerning the allocation of research funds) “It is folly to use as one’s guide in the selection of fundamental science the criterion of utility. Not because (scientists)… despise utility. But because. .. useful outcomes are best identified after the making of discoveries, rather than before.” — John Polanyi, Speech to the Canadian Society for the Weizmann Institute of Science, Toronto (1996-06-02)

“Faced with the admitted difficulty of managing the creative process, we are doubling our efforts to do so. Is this because science has failed to deliver, having given us nothing more than nuclear power, penicillin, space travel, genetic engineering, transistors, and superconductors? Or is it because governments everywhere regard as a reproach activities they cannot advantageously control? They felt that way about the marketplace for goods, but trillions of wasted dollars later, they have come to recognize the efficiency of this self-regulating system. Not so, however, with the marketplace for ideas.” — John Polanyi, Quoted in Martin Moskovits (ed.), Science and Society, the John C. Polanyi Nobel Lareates Lectures (1995)

Instead of listening to today’s voice along this lineage, John Polanyi, Canada’s current federal¹ government listens to Dean Roger Martin.

“What makes a country prosperous is not investment in science and technology. It is businesses producing high paying jobs by having unique products and processes that a customer needs.” — Roger Martin, Canada will shrivel under business-school neglect, dean says , The Globe and Mail (2011-03-16)

Canada, like Texas, doesn’t shrivel.

The report of the expert panel, the “Jenkins Report”, advising the government on the effectiveness of federal support of business research and development recommended² the creation of a new Industrial Research and Innovation Council (IRIC) and separately recommended³ that the NRC be ramified “into a constellation” of new R&D centres, a natural progression following the birth of the Tri-council. Rather than constellating the NRC, Budget 2012 does the opposite with plans for the role of the proposed IRIC to be carried out by a more centralized, business-focused, NRC:

…the Government will consider ways to better focus the National Research Council on demand-driven research, consistent with the recommendations of the Expert Panel. - Budget 2012

The policy implementations of Budget 2012 for CIHR, NSERC, SSHRC will be announced today in a meeting to Vice Presidents [Research] of Canadian universities. Drifting along a decade-long trend, a transfer of funds away from research investment programs distributed through competitive merit review toward new programs aimed at business recipients without competitive review is anticipated. Canadian business innovation gaps will not be solved by cutting funds supporting academic research and education activities.

The wisest investment strategy of federal dollars should rest upon systems involving the best expertise and considerations of the disruptive impact of basic research.

Footnotes:

The current government of Ontario is similar: after earmarking \$50M to the Perimeter Institute in 2011, \$42M was cut out of research in 2012.
"Recommendation 1: Create an Industrial Research and Innovation Council (IRIC), with a clear business innovation mandate (including delivery of business-facing innovation programs, development of a business innovation talent strategy, and other duties over time), and enhance the impact of programs through consolidation and improved whole-of-government evaluation."
"Recommendation 4: Transform the institutes of the National Research Council (NRC) into a constellation of large-scale, sectoral collaborative R&D centres involving business, the university sector and the provinces, while transferring NRC public policy-related research activity to the appropriate federal agencies."

Resources:

Misaligned Incentives in Canadian Science Policy

Thu, 12 Apr 2012 18:53:04 +0000

Budget 2012 continues to shift Canadian federal investment away from basic research toward industrial applied research. This shift is politically expedient: the redirection of funds can be discussed with tantalizing justifications based on job creation, targeted investment, streamlining discovery, and so forth. The shift resonates with a public concerned about frivolous expenditures of dollars collected through taxation. The late Senator from Wisconsin, William Proxmire, advanced this line of political rhetoric by issuing Golden Fleece Awards for science projects he lampooned as unworthy of government investment. Why should the government waste taxpayer money so that scientists can pursue their curiosity? Although slightly slanted with the operative verb “waste”, this is an entirely reasonable question which the scientific community must strive to answer.

Why should the government invest in basic research?

This question was eloquently answered by Vannevar Bush in his report Science: The Endless Frontier to President Roosevelt from July 1945. After my reading of the 2012 Budget, I thought it timely to share some relevant extractions (all emphasis added):

Basic research leads to new knowledge. It provides scientific capital. It creates the fund from which the practical applications of knowledge must be drawn. New products and new processes do not appear full-grown. They are founded on new principles and new conceptions, which in turn are painstakingly developed by research in the purest realms of science. (p. 16)
The simplest and most effective way in which the Government can strengthen industrial research is to support basic research and to develop scientific talent. (p. 17)

One of the peculiarities of basic science is the variety of paths which lead to productive advance. Many of the most important discoveries have come as a result of experiments undertaken with very different purposes in mind. Statistically it is certain that important and highly useful discoveries will result from some fraction of the undertakings in basic science; but the results of any one particular investigation cannot be predicted with accuracy. (p. 15)

Basic research is performed without thought of practical ends. It results in general knowledge and an understanding of nature and its laws. This general knowledge provides the means of answering a large number of important practical problems, though it may not give a complete specific answer to any one of them. The function of applied research is to provide such complete answers. The scientist doing basic research may not be at all interested in the practical applications of his work, yet the further progress of industrial development would eventually stagnate if basic scientific research were long neglected. (p. 15)

Industrial incentive systems are misaligned with basic research

Basic research is the soil from which “innovation” and “commercialization” grow. Where should these foundational studies take place? Should they be carried out by industry or by some cleverly designed industrial-academic partnership? The answers according to Vannevar Bush are illuminating. The incentive systems for industry are rarely aligned with basic research. (A notable exception was Bell Labs, where scientists free to explore basic research produced stunning advances.)

Industry is generally inhibited by preconceived goals, by its own clearly defined standards, and by the constant pressure of commercial necessity. Satisfactory progress in basic science seldom occurs under conditions prevailing in the normal industrial laboratory. There are some notable exceptions, it is true, but even in such cases it is rarely possible to match the universities in respect to the freedom which is so important to scientific discovery. (p. 16)
…Research is the exploration of the unknown and is necessarily speculative. It is inhibited by conventional approaches, traditions, and standards. It cannot be satisfactorily conducted in an atmosphere where it is gauged and tested by operating or production standards. Basic scientific research should not, therefore, be placed under an operating agency whose paramount concern is anything other than research. Research will always suffer when put in competition with operations. (p. 26)

The benefits of basic research do not reach all industries equally or at the same speed. Some small enterprises never receive any of the benefits. (p. 17)

Budget 2012 and the prior decade of mission drift

Over the past decade, there has been a shift in the federal investment in science. The new system aims to industrialize the research activities of university and government scientists. Commercialization, innovation, job creation, whatever you want to call it, is a preconceived goal which constrains the freedom necessary for the unanticipated, the disruptive breakthroughs brought by basic research advances. Despite the recent assurances by Ministers Flaherty and Goodyear that “blue sky” research will continue to receive federal investment, there has been a steady shift toward a system with explicit commercialization incentives instead of one with the freedom open to transformational discovery.

The publicly and privately supported colleges, universities, and research institutes are the centers of basic research. They are the wellsprings of knowledge and understanding. As long as they are vigorous and healthy and their scientists are free to pursue the truth wherever it may lead, there will be a flow of new scientific knowledge to those who can apply it to practical problems in Government, in industry, or elsewhere. (p. 10)
The history of medical science teaches clearly the supreme importance of affording the prepared mind complete freedom for the exercise of initiative. It is the special province of the medical schools and universities to foster medical research in this way - a duty which cannot be shifted to government agencies, industrial organizations, or to any other institutions. (p. 13)

Canada's Research Policy is Misaligned with Scientist's Incentives

The intense desire to deeply understand is the incentive for scientists, especially young scientists, to carry out research. Newton did not have to anticipate the industrialization of space to justify his study of gravity. Darwin did not have to anticipate applications of his studies in the pharmaceutical industry. Einstein did not have to forecast the Global Positioning System to justify investment in the theory of relativity. Canadian scientists aren’t free to pursue their breakthroughs. Instead, we are expected to cultivate relationships with industrial partners through government funded “first dates” and plan commercialization of our ideas in advance of their discovery. Instead of the freedom to pursue the curiosity that emerged from tens of thousands of hours of study, NRC scientists have been promoted and can now play the role of “business concierge”.

Where will these new products come from? How will we find ways to make better products at lower cost? The answer is clear. There must be a stream of new scientific knowledge to turn the wheels of private and public enterprise. There must be plenty of men and women trained in science and technology for upon them depend both the creation of new knowledge and its application to practical purposes. (p. 15)
To serve effectively as the centers of basic research these institutions must be strong and healthy. They must attract our best scientists as teachers and investigators. They must offer research opportunities and sufficient compensation to enable them to compete with industry and government for the cream of scientific talent. (p. 16)

The federal government’s research incentive system is no longer aligned with the intrinsic motivations of scientists.

Further Resources

A Report on the 2012 NSERC Discovery Grants Results for Toronto Math

Fri, 06 Apr 2012 18:51:57 +0000

Fifteen¹ faculty members from the Department of Mathematics at the University of Toronto submitted proposals to the 2012 NSERC Discovery Grants competition. Of these, one was a first time applicant (En), two (Ga, Ia) applied after a successful appeal of 2011 results, and one (Cd) was an appellant whose appeal was denied but could reapply because the 2011 award was for zero dollars. The first table below shows the 2012 results (in thousands of dollars per year) with 2010, 2011 award amounts for those researchers. The second table shows similar data for Toronto mathematicians in the 2011 competition, including the amounts for researchers (1d, 2d, 3d, 4d) whose appeals of 2011 results were rejected. The average for 2012 awards to Toronto mathematicians was 153% the average for 2011.

Average Grant Amount (Toronto Math)

2006: \$27k/y
2007: \$26.3k/y
2008: \$26.5k/y
2009: \$25k/y
2010: \$25.3k/y
2011: \$19.3k/y
2012: \$29.5k/y

Instability Visualized

Viewing the results from the perspective of researchers in these competitions reveals instability in the Discovery Grants evaluation and appeals processes:

Imagine the experience of researcher 2d. This person had five years at \$42k/y, was cut to \$30k/y in 2009, and successfully appealed that outcome. The result of the appeal was a one year reinstatement of the previous grant level at \$42k/y and permission to reapply to the 2011 competition. In 2011, this researcher’s grant was hacked to \$18k/y so this person files another appeal. The 2011 appeal is rejected.
Contrast the experience of professors 4d and Ga. Both launched their Canadian research careers and entered the Discovery Grants competition for the first time in 2011. Ga’s appeal of the \$13k/y result from 2011 was successful and the 2012 competition led to a new result of \$30k/y for the next five years. 4d’s 2011 appeal was denied so this researcher is locked in for five years at \$11k/y. Which of these researchers is likely to have better HQP numbers at renewal time five years from now?
The experience of 2011 appellant Ia is also a bit strange. After a long run of celebrated research funded at the \$40k/y level, this researcher’s funding level was dropped in 2011 to \$15k/y. That outcome was successfully appealed and the 2012 outcome was \$35k/y.
The 2011 appeals by 1d, 2d, 3d, and 4d were all turned down so these researchers are locked in at relatively low funding levels for the next five years.
NSERC deviated from standard policy (a “pilot program”) in their handling of the 2011 appeals of Toronto mathematicians. Toronto’s appeals were evaluated by multiple appeals advisers while appeals from other universities were evaluated by one. There is evidence² showing that Toronto appeals were denied even when one of the appeals advisers advocated for granting the appeal.

Consistent Results on Appeals Cases show 2011 was Anomalous

Proposals by three Toronto researchers were evaluated in both the 2011 and 2012 competitions. The outcomes for these proposals provide a comparison³ between the accuracy of the merit evaluations by the 2011 and 2012 Evaluation Groups and bin-to-funding assignment by the Executive Committee and NSERC staff. Here is the data, including the percentage adjustment from 2011 to 2012:

These three cases provide further evidence, consistent with the message in the public statement signed by over 300 Canadian researchers, that the 2011 evaluations were anomalous. Despite the consensus opinion from the Canadian math/stats community, a public message from a majority of the 2011 Evaluation Group, and advice from top administrators that the 2011 anomalies required an altered appeals process, NSERC chose not to reevaluate the scientific merit of proposals when considering whether to grant or deny an appeal. The grounds for a successful appeal required evidence of administrative errors; evidence of error in merit evaluation was not considered germane.

The Math-NSERC Liaison Committee is collecting data from department chairs about the 2012 competition for Section 1508. It might turn out that 2012 will be viewed as more consistent with expectations, a more accurate evaluation compared to 2011. This would be encouraging. However, the unsuccessful 2011 Toronto “pilot program” appellants (researchers 1d, 2d, 3d, 4d) face the next five years with inadequate funding to support their research programs.

Footnotes:

The names of the faculty members are suppressed. The 2012 applicants will be referenced with codes A, B, C, …, H; 2011 applicants (without successful appeal) will be referenced using 1, 2, …, 7. The appended small case letters indicate whether the researcher had an appeal granted (a), an appeal denied (d), or was a new applicant (n).
This evidence is contained in the reports of the appeals advisers provided to the appellants by NSERC and also in documents obtained by some of the appellants through formal requests under the Access to Information Act.
It would be useful to know the data (number, success rate, basis for granting) for Discovery Grant appeals submitted to NSERC over the past few years. As far as I can tell, NSERC does not provide this data.

Anticipating the 2012 NSERC Discovery Grants Competition Results

Mon, 26 Mar 2012 18:50:38 +0000

2011 Discovery Grants Competition Aftermath

Anomalies in the results of the 2011 NSERC Discovery Grants competition provoked a flurry of activity nearly one year ago. My blog post from April last year reported on surprising results for several of my colleagues at Toronto. An email flurry among Canadian mathematicians culminated in a late April public statement which was eventually signed by 336 Canadian researchers, including 35 Fellows of the Royal Society and 27 Canada Research Chairs. In late May, a majority of the Evaluation Group for Section 1508 released a public letter to NSERC President Suzanne Fortier reporting on a “lack of fairness”, a “lack of transparency” and that their “confidence in the program, as currently administered, is regrettably shaken”. President Fortier gave a presentation to the Canadian Mathematical Community at the Summer CMS Meeting. President Fortier’s slides have not been released but there are notes and rebuttals by Walter Craig. Claims of grade inflation have also been analyzed.

Toronto Math Appeals of 2011 Results

Last May, seven Toronto mathematicians submitted appeals of their 2011 results. As far as I can determine, NSERC does not provide statistics on the number of appeals but seven math appeals was unprecedented from Toronto. Appeals are normally resolved within two to three months. As of late September, the decisions on the appeals were not yet announced. Toronto Math made email and telephone inquiries asking that decisions be made soon since successful appellants would need time to prepare new proposals in advance of the November 1 submission deadline.

Email correspondence revealed that Toronto’s appeals were handled differently than those from other universities. In particular, NSERC’s Isabelle Blain informed me that Toronto’s appeals were processed through a “pilot” program involving two or more appeals advisers. Three of the appeals decisions were announced in late September and the remaining four appeals were announced on October 19, 2011: 2 of 7 appeals were granted. The good news for the successful applicants was bittersweet since it set in motion a rushed effort to prepare proposals for the 2012 competition. Unfortunately, the reports from the appeals advisers for one of the successful appellants could not be consulted while preparing the 2012 proposals since they were not provided by NSERC until after the submission deadline. Toronto appeals were rejected in circumstances when one or more of the appeals advisers advocated for granting the appeal. Some Toronto appellants perceive the “pilot” appeals process to have been unfair. Conversations along these lines continue….

Repercussions: Uncertainty, Frugality and HQP

In the wake of the 2011 anomalies, many Toronto mathematicians who face renewal of their NSERC Grant in 2012, 2013 and even in 2014 are hesitant to commit funds toward hiring postdocs or toward training graduate students. Uncertainty has provoked frugality. Running a successful, even world leading, research program is no longer sufficient to justify an NSERC Discovery Grant. The new evaluation metrics, introduced by NSERC in 2009, require evidence that the principal investigator is successful at training highly qualified personnel (HQP). Taking frugal actions now will limit HQP production over the next few years resulting in a lower or zero grant next time: a vicious cycle which obstructs research advancement and training of young investigators. With the goals of breaking this cycle and maintaining the research vibrancy postdocs bring, the Department of Mathematics at Toronto has raised its funding support level (by 33%) for postdoctoral hires.

Looking Forward: Continued Instability

The 2012 Discovery Grants Competition results will be announced in the next week or two. Hearsay and anecdotal reports about the evaluation process have been reassuring. The Evaluation Group received recommendations jointly authored by the Long Range Plan Committee Chair Nancy Reid and the Math-NSERC Liaison Committee Chair Walter Craig. Similar recommendations were made in a December letter by a group of physicists to President Fortier. The Long Range Plan should help heal the rifts between the mathematics and statistics communities of Canada when it is released later this year. There are reasons to be optimistic in the short term.

In the intermediate term, there is likely to be continued instability with the peer review system at NSERC. Despite the 2011 anomalies for math and the ensuing kerfuffle, receiving similar complaints and suggestions for improvements by physicists, and faced with public criticisms by astronomers, chemists and engineers, NSERC staff insists that the new system is working well. The new evaluation system, the “conference model” with its “bins” and HQP touchstone, was introduced in 2009 and will be the subject of a five year review in 2014. Whether led by current or future NSERC staff, or by an energized scientific community, I anticipate substantial changes, and therefore instabilities, to the peer review process at NSERC in 2014 or 2015.

Shrinking Funds for Basic Research

Beyond the fairness of the evaluation process, there is the issue of shrinking federal investment in basic research. According to NSERC's 2010-2011 tables, expenditures on Discovery have been flat to decreasing when viewed in constant 2000 dollars over the past decade (data taken from Table 1). In contrast, the total NSERC budget has grown with the expansion occurring in programs aimed at commercialization rather than basic research.

Meanwhile, strategic recruitment leveraged by the CRC and CERC programs has increased the number of faculty submitting Discovery Grant applications (data taken from Table 27). Over the same time period, the number of successful applications has decreased (data taken from Table 27).

Table 50 shows that mathematicians do not really benefit from programs outside of the Discovery Grants. Mathematics, the “poetry of logical ideas”, is profoundly relevant to various industries. Canada’s research mathematicians need to find ways to share their expertise and leverage their Discovery Grant funds the way other disciplines do.

IAS Workshop on Symplectic Dynamics 2: Friday

Fri, 16 Mar 2012 18:49:10 +0000

IAS School of Mathematics

Workshop web page

Friday: 2012-03-16

9:00 - 10:00 James Colliander, University of Toronto, “Big frequency cascades in the cubic nonlinear Schroedinger flow on the 2-torus” abstract
10:15 - 11:15 Marcel Guardia, IAS, “Growth of Sobolev norms for the cubic defocusing nonlinear Schroedinger equation in polynomial time” abstract
11:30 - 12:30 Yann Brenier, University of Nice, “Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics” abstract

James Colliander: Big frequency cascades in the cubic nonlinear Schrödinger flow on the 2-torus

(chalk talk)

(joint work with M. Keel, G. Staffilani, H. Takaoka, T. Tao)

I prepared slides but decided to give a chalk talk. The slides are located here: http://uoft.me/nls-cascade. The paper discussed in this talk is located here.

(See also: The thesis of Zaher Hani has advanced along these lines and is surveyed on his slides from the Ilde de Berder Workshop.)

The construction of the frequency civilization is partly conveyed by the following cartoon. Notice that the underachieving child frequency in the cartoon is always sent to the zero frequency. This violates the injectivity requirements used in our construction of the set $\Lambda$.

The next cartoon is meant to convey a traveling wave through the generations in the civilization. This wave is constructed by concatenating heteroclinic orbits in the toy model evolution.

The idea that the orbits could be concatenated reminded my coauthors of this famous commercial:

http://www.youtube.com/watch?v=KXA8g90g7so

Marcel Guardia: Growth of Sobolev norms for the cubic defocusing nonlinear Schrödinger equation in polynomial time

(joint work with Vadim Kaloshin; we have a preprint; slides from the talk; 32 pages)

This talk is strongly related with the previous talk.

$NLS_3^+ (T^2)$. Energy and Mass are conserved. The problem is globally well-psed in time Bourgain 1993.

Transfer of Energy

Fourier series of $u$.
Can we have a transfer of energy to higher and higher modes ass $ t \rightarrow + \infty$?
This is quantified with the growth of Sobolev norms.

We need to move mass toward high frequencies in a careful way to satisfy the mass and energy constraints.

Theoreom (Bourgain 1993): As $t \rightarrow + \infty$, the $H^s$ norm is upper bounded by $\leq t^{2(s-1)+} | u(0) |_{{H^s}}.$

This result has been improved or applied to other Hamiltonian PDEs by various authors.

Question (Bourgain 2000): Are there solutions $u$ such that for $ s>1$ such that $$| u(t)|s \rightarrow \infty $$ as $ t \rightarrow + \infty? $ Moreover, he conjectured that the growth should be subpolynomial in time: $ | u(t)|{H^s} \ll t^\epsilon$.

The second part was partly conjectured because of insights related to Nekoroshev type theorems for NLS.

Kuksin studied the growth of Sobolev norms for NLS for large initial condition. For such data, a change of coordinates recasts the dynamics into

i \dot{w} = - \delta \Delta w + |w|^2 w, ~ \delta \ll 1. $$

Theorem (CKSTT 2010): $\exists $ big frequency cascades in the $NLS_3^+ (T^2)$ flow.

The solutions have small intial mass and energy. They remain small as time involves whereas the s-Sobolev norm grows considerably.

Theorem: (long statement, I’m reading instead of typing.)

The mass is small but the $H^s$ norm is initially large. They can then grow it up to any big threshold over a polynomially related time interval.

Remark: One might view this equation as a perturbation (when the data is small) of the (integrable) linear Schr"odinger. It is well know that the Nekoroshov type results for PDEs often loses the exponential estimates and becomes polynomial. Our result is consistent with this.

Remark: Our result deals with a different regime than the Bourgain subpolynomial conjecture. Our result is rather fast, but it could perhaps slow down over infinite time. Our construction involves a finite number of modes. If we try to build something on an infinite number of modes, the transfer mechanism might slow down.

Comments:

One can tensor this up to obtain similar results on $T^d, d \geq 2$.
We can obtain more detailed information about the distribution of the Sobolev norm of the solution $u$, among its Fourier modes when $t = T$. In particular, the high Sobolev norm is carried by two high achievers at the last stage. The high Sobolev norm is essentially localized in two modes.

Main Ideas in the Proof:

$I$-team introduced a finite-d toy model.
This toy model approximates well certain solutions of NLS
Our contribution is the analysis of the toy model. Using dynamical system tools, and a careful choice of the initial conditions, we find a faster motion.
The solutions of NLS can be proven to approximate well the solutions for the toy model for long time.

Reduction to the toy model.

$FNLS$
$RFNLS$
Construct $\Lambda$.
Toy Model ODE

For $N$ big enough, the set $\Lambda$ can be chosen to have the “wide diaspora property.” This is partly why we don’t have an infinite cascade. The construction only involves a finite number of modes. We want to quantify everything in terms of the number $N$ of generations. At the end we have $N \thicksim \log K$. We have to quantify everything.

Toy Model Theorem: There exists an orbit in the toy model which moves from the first generation to the last. Their statement includes quantifications! They compute the time of this transfer process.

To make things happen quickly, they want to make the transfers as fast as possible. This development uses a different orbit construction than the one performed by CKSTT.

Dynamics of the Toy Model:

ODE explicitly written out.
Each 4-d plane is invaraint.
Dynamics in each 4-d plane is given by a simple Hamiltonian involving nearest neighbor interactions.

Nice picture of invariant planes intersecting to form something like a polyhedra with a curve following along nearby invariant lines. “Of course, we are not in the plance but we are nearby it.” Of course to do this, we need to understand the dynamics in each of these planes. To obtain these orbits, we use hyperbolicity. But these planes have certain normal positive Lyapunov exponents so one has to be very careful. If we just move away from these planes, we lose control.

Dynamics in $L_j$:

To construct such orbits, we need to understand dynamics in each $L_j$.
Hamiltonian $h_j$ and $M_j(b_j, b_{j+1}) = |b_j|^2 + |b_{j+1}|^2$…..ack slide changed.
Contains two periodic orbits.
Periodic orbits in $L_j$ are hyperbolic.
Stable and unstable invariant manifolds of the periodic orbits coincide.
Call $\gamma_j$ the heteroclinic connection between the two dimensional manifold asymptotic to $T_j$ as $ t \rightarrow - \infty$ and asymptotic to $T_{j+1}$ as $ t \rightarrow \infty$.

Key Problem: The Shadowing

(nice picture)

We put sections transveral to the flow.
We study local maps: dynamics close to the periodic orbits $T_j$. Global maps: study dynamics close to the heteroclinic connections $\gamma_j$.

Local and Global Maps:

Shadowing for global map is basically applying (refined) Gronwall estimates.
Local map is more delicate: periodic orbits are of mixed type. Hyperbolic eigenvalues are resonant.
This resonance complicates the analysis of the local maps.

We need to choose very carefully which orbits we study.

The Model Problem:

After some reductions, we have a Hamiltonian of the form:

$$ H(p,q) = p_1 q_1 + p_2 q_2 + H_4 (p,q) $$ where $H_4$ is a degree 4 homogenous polynomial, the variables “1” correspond to the variable $b_{j-1}$ ….slide changed.

Analysis of map from a section $\Sigma_+$ to $\Sigma_-$.

Dynamics of the linear saddle (Kill the $H_4$ and see what happens.).

Dynamics of the resonant saddle:

System is not well approximated by its linear part due to the resonance.
For typical initial conditions, we have a resonat affect creating logarithmic (in $\delta$ ) corrections to the transfer across hetereoclinic connections.
We need $~N$ transitions.
The number of logarithms becomes exponential in $N$.
We need to stay close to the periodic orbits to control the shadowing
This implies we need to start….slide change

We use the beautiful Shilnikov trick. The worst term that was developing with logarithms is now computed more accurately in terms of some function $g(p_0, q_0)$. This transfers the resonant saddle dynamics into essentially the dynamics of the linear saddle, provided that we carefully choose the domain of the map. This is kind of delicate and needs to be iterated through compositions.

Composing the local and the global maps:

We need to compose the local and global maps.
We define sets $U_j$ in the transversal secions and we show that the dynamics moves one into the other. (This is the “perfect shot”.)
To avoid deviations at each local map, we need to impose a restriction at every step.
“Product-like” step.

Product-like structure sets.

We start with a polydisk.
At each step, we impose a condition on the mode $b_{j-1}$.
Inductively, we rstrict the domain on previous domains involving conditions on previous mode involving the Shilnikov function $g$.
Since the restricitons involve a different mode at each step, the conditions are compatible.

Composing the local and global maps produces the toy model result. The detailed discussion partly explains the time quantification.

Approximating solutions of NLS:

Last step obtain a solution of NLS close to the solution of the toy model.
We modify the set $\Lambda$ from the $I$-tema so that the modes out of $\Lambda$ only gets influenced by few modes in $\Lambda$.
Each $b_j$ is excited only for a short period of time.
A mode out of $\Lambda$ only receives mass from $\Lambda$ during a short time.
This implies that the spreading of mass to modes out of $\Lambda$ is very slow.
We obtain an orbit for NLS that undergoes the growth of Sobolev nroms in polynomial time.

Yann Brenier: Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics

(chalk talk; the slides once linked here are no longer available.)

It’s a good time for all of us to thank the organizers for this meeting. (Applause!)

Related to a question posed by Shnirelman from 1985.

System of interacting particles along the real line with sticky collisions. When the particles hit, they merge and continue with the same momentum. This is an inelastic, sticky collision. This is clearly

dissipative
nonreversible in time

Shnirelman’s Question (1985): Can we modify the action principle to handle these dissipative collisions?

Unfortunately, the paper is hard to find. You can think of the collision in a higher dimensional space and keep track of the energy in the extra variables.

G. Wolansky (2008 ?)

In this talk, I want to provide some ideas that come from ideal fluids. This seems strange because this problem is highly compressible, etc.

This talk is about a proposal for a modified action suggested by ideal fluid mechanics.

Arnold’s geometric interpretation (1966) of Euler equation for incompressible fluids (1755).

Let $D = [0,1]^3$. Let $VPM (D) = [ volume ~ preserving ~ maps ~ of ~ D]$. This may be viewed as a subset of $H = L^2 (D, R^3)$. Geodesics along VPM are (formally) the solutions of the Euler equations.

There is a discrete subset of $VPM (D)$ are the permutation maps $S =P_N (D)$. Partition the unit cube into a collection of $N$ subcubes $Q_i$ each with center of mass $A_i$. You would like to do some kind of discrete fluid mechanics by exchanging these cubes. There is a folklore of approximating geodesics with these kinds of maps. This is used in some works in computational geometry. How to define approximate geodesics along $P_N (D)$?

More generally, let $H$ be a Euclidean (or Hilbert) space. You have a closed bounded subset $S$. Introduce a potential $$\Phi [x] = \frac{d^2}{2} (x,s) = \inf_{s \in S} \frac{|x-s|^2}{2} = \frac{|x|^2}{2} - R(x). $$ Here $R$ is the Legendre transform: $$ R(x) = \sup_{s \in S} (x|s) - \frac{1}{2} |s|^2. $$ Convex, Lipschitz, usually not smooth.

Approximate minimizing geodesics are found by minimizing between two given points $A, B \in H$ by $$ \int_0^1 (\frac{1}{2} |\frac{dx}{dt} (t)|^2 + \frac{1}{2\epsilon} \Phi [x(t)] ) dt $$ satisfying $X(0) = A, X(1) = B$. If $S$ is a smooth manifold this converges to geodesics Rubin-Ungar 1957 (Yann’s birth year!).

These ideas were applied by David Ebin to fluids.

A simpler example than the one appearing in Shnirelman’s question…

Take $H = R^2$. Let $S$ be the St. George cross. He writes the coordinate axes in $R^2$ in red and forecasts that a joke will soon come up…

Whenever $\Phi$ is smooth about $X$, we have $\nabla \Phi (x) = x - \pi_S (x)$ (the closes point to $x$ inside $S$, not necessarily unique). The bad set $N$ where differentiability fails is both meager and has lebesgue measure zero in finite-d case. This has to do with the regularity of Lipschitz functions.

What is the bad set related to the St. George cross? Of course, it is the St. Andrew cross, the flag of Scotland! (He draws that in blue.) You can also reverse the picture so that the bad set becomes the St. George cross if you prefer to view it that way…..

If $x \in H \backslash N$, we have $\phi (x) = \frac{1}{2} |x - \pi_S (x)|^2 = \frac{1}{2} |\nabla \phi (x)|^2$.

Look at the action (for simplicity $\epsilon = 1$) for a “good curve” $ t \rightarrow x(t)$. Namely a curve for which $x(t) \in H \backslash N$ for a.e. time, the action reads $$ \int_0^t (\frac{1}{2} |\frac{dx}{dt}|^2 + \frac{1}{2} |\nabla \Phi [x(t)]|^2 ) dt. $$

So, obvious minimizers are those good curves that satisfy the first order equation $$ (FO) ~ \frac{dx}{dt} = \nabla \Phi [x] = x - \nabla R [x]. $$

This is a so-called gradient flow of a Lipschitz convex function (up to the first term which can be absorbed). These objects have been studied.

The theory of maximal monotone operators does the job (cf H. Brezis book) in the sense that this is completely well-posed in $H$. We know from that theory that $ x \in C(R_+; H)$, Lipschitz in $t$, and $$ \frac{dx}{dt} (t+0) = x(t) - {d^0 R[x]} $$ which is sometimes called the minimal selection gradient or “mean” gradient (studied in the Italian school).

Example. Differentiate $|x|$. The subgradient fills in the vertical line. The minimal gradient has value zero at $x=0$. This is a nice theory but it gives us very bad curves.

If you start on this St. George cross example, he describes the dynamics and interprets this as a dissipative mechanism. This has little to do with the action principle but it does have dissipation. So, we might take some inspiration from this example….this is a proposal for a modified action.

Modified action:

$$ \int_0^t \frac{1}{2} |\frac{dx}{dt} - d^0 \Phi [x(t)]|^2 dt. $$ Minimizers of the modified action are very likely to be bad curves.

Some rats were confined in a box by electric shocks and another which is very hot. But, if you dig a small channel between the other two boxes. It turns out the rats can survive longer by moving back and forth between the two boxes. I hope it is not a true story….

The dissipation is not incompatible with the arrow of time if you order the data.

Now, I’d like to go back to permutations and fluids. What kid of equation do I get?

Remember the box, broken up into the subcubes. Consider the set $S$ to be the permutations of all the centers. Let $H$ denote $R^{dN}$. In the $d=1$ case, you get a friendly approximate geodesic equation through the classical (nonmodified) action. We are then describing $N$ particles on the line. $$ \epsilon \frac{d^2 x_i}{d t^2} = x_i - \frac{1}{2N} \sum_{j=1}^N ~{\mbox{sgn}} (x_i - x_j). $$ This is like a gravitating parallel pancackes according to Newton gravity plus a repulsive background. This type of model was studied by people like Zeldovich. The repulsive effect is natural in that context. By approximating the incompressible Euler this way, it is nice that you get a model that is reasonable from the point of gravity.

In higher dimensions, the model is NOT consistent with Newtonian gravitation but is instead consistent with a Monge-Ampere correction to Newton’s gravitation. You get something like $\Delta \phi = \rho -1$ and then eventually find something like $ {\mbox{det}} (I + D^2 \phi) = \rho.$ I am not yet certain if this is geometrically reasonable. It is related to Born-Infeld correction to Maxwell’s equations.

So, what is the point? If you modify the action, you can recover interaction with sticky collision.

This is the so-called “Dust” in the Russian literature. These are elementary ideas that explain why matter has clumped in cosmology. Sluggish motions in the early universe moves like honey. Tiny fluctuations of qunatum origin and these create a Jeans instability which tends to concentrate matter. This is at a very large scale and concentrated on a llower dimensional fractal set.

IAS Workshop on Symplectic Dynamics 2: Thursday

Thu, 15 Mar 2012 18:47:33 +0000

IAS School of Mathematics

Workshop web page

Thursday: 2012-03-15

9:00 - 10:00 Peter Topalov, Northeastern University, “Qualitative features of periodic solutions of KdV” abstract
10:15 - 11:15 Jiansheng Geng, Nanjing University, “Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential” abstract
11:30 - 12:30 Massimiliano Berti, UNINA, “Quasi periodic solutions of Hamiltonian PDEs” abstract
2:30 - 3:30 Ralph Saxton, University of New Orleans, “The generalized inviscid Proudman Johnson equation” abstract
4:30 - 5:30 Dongho Chae, Sungkyunkwan University, “On the blow-up problem for the Euler equations and the Liousville type results in the fluid equations” abstract

Peter Topalov: Qualitative features of periodic solutions of KdV

I need to do some detailed setup to expose the ideas I want to describe. We will discuss the KdV equation.

$$ q_t - 6 q q_x + q_{xxx} = 0.$$

Let’s impose periodic boundary conditions. We impose the initial condition $q|{t=0} = q0 \in H^N (T) $. This parameter $N$ will change at different times in the context of the talk, depending upon the theorem we are considering.

$$ H_{KdV} (q) = \int_0^1 (q^3 + \frac{(q_x)^2}{2} ) dx. $$

What is the symplectic (more precisely the Poisson) structure? The phase space where the evolution will happen in $H^N$. For two functions $F,G: H^N \rightarrow R,$ we have the Gardner bracket $$ { F, G } = \int_0^1 \partial_q F \partial_x (\partial_q G) dx. $$ (Ack….I am having trouble making curly brackets show up in the Gardner bracket even when I try to escape using a slash.)

Linearizing around $q=0$, we find $q_t =q_{xxx}$ which we can solve explicitly to find the evolution for the Fourier coefficient: $$ \dot{\hat{q_k}} = - (2 k \pi i)^3 \hat{q_k} = (2 k \pi)^3 i \hat{q_k}. $$ We can solve this directly to find $$ \hat{q_k}(t) = \hat{q_k} e^{i (2k\pi)^3 t}. $$ He draws a collection of complex Fourier planes and draws circles representing the motions of the Fourier coefficients.

Let’s see what the Poisson structure looks like when the dynamics are viewed in terms of the Fourier coefficients.

We compute the Gardner bracket of two Fourier coefficients: $$ { \hat{q_k}, \hat{q_l} } = \int_0^1 e^{-2 k \pi i x} (e^{-2 k \pi i x})’ dx = - 2 l \pi i \delta_{k, -l}. $$ (missing curly brackets on left side.)

We fix attention to zero mean initial data. We will look at $H^N_0$ where the subscript reminds us that we are looking at the zero mean setting.

We define $z_k = \frac{\hat{q_k}}{\sqrt{|k| \pi}}$ and then observe that $z_k = x_k + i y_k$ gives us Darboux coordinates $x_k, y_k$.

We have a mapping $\Phi_L : H^N_0 \rightarrow h^{N+\frac{1}{2}}$. Let’s see why this $\frac{1}{2}$. We take an element of phase space $q$ and apply $\Phi_L$ and this takes us to the associated Darboux coordinates $z_k = \frac{\hat{q_k}}{\sqrt{|k| \pi}}$ and the division by $|k|$ explains the $\frac{1}{2}.$

Remarks about this map $\Phi_L$:

diffeomorphism
canonical
linearizes the flow

Return to KdV.

Theorem 1: $\exists ~ \Phi: H^N_0 \rightarrow h^{N + \frac{1}{2}}$ such that

$\Phi$ is a diffeomorphism;
$\Phi$ is canonical;
$z_k (t) = z_k e^{i \omega_k (q) t}.$
(New) $\Phi = \Phi_L + A; ~\Phi^{-1} = \Phi_L^{-1} + B$ where $A$ is 1-smoothing. What this means is that $A, B$ are bounded maps such that $$A: H^N_0 \rightarrow h^{N + \frac{3}{2}};$$ $$ B: h^{N + \frac{1}{2}} \rightarrow H^{N+1}.$$

In 3. the phases depend only upon the initial data but for some reason I don’t want to write $q_0$ right now.

1., 2., 3. were proven by Kappeler-Poschel-Makarov for $N \geq 0$. For the interval $-1 \leq N \leq 0$, 1.,2.,3. was established by Kappeler-Topalov.

Item 4. is new and recently proven by Kappeler-Schad-Topalov (I didn’t catch the name…). This advance may be viewed as a globalization of a local statement obtained by Kuksin-Perelman.

Consider the KdV evolution moving through phase space. We can also consider the linearized evolution. We are interested in the difference. Denote by $S_t (q)$ the KdV evolution. We can do something a little bit different: $$ S_t (q) - \sum_{k \neq 0} (\hat{q_k} e^{i \omega_k (q) t}) e^{2k \pi i x} = R_t (q). $$

Theorem 2:

$R_t: H^N_0 \rightarrow H^{N+1}_0$ is continuous (even analytic on the Casimir $[q] = 0$).
$\forall ~ q \in H^N_0$, we can consider the orbit $[R_t(q): t \in R] \subset H^{N+1}_0$ is relatively compact.
$\forall ~ M > 0, [R_t (q): t \in R, \| q \|{H^N} \leq M] \subset H^{N+1}0$ is bounded.

In particular, from 2., the norms are relatively bounded.

I want to say something about the proof. The overview involves an expansion of the flow maps using the structure in Theorem 1, item 4. The core of the analysis is in the spectral theory of the Shcrodinger operator.

Jiansheng Geng: Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential

(joint work with J. You an Z. Zhao)

We study a nonlinear Schrodinger equation on the lattice and show there exist quasiperiodic solutions. $$ i \dot{q_n} + \delta( q_{n+1} - q_n) + V_n q_n + |q_n|^2 q_n = 0, n \in Z.$$

Here $\delta $ is small. $V_n (x) = V(n \tilde{\alpha} + x)$ with $V$ a nonconstant real analytic function on R/Z and $\alpha$ satisfying a Diophantine equation.

Eliasson 1997, Acta

Slides moving fast…

Theorem: For small enough $\delta$, this equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions for a.e. $x \in R/Z$.

Also works in the nonlinear case.

Choffrut: What is Whitney smooth? A: Some discussion… Kaloshin: The function is defined on a Cantor set and you need to define what it means to be smooth. You can’t differentiate so you have to do something to understand smoothness….this is the idea of Whitney smooth.

Related works (Linear case):

Belissard-Lima-Scoppola 1983 CMP
Fröhlich-Spencer-Wittwer 1990 CMP
Chulaevsky-Dinaburg 1993 CMP
Eliasson 1997 Acta

Related works (Nonlinear case):

Töplitz-Lipschitz property

Eliasson-Kuksin 2010
Geng-Xu-You 2011

Slides are quite dense, too technical for me to convey here. Abstract KAM theorem.

Massimiliano Berti: Quasi periodic solutions of Hamiltonian PDEs

Nonlinear Wave Equation

$$ (NLW):~ u_{tt} - \Delta u + V(x) u = \epsilon f( \omega t, x, u).$$

$\omega$ diophantine.

Question: Do $\exists$ quasiperiodic solutions of NLW ro $\epsilon \neq 0$?

Linear wave equation: ($\epsilon = 0$.)

Solutions are built by superposition.

Eigenfunctions are orthonormal in $L^2$: “Normal Modes”
Eignevalues $\lambda_j \rightarrow + \infty$: the $\sqrt{\lambda_j}$ are the “Normal frequencies”.

All these linear soutions are periodic. Their superpositions are quasiperiodic. Do these persist when we turn on the nonlinearity.

We look for quasiperiodoc solutions. This leads to an equation for qp solutions:

$$ (\omega \cdot \partial_\phi)^2 u - \Delta u _ V(x)u = f. $$

We can approach this existence question as a bifurcation problem.

We make a NON-RESONANT assumption:

$$ | (\omega \cdot l)^2 - \lambda_j | \geq \frac{\gamma}{1 + |l|^\gamma}, ~ \forall (l,j). $$ The inverse operator is unbounded so the classical implicit function theorem fails. We need a replacement, some kind of Quadratic scheme.

We use a Nash-Moser IFT: Newton method + “smoothing”

The advantage is the rapid convergence. The disadvantage is that we have to invert in a whole neighborhood of the expected solution.

Literature

$d=1$

Kuksin 89, Wayne 90; 2nd order Melnikov non-resonance conditions OK. Dirichlet conditions to ensure simplicity of eignevalues.
Craig-Wayne 93 periodic solutions
Bourgain 94 quasiperiodic solutions

Lyapunov-Schmidt, f analytic, Netwon Method. 1st order Melnikov conditions.

$d \geq 2$

Eigenvalues of $\Delta + V(x)$ appear in clusters of increasing size.
If $d \geq 2$, the eigenfunctions of $-Delta + V(x)$ are NOT localized wrt exponentials! (**Feldman-Knönner-Trubowitz**)

Often, these issues motivate the study of “pseudo-PDEs.”

Newton Method

Bourgain 98 Annals 05 Annals
Wang 10, 11

KAM theory

…Processi, Berti…Craig-Wayne…ack slide changged.

Nash-Moser

Eliasson 89

Berti-Bolle 2011 (to appear in JEMS)

Existence: (Summary of statements; slides are more precise) Under some conditions on $f$, there exists a Cantor like set $C_\epsilon$ of asymptotically full Lebesgue measure. “This is a classical KAM-like statement.”

Regularity:

The Cantor-like set is not technical, e.g. CKSTT 2010 Inventiones.

Pre-assigned direction of tangential frequencies

Geng-Ren 2010
Berti-Biasco CMP 2011
Bambusi-Berti-Magistrelli, JDE 2011

Weaker non-resonance condition

simpler technique

Many of these results should carry over to spheres, Zoll manifolds, Lie groups, homogenous spaces: symmertries and properties of eigenfunctions and eigenvalues are key properties! Related to Birkhoff normal form results by Bambusi, Delort, Grebert, Szeftel for spheres and Zoll manifolds.

For periodic solutions, see Berti-Procesi Duke 2011

Idea of Proof

Small divisors.

Töplitz matrices.

Difficulties:

T has only a polynomial decay off the diagonal.

Smoothing operators; finite-d projectors. TAME estimates are needed. We need estimates on the inverse operator on high regularity Sobolev spaces. Counterexaple of Lojaciewitz-Zehnder! This example shows identifies a parameter boundary in the Newton iteration scheme.

Step 1. $L^2$-estimates: Lower bounds for the eigenvalues.

Step 2. “Separation Properties” of small divisors

Locations where the divisors are small become more and more rare. There emerge “irrational” conditions on the slope $\omega$. These conditions are not needed for the Schrödinger equation. The dispersive relationship is different and helps you here.

KAM

Nash-Moser via the 1st Melnikov conditions. This is in some sense the minimal assumption. This approach works well in case of multiple eigenvalues. However, it has the disadvantage that it requires studying the linearized equation with non-constant coefficients.

Other strategy: impose stronger nonresonant conditions of 2nd Melnikov type (as usual in KAM). This has the advantage that we have a linearized equation with constant coefficients. There exists a torus and a reducible normal form.

Question: Do quasiperiodic solutions persist for nonlinearities which involve derivatives? Important physical applications.

Kuksin 1998
Kappeler-Pöschel 2003
Liu-Yuan 2010 for Hamiltonian DNLS (Benjamin-Ono)

Theorem (Berti-Biasco-Procesi 2011): DNLW has a Cantor-like family of quasiperiodic solutions. These qp solutions have zero Lyapunov exponents and the linearized equations can be reduced to constant coefficients.

Ideas of proof. View this as an infinite dimensional Hamiltonian system. Use conservation of momentum (Geng-You).

Birkhoff Normal form step, reduction to action-angle variables. Then apply an abstract infinite-d KAM theorem.

The Hamiltonian vector field is BOUNDED and “Quasi-Töplitz”.

Procesi-Xu 2011 (introduced Quasi-Töplitz)
Eliasson-Kuksin (similar notion Töplitz-Lipschitz)

Quasi-Töplitz functions

see slides….there is an algebraic closure property of this class under the normal form manipulations.

DNLW

Not Hamiltonian but “reversible” PDE. This is a relaxed setting but which rules out certain nonlinearities like $y_t^3$.

Real coefficients condition which excludes $y_x^3$.

Moser, Arnold, Sevriuk. Algebra of classical reversible KAM theory works out on this PDE as well. The asymptotic expansion of the normal frequencies controlled similarly as in the Hamiltonian case, in analogy with the quasi-Töplitz framework.

Ralph Saxton: The generalized inviscid Proudman Johnson equation

(joint work with Aleajandro Sarria)

This is the Proudman-Johnson (PJ) equation:

$$ (\partial_t + u \partial_x) \partial_x u = \lambda u_x^2 - (\lambda+1) \int_0^1 u_x^2 .$$

This equation comes from the n-dimensional Euler equations. The solutions we consider coming from Euler are unbounded as we go toward spatial infinity so these are infinite energy. He describes some further modeling assumptions culminating into a collapse of Euler into the Proudman-Johnson equation.

History:

Childress, Lerley, Spiegel, Young 1989
Saxton-Tiglay 2008, Okamoto 2009
Okamoto-Zhu 2000
wunsch 2009
A. Constantin 2000

Diverse phenomena as $\lambda$ varies.

Dongho Chae: On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations

Contents

On the blowup problem for Euler
Liouville type equations for fluids

Euler Blowup Problem

Euler 1757

Euler equation on $R^N$.

Kato, Temam, Bouguignon-Brezis. Local existence in $H^m (R^3)$ with $m>5/2$. Do singularities form?

Beale-Kato-Majda 84 Critereon: If there is blowup at time $ {T^*}$ then$$ \int_0^{T^*} \| \omega (s) \|_{L^\infty} ds = \infty. $$
Constantin-Fefferman-Majda 1996 critereon.
Refinements using Triebel-Lizorkhin type spaces. Interpolations.

On the self-similar blowup scenarios:

Self-similar blowup is a popular scenario in search for finte time singularity in nonlinear PDE.
E has a scaling property: $$ v^{\lambda, \alpha} = \lambda^\alpha v(\lambda x, \lambda^{\alpha + 1}t), ~ p = \lambda^{2\alpha} (same).$$

We consider the possibility of self-similar blowups for E.

Energy conservation suggests choosing $\alpha = \frac{N}{2}

Substitute a self-similar ansatz into E to obtain a system called SSE, the self-similar Euler equation. In the Navier-Stokes case, this system is called the Leray system. Leray asked if there exist self-similar blowup solutons for the Navier-Stokes equations in 1930.

Negative answers to Leray’s questions.

$V \in L^3 (R^3)$. Necas-Ruzicka-Sverak 1996
$V \in L^p (R^3), p>3$. Tsai 1998.
Theproofs rely upon maximum principle based arguments, which are not available in the context of the Euler equation.

Theorem (Chae 2007):

Let $V$ be a solution of SSE satisfy

$V \in [C^1 (R^3)]^3$ vanishing near infinity.
There exists $p_1 >0$ such that $\Omega - \nabla \times V \in \bigcap_{0<p<p_1} L^p (R^3).$ Then $V=0.$

The proof of this theorem used the “back to label map” due to Constantin. Recently, I found a much simpler elementary proof.

Chae-Shvydkoy 2012

This is the Euler version of the Navier-Stokes $L^3$ result of Necas-Ruzicka-Sverak.

Nonexistence of asymptotically self-similar blowup Giga-Kohn 1985. See Chae 2007.

Liouville Type Results for Navier-Stokes

Compare with Galdi. Slides are very detailed, provides a survey of the field.

Princeton Math Colloquium: Soliton Resolution for the Radial Energy Critical Focusing Nonlinear Wave Equation

Wed, 14 Mar 2012 18:45:45 +0000

(Apologies for typos or misquotations…; please help me improve the post. –J. Colliander)

Princeton Math Colloquium 2012-03-14

Presenter: Carlos Kenig

University of Chicago

Title: A case study for critical non-linear dispersive equations: the energy critical wave equation

Abstract: We will discuss recent work on the energy critical wave equation. The issues studied are global existence, scattering, finite time blow-up, universal profiles at blow-up and soliton resolution. This is viewed not as an isolated series of results, but as a way of approaching many similar critical non-linear dispersive equations.

We hope this is a model case for how nonlinear dispersive wave equations should proceed. This is a model case. Techniques of harmonic analysis have been introduced into the subject. The focus of those studies concentrated on local well-posedness. In the last fifteen years, there has been a new concentration on global-in-time, blowup aspects.

Linear wave equation

We start with a review of the linear wave equation.

$$ (LW) \partial^2_t w - \Delta w = h, data (w_0, w_1).$$

We write the soluton as $$ w(t) = S(t)(w_0, w_1) (t) + D(h)(t). $$

Finite speed of propagation. Picture of a cone.

An important estimate in these studies is the Strichartz estimate.

Critical focusing NLW

$$ \partial^2_t u - \Delta u = u^5, data (w_0, w_1). $$

Defocusing has $ - u^5 $. Reviews scaling and explains criticality.

Small data theory for NLW

Small (in $\dot{H}^1 \times L^2$) initial data evolves uniquely into a global-in-time solution which asymptotoically approaches (in same topology) into a linear solution as $ t \rightarrow \pm \infty$. Moreover, for any data, we have short time existence and a maximal time interval of existence.

This problem has an energy which is constant throughout the solution lifetime. Focusing vs. defocusing. There is competition, in the focusing case, between these two terms.

In the defocusing case, work of Struwe, Grillakis, Shatah-Struwe, Bahouri-Shatah proves that for any finite energy data, the solution exists globally and scatters. Another factor, Grillakis showed persistence of higher regularity.

In the focusing case, this fails. Levine 1974 showed that if $E(u_0, u_1) \leq 0$ then the solution can not exist globally in time (in either direction). (This is done by obstruction.) Recently, Kreiger-Schlag-Tataru 2009 constructed solutions for which $T_+ < \infty$ but for which the energy norm remains bounded. Also, in the focusing case, the elliptic equation admits a nonnegative solution $W$. This is a solution which is independent of time. This solution satisfies $\Delta u + u^5 = 0$. This solution is called the ground state. The reason for this terminology is that it arises as the optimizer in a Sobolev inequality and is charaterized variationally. This elliptic equation has been studied in connection with the Yanage problem in differential geometry. The formula for $W$ is explicit $$ W(x) = \frac{1}{(1 + |x|^2/3)^{1/2}}.$$ $W$ is the unique nonnegative solution of the elliptic equation (Gidas-Ni-Nirenberg 1979) and the only $\dot{H}^1$ solution (Pohozaev 1965). It does not scatter to a linear solution. It is “non-dispersive.” Donninger-Krieger 2012 have constructed global-ini-time solutions which are bounded in the energy class, are radial, and don’t scatter to either a linear solution or to $W$. There are other objects.

Recall a collection of recent results

Theorem (Kenig-Merle 2008): If $E(u) < E(w)$ then: 1. If the gradient is smaller, we have global existence and scattering. 2. If the gradient is bigger, we have breakdown in (both directions of) finite time. The case of equality is impossible under the hypothesis.

The proof of this result is an application of a general method. We call this a concentrated compactness rigidity method. Provides a brief summary of these ideas: small data, variational aspects, critical element extraction, rigidity.

Theorem (Duyckaerts-Merle 2008): There exists $W_-, ~ W_+$ radial which have the same energy as $W$. Then, there are three statements which provide a characterization of the dynamics.

(original statement of DM had an extra hypothesis which is now removed with inputs from Duyckaerts-Kenig-Merle 11, Krieger-Nakanishi-Schlag 11)

Now, we need to go beyond the energy threshold of $W$.

Existence of Type II blowup solutions

These solutions exist for a finite time but their critical norm remains bounded.

Krieger-Schlag-Tataru 2009

Describes a singularity formation process along a rescaled $W$ profile plus a continuous remnant.

We next show this is a “universal” phenomena.

DKM 2009, 2010:

(Radial) Any Type II blowup (critical norm stays bounded) solution with just a little bit bigger size than $W$. Then the soution looks like a rescaled $W$ plus errors. All type II blowups look like rescaled $W$.

Corresponding nonradial result: Slightly larger (by $\eta_0$) in the $\sup$ sense as above. After a rotation and a translation, we find that the solution looks like a rescaled and translated $W_l$ plus a small error. The rescaling rate is understood. And we have $$\lim_{t \uparrow 1} \frac{x(t)}{1-t} = l e^1 .$$

So, the possible profiles in the nonradial case are $W$ or its Lorentz transformations.

There also exist explicit solutions which blow up in finite time. These are built with ODE techniques and are independent of $x$. You can then chop them up with finite speed of propagation. These solutions have exploding critical norm. These are called Type I solutions. Conceivably, there are solutions which are Type I along one sequence of times and Type II along a different sequence of times.

DKM 2011: $W_+$ turns out to be Type I.

Soliton resolution for radial solutions of NLW

For a long time, there has been a widespread belief in the math physics community that large, global in tie, solutions of dispersive equations, asymptotically in time, they decouple into a sum of finitely many modulated solitons, a free radiation term and a term that goes to zero at infinity. This is a kind of philosophy that guides the research. So far, this has only been proved for the integrable KdV equations. Eckhaus-Schurr carried this out using the completely integrable structure. Corresponding results for mKdV can be obtained via the Miura transform. Heuristic arguements for this conjecture, in the case of cubic NLS in 1D, were given by Ablowitz-Segur 1976 and Zakharov-Shabat 1971. These are all subcrticial equations, for which one expects that these decompositions are stable, unlike in critical equations.

For more general equatiosn, so far, results have been found for data close to the soliton, in subcritical nonlinearities due to several authors. Buslaev-Perelman 1992 NLS, Soffer-Weisntein 1990, Marterl-Merle 2001 for gKdV) For corresponding results with blowup there are works by Merle-Raphael, Martel-Merle. There have also been large solution results for critical equivariant wave maps onto the sphere due to Christodoulou-Tahvildar-Zadeh, Shatah-T-Z, Struwe. They sow convergence along some sequence of times converging to the blwup time locally in space to a soliton (harmonic map).

In the finite time blowup case, for the 1d nonlinear wave equation, Merle-Zaag have obtained reusults of this kind using a Lyapunov functional tool.

In critical elliptic problems, such as the ones mentioned earlier, in domains excluding a small ball, considering radial solutions, there have been obtained results on decompositions inot “towering bubbles” (the analog of a finite sum of modulated solitons), as the size of the ball goes to zero. Musso-Pistoia 2006

The first general results for radial solutions of NLW were for type II solutions and held only for a sequence of times. We now have the full soliton resolution for radial solutions of NLW.

Theorem: Let $u$ be radial solution of NLW. Then, one of the following holds true:

Type I blowup.
Type II blowup with a full decomposition as a finite sum rescaled $W$, with ordered scaling speeds.
Radiation plus a sum of modulated solitons.

Consequence: Any solution which exists globally in time is bounded.

Remark 1: When the existence time is finite, the limit of the norm exists. It is either divergent or it is bounded. There are no mixed asymptotics. We do not yet have solutions which require more than one bump but we expect that they exist. For 1d NLW similar constructions were made by Cote-Zaag 2011. See also Musso-Pistoia 2006.

Remark 2: Quantifications. Each bump absorbs some energy.

Discussion….no claim of stability. This is not generic.

Ideas of the proof (global case)

The fundamental new ingredient of the proof is the following dispersive property that all radial solutions to NLW (other than 0 and $\pm W$ ) must have: $ \exists ~ R >0,, ~ \eta > 0$ such that for all $ t \geq 0$ or all $ t \leq 0$ $$ \int_{|x| > R + |t|} |\nabla_{x,t} u (x,t)|^2 dx \geq \eta. $$

IAS Workshop on Symplectic Dynamics 2: Wednesday

Wed, 14 Mar 2012 18:44:32 +0000

IAS School of Mathematics

Workshop web page

Happy Einstein Birthday!

Wednesday: 2012-03-13

9:00 - 10:00 Wilfrid Gangbo, Georgia Institute of Technology, “Lifting absolutely continuous curves from P(Td) to P2(Rd)” abstract
10:15 - 11:15 Jonatan Lenells, Baylor University, “Geometry of diffeomorphism groupos, complete integrability and optimal transport” abstract
11:30 - 12:30 David Ebin, SUNY, “Groups of diffeomorphisms and geodesics on them” abstract
2:30 - 3:30 Susan Friedlander, University of Southern California, “Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even”. abstract

Wilfrid Gangbo: Lifting absolutely continuous curves from $P(T^d)$ to $P_2(R^d)$

(chalk talk; joint work with A. Tudorascu)

This work extends earlier work on the space of probability measures on the torus $P(T)$ to analogous results on $P(T^d)$. The earlier work used the embedding $P(T) \rightarrow L^2 (0,1)$ but we don’t have this embedding in the higher dimensional case.

Let $P_2 (R^d)$ be the set of Borel measures on $R^d$ with finite second moment $\int |x|^2 \mu (dx) < \infty$. We say $\mu_0 \thicksim \mu_1$ if and only if $\int F d\mu_0 = \int F d \mu_1 ~ \forall F \in C(T^d), ~ \forall F \in C(R^d), ~ \forall F(x+z) = F(x), z \in Z^d$.

I define $P(T) = P(T^d)/\thicksim$.

Let $\gamma$ be a measure on $R^d \times R^d$ which satisfy $\pi_1$ # $ \gamma = \mu_0$ and $\pi_2$ # $ \gamma = \mu_1$. More generally, we write $$W^2_2 (\mu_0, \mu_1) = \inf_\gamma \int_{R^d \times R^d} |x-y|^2 \gamma (dx, dy).$$

Problem: Data: $v:(0,T) \times T^d \rightarrow R^d$ and $t \rightarrow \sigma_t \in P(T^d)$. Assume that $\partial_t \sigma_t + \nabla \cdot (\sigma v) = 0$ (in the sense of distributions). Can we find $t \rightarrow \hat{\sigma_t} \in P_2 (R^d)$ and $\hat{v}: (0,T) \times T^d \rightarrow R^d$ such that $\partial_t \hat{\sigma} + \nabla \cdot (\hat{\sigma} \hat{v}) = 0$ in the sense of distributions. Here $\hat{\sigma_t} \thicksim \sigma_t.$

Such a lift becomes important if I want to associate the rotation number. I want to write $$ \frac{d}{dt} \int_{T^d} x d \sigma_t = \int_{T^d} v_t d \sigma_t$. $$

Weak KAM: (A small fraction of what is known) $M= T^d$. Let $h: T^* M \rightarrow R$. Let $w(z_0, z_1) = z_0 (J z_1)$ where $J$ is the usual matrix satisfying $J^2 = - Id$. Let $X_h$ denote the associated Hamiltonian vector field: $$ \dot{\phi} = X_h (\phi), \phi_0 = (x_0, p_0). $$ The associated flow is denoted $\phi_t = (x_t, p_t).$

Existence of weak Lagrangian Tori: $\overline{h}: R^d \rightarrow R$ is the effective Hamiltonian, $ c \in R^d$.

$\exists ~ u \in C(t^d)$ with $h(x, c+ \nabla u) = \overline{h} (c)$ (viscosity)
$u_c (x) = c \cdot x + u(t).$ ($\partial u_c$ is invariant under $\phi$.)
$\forall ~ x_0 \in T^d ~ \exists v_0 $ such that if $(x_t, p_t) = \phi_t$ then $\forall ~ T$ $$ u(x_T) - u(x_0) = \int_0^T [l(x,\dot{x}) + c \cdot \dot{x} + \overline{h} (c)] dt$$ $$ \lim_{t \rightarrow \infty} \frac{\hat{x_t}}{t} = - \nabla \overline{h} (c). $$

General Fact: $M \rightarrow $ compact.

Specific to finite $d$:

Given $x \in W^{1,2} (0,T, T^d)$ and take two lifts $\hat{x}, \hat{y} \in W^{1,2, R^d}$. We then find that $$ \hat{x_t} - \hat{y_t} = n \in Z^d $$ because we have $$ \lim_{t \rightarrow \infty} \frac{\hat{x_t}}{t} = \lim_{t \rightarrow \infty} \frac{\hat{y_t}}{t} $$

Obstacle in infinite dimensions:

Let $M_0 = P(T^d)$ and in the sense of distributions we have $$\partial_t \sigma + \nabla \cdot (\sigma v) = 0.$$

If $\nabla \cdot (\sigma w) = 0$ then $v+w$ is another velocity.

(wash board…)

Let $\mu \in P_2 (R^d)$ and define the tangent space $T_\mu P_2 (R^d)$ and also the space $T_\mu P(T^d)$. These are defined with $L^2$ closures.

Pseudo symplectic form: ….going faster and I’m not keeping up with the typing.

Theorem (Gangbo-Kun-Pacuni 2011):

$\Omega$ is a closed skew symmetric nondegenerate 2-form.
$\exists ~ X_H$ such that $ -dH = \Omega (X_H, \cdot)$.
$\dot{f} = X_H (f) \iff \partial_t f + \nabla_x (vf) = \nabla_v (f (\nabla V + \nabla W * \rho)).$

This is a nonlinear Vlasov equation.

I want to state the analog of the weak KAM theorem in our context.

Theorem Let $\overline{H}$ be the effective Hamiltonian of $H$ restricted to $R^d$ and let $c \in R^d$.

$\exists ~ U: P(T^d) \rightarrow R$ such that (in viscosity sense) $$ H(\mu, c + \nabla_w H) = \overline{H} (c). $$
Given $\sigma_0 \in P(T^d) ~ \exists ~ v_0: T^d \rightarrow R^d$ such that if $f_0 = \sigma_0 \delta_{{v0}}$ then $$ f_t = \sigma_t \delta_{{vt}}, $$ $$ U(\sigma_t) - U(\sigma_0) = \int_0^T [ L(\sigma_t, v_t) + \int_{R^d} v_t \cdot c d\sigma_t + \overline{H} (c)] dt. $$
We also have $$ | \frac{1}{T} \int_0^T dt \int_{R^d} v_t d\sigma_t +\nabla \overline{H} (c) | \leq \frac{const}{\sqrt{T}}. $$

Corollary: If $(\hat{\sigma}, \hat{v_t})$ is an appropriate lift then $$\lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T ( \int_{R^d} x d\sigma_t) dt = - \nabla \overline{H} (c).$$

Questions:

Mather: This has connections with fluids?

Answer: Kinetic theory. Consider the system $ \partial_t^2 x = \frac{1}{N} \sum_{j=1}^N W(x_i - x_j) - \nabla V(x_i)$. When we consider the $N \rightarrow \infty$ limit, we can move the weak KAM theory from this $N$ particle system to the infinite particle case by moving to the setting of measures. This framework lets us prove convergence of discrete models to the PDE case.

Jonatan Lenells: Geometry of diffeomorphism groups, complete integrability and optimal transport

(pdf slides; Happy $\Pi$ day!; Einstein’s birthday)

(joint work with B. Khesin, G. Misiolek, S. Preston)

Outline

A new equation
Geometry of $Diff(M)$
A sphere
Optimal Transport
Geometric Statistics

A new equation

$$ \rho_t + u \cdot \nabla \rho + \frac{1}{2} \rho^2 = \frac{- \int_M \rho^2 d \mu}{2 \mu(M)}. $$

$M$ is a compact Riemannian manifold.
$\mu(M)$ is the volume of $M$.
This is an exciting equation because it is completely integrable for any $M$.
This is a geodesic equation on $Diff(M)/Diff_\mu (M)$.
Describes $\dot{H}^1$-optimal transport
reduces to the Hunter-Saxton equation for $M=S^1$. (derived in the context of liquid crystals in the early 90s.)

Euler-Arnold Equations. Summary of those ideas.

abc-metric. You can add some other terms to the original $L^2$ inner product involving $L^2$ inner products involvling codifferentials and the musical isomorphism. A lot of different equations arise as you take different values of the parameters. Writing down the associated abc Euler-Arnold equation generates a big equation which can be specialized into various equations. To obtain the $\dot{H}^1$ metric, we cancel away the terms associated with factors a and c. We simplify by setting $a=0, b= \frac{1}{4}, c = 0$.

The equation induced by these choices has some degeneracy issues. These can be resolved by quotienting out part of the phase space. The function $u$ is not uniquely determined but its coset is uniquely determined. (Similar issues arise in Hunter-Saxton.) The equation we are considering here is a geodesic equation on a (quotiented) Diffeomorphism group.

Jacobian determinant.

A sphere

Theorem (Khesin-Misiolek-Lennels-Preston): The map which takes the coset $[\eta]$ to its associated Jacoobian $\sqrt{Jac_\mu \eta}$ is an isometry onto a subset of the sphere.

This isometry lets them transport all the questions about the geodesic equation on this complicated Diff phase space into corresponding questions about geodesics on the sphere. Since we understand the sphere well, we can conjugate results there using the mapping to obtain explicit solution formulae for the geodesic equation. Magical integrability!

Preceding works.

Khesin-Misiolek 2003: Showed Hunter-Saxton may be viewed within the Euler-Arnold framework.

Lennels 2006: Recognized the image of the map as a portion of the sphere.

Optimal Transport

Optimal Transport. Wasserstein distance between two probability measures.

Moser 1965, Ebin-Marsden 1970, Otto 2001, also Benamou-Brenier.

The $\dot{H}^1$ optimal distance induces what they call the spherical Hellinger distance since it resembles the Hellinger distance used in probability theory

Geometric Statistics

Statistical model.

Fisher-Rao information metric.

Theorem (KMLP): The $\dot{H}^1$ metric coincides with the Fisher-Rao metric when restricted to any k-dimensional submanifold of the (quotiented) $Diff$.

This is another reason why we think this metric is important. It arises from many different points of view.

Summary

We found a new integrable PDE.
The PDE is a geodesic equation on a quotieted diff with $\dot{H}^1$ metric.
A sphere
One can understand what is going on using the optimal transportation point of view using the $\dot{H}^1$ metric.
This metric coincides with a basic metric arising in geometric statistics.

Open problems

Global weak solutions of the PDE? All solutions break in finite time because you hi the boundary of the diffeomorphism coset. However, there is no problem when you view the dynamics on the sphere. The motion along the great circle may be continued. This type of development has taken place already in the context of the Hunter-Saxton equation. The process appears to be more complicated in this more general context since the Jacobian can vanish.
Transfer results from geometric statistics into this diffeomorphism quotient. Then reinterpret these objects in the setting of PDE. Amari-Nagaoka 2000 alpha-connections, dual connections.
Develop an optimal transport theory based on the $\dot{H}^1$ theory.
Find a Lax pair.
Find a bi-Hamiltonian structure.
Analyze the associated two-component equation (c.f. Lennels-Zhao 2011). There is a 2-component Hunter-Saxton so that object suggests we might find a corresponding generalization. This has been observed by LZ.

David Ebin: Groups of diffeomorphisms and geodesics on them

(joint work with Stephen Preston)

Maps from a manifold to itself. Discussion of various topologies of such maps.

Volume preserving maps. Diffeomorphisms (Volumorphisms)
Even dimensional manifolds with a symplectic form. We can consider the maps which preserve the symplectic form. (Symplectomorphisms)
For odd dimensional manifolds, we can consider maps which preserve the contact form. (Contactomorphisms)

In all these cases, we can discuss the geodesics…..ack low battery.

Boothby-Wang fibration.

Susan Friedlander: Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even

(joint work with Vlad Vicol, Walter Rusin, Francisco Gancedo, Weiran Sun)

Homage to Oscar Wilde…

Amain theme is that there is a difference in behavior of solutions in Active Scalar Equations when the associated Fourier multiplier is even versus odd.

Active Scalar Equations; Incompressible Fluids

$$\theta_t + u \cdot \nabla \theta =0. $$ $$ \nabla \cdot u = 0. $$

$$ u = O [\theta], ~ PDO $$

$R^d$ or $T^d$. Even or odd Fourier multiplier symbol. The results I’ll describe are not influenced by the presence of a physical boundary. The emphasis will be on examining the influence of the operator O on the properties of the PDE.

Consider $u_j = \partial_i T_{ij} \theta, ~ \nabla \cdot u = $. Here $T_{ij}$ is a $d \times d$ Calderon-Zygmund operators.

ODD Symbol: Locally well-posed in Sobolev spaces. Commutator in energy estimates. **Chae et. al, Friedlander-Vicol.
EVEN Symbol: Lipschitz ill-posed in Sobolev spaces. Friedlander-Vicol; Nonuniqueness for $L^\infty$-weak solutions. Techniques from convext integration. Shuydkoy.

Recent reviews of results for certain active scalar equations. “Regularity and blowup for active scalars” Kiselev 2010.

SQG Equation: $u = R^\perp \theta$, symbol $\frac{i (k_2, -k_1)}{|k|}$.

Constantin-Majda-Tabak 1994, Resnick 1995 (Chicago thesis; unpublished), Wu, Cordoba, Chae, Iyer, Ju, Fefferman

The SQG equation had been known in the geophysics community before its introduction to the mathematical community by Constantin et.al.

Local existence for smooth initial data BUT global existence of smooth solutions is OPEN (just as it is open for 3D Euler). Cordoba and Fefferman have ruled out the existence of certain solution scenarios.

“Modified” SQG Equation (Okhitani)

Insert a power of $(-\Delta)^{1/2} = \Lambda$ in the map $\theta \rightarrow u$ so that $$ u = \nabla^\perp \Lambda^{\beta-2} \theta $$ where $1 < \beta \leq 2. Chae, Constantin, Cordoba, Ganceda, Wu 2011. Local existence of smooth solutions in $H^s$, global existence of weak solutions.

Note: result holds more generally when the symbol is ODD and order $\leq 1$.

IPM equation: singular integral operator with EVEN symbol

Darcy’s law.

$$ u = R^\perp R_1 \theta. $$

Cordoba-Gancedo-Orive 2007

Regular initial data, local existence, weak solutions, SQG and IPM present different behaviours. Global existence of smooth solutions is OPEN.

Even symbol: $$( \frac{k_1 k_2}{|k|^2}, \frac{-k_1^2}{|k|^2}).$$

There is very different behaviors among these equations for rough data. “Patch type initial data”

SIPM equations (even, unbounded)

$$ u = R^\perp R_1 \Lambda^\beta \theta$$

Friedlander-Gancedo-Sun-Vicol (2012)

Locally Lipschitz ill-posed in $H^s, ~ s>2$. Proved for $0 < \beta \leq 2$ in $T^d \times [0,\infty]$
Locally well-posed for some “patch-type” weak solutions. Proved for $0<\beta <1 $ in $R^2$.

(Discussion: the notions of “wellposedness” changes between the previous two bullet points.)

Symbol: $k_1 k^\perp |k|^{\beta -2}$

Magnetogeostrophic (MG) equations

Friedlander-Vicol 2011

Long symbol $M$, even, unbounded, 3D. $ u = M\theta$. Here $M$ is a vector operator that defines a 3-vector $u$.

Cauchy problem is ill-posed in Hadamard sense in Sobolev spaces. There is no Lipschitz solution map.

Ill-posedness: singular, even, symbol

Active scalar equation. Special direction with index $d$, often associated with gravity. A list of many conditions on the $d$th component $S_d$ of the Fourier multiplier operator….allowing them to build eigenfunctions to show Lipschitz failure.

Definition: Locally Lipschitz $(X,Y)$ well-posed.

$$| \theta_1 (\cdot, t) - \theta_1 (\cdot, t)|X \leq K | \theta1 (\cdot, 0) -\theta_2 (\cdot, 0) |_Y. $$

The spaces $X,Y$ are often chosen to be $(H^r, H^S)$.

Theorem: Under the many assumptions on $S_d$, the active scalar equation is Lipschitz $(H^r, H^s)$-illposed for any $r > R, s \geq r+1$.

Linear problem

Linearize around $\theta_0 = \sin m x_d $. Write out a Fourier series. Crank out a recurrence relation.

Continued fractions, characteristic equation. These ideas where used by Michalkin and Sinai to show unstable eigenvalues for the shear flow for Navier-Stokes equations.

Ill-posedness of the nonlinear problem

Follows a proof by contradiction.

Effects of dissipation: MG

Dissipation: $ \nu (-Delta)^{1/2}$

Using De Giorgi techniques, Caffarelli-Vasseur proved critical SQG. What can we say about the MG equation?

Case $1/2 M \gamma < 1$: LWP in $H^s$, for $ s> \frac{5}{2} + (1 - 2\gamma)$. Well-prepared initial data.
Case $ 0 < \gamma < 1/2$: Diffusion is too weak to overcome the continued fraction construction.
Case $\gamma = 1/2$: Unique global solution when the initial data and source are small in a suitable sense, then there exists a unique golbal solution. However, if the data are large in this respect then we can run the ill-posedness construction. This reveals a very precise dichotomy.

IAS Workshop on Symplectic Dynamics 2: Tuesday

Tue, 13 Mar 2012 18:42:10 +0000

IAS School of Mathematics

Workshop web page

Tuesday: 2012-03-13

9:00 - 10:00 Laszlo Szekelyhidi, University of Leipzig, “The h-principle for the Euler equations” abstract
10:15 - 11:15 Camilo de Lellis, University of Zurich, “The h-principle for the Euler equations Part 2” abstract
11:30 - 12:30 Vladimir Sverak, University of Minnesota, “On the long-time dynamics of some infinite-dimensional Hamiltonian systems” abstract
2:30 - 3:30 Antoine Choffrut, University of Leipzig, “On the local structure of the set of stationary flows to the 2D incompressible Euler equations” abstract
4:30 - 5:30 Thomas Kappeler, University of Zurich, “Symplectic techniques for integrable PDEs” abstract

Laszlo Szekelyhidi: The h-principle for the Euler equations

(chalk talk; joint work w. Camillo De Lellis)

“What I will speak about has nothing to do with symplectic and nothing to do with dynamics.” Hofer: very good.

Euler

$$\partial_t v + \nabla \cdot (v \otimes v) + \nabla p = 0; \nabla \cdot v = 0.$$

The spatial dimension $n=2,3$, certainly $>1$. We will speak about weak solutions $v \in L^2_{loc} (T^n \times [0,T])$ and we throw all derivatives onto test functions.

Why look at weak solutions? The equations tell you conservation of mass and momentum. The derivation is done from a continuum analysis so this formulation is natural. For $n=3$, another reason is the relationship with turbulence (K41, O49). The story starts with anomalous dissiipation. The observation is that if you consider $NS_\nu$ with small $\nu$ then formally, the dissipation rate $$ \nu \int |\nabla v|^2 dx > \epsilon. $$ This is observed in experiments as $\nu \rightarrow 0$. If you plot $\log k$ vs. $\log E(k)$ there are three different regimes: a low frequency regime related to the geometry of the domain, an inertial range with slope $-5/3$ and then a rapid dissipation at high frequencies. Bob Kohn once said that a log-log plot always looks linear. ….discussion with Peter and Camillo….”Bob Kohn is not here so let’s leave him alone.” K41 looked at ensemble averages and he derived the $-5/3$ scaling law. O49 was thinking of a single solution. He said that if we beleive in this kind of log-log picture then in the intermediate regime we are far from dissipation so the nonlinear term is responsible for the $-5/3$ decay. Is it possible to see a single solution that displays this type of decay. If you translate this point of view into a regularity statement and look for $ v \in L^\infty_t C^\alpha_x$ then when $\alpha > 1/3$ we have energy conservation and for $\alpha < 1/3$ then you have anomalous dissipation possible. Klainerman: Is this $1/3$ easy to see? Discussion: Yes, it is just scaling, look at Fourier coefficients….

Eyink, Constantin-E-Titi solved the $\alpha > 1/3$ part of Onsager’s conjecture. There is basically nothing known in the $\alpha < 1/3$.

Spencer: Uniqueness in that range? Answer: No you need Lipschitz to see uniqueness so there remains a big gap.

Studying weak solutions puts us ina different framework than the study of smooth solutions, long time behavior for 2D,etc. This is a different world.

Theorem (Scheffer-Shnirelman): There exists a nontrivial weak solution with compact support in time.

This solution can be thought of as having initial data zero, then it is not zero and after a while, it is zero again. (This solution is far from regularity $1/3$.)

h-principle (Gromov)

This theorem can be viewed as a statement of the form of the h-principle. This principle should be viewed as a different tpe of statement related to Hadamard ill-posedness.

Theorem (Nash-Kuiper): Any strictly short smooth embedding of (compact) $M^n \rightarrow R^{n+1}$ can be uniformly approximated by $c^1$ isometric embeddings.

For a geometer, this is viewed as a completely wrong theorem. It seemingly contradicts the classical rigidity of the 2-sphere. Any isometric embedding of the 2-sphere into $R^3$ is the standard embedding. However, that theorem requires curvature so needs $C^2$.

Berti: What is short? Answer: Distances in the image are shorter than distances in the domain. So, Lipschitz with constant less than 1.

Two conditions: A topological global condition, an embedding. A local condition, isometric.

global: embedding
local: isometric

General statement. If you can satisfy the global constraint, you can twist it satisfy the local statement. Another example is Gromov’s theorem saying 2 forms can be converted into symplectic forms.

An idea of the proof of this statement (Nash): $M^n \rightarrow R^{n+2}$. This is basically about a single chart so let’s look instead at $\Omega \subset R^n$ and we consider the embedding $\Omega \rightarrow R^{n+2}$. We have $$ \nabla u^T \nabla u = g $$ and strictly short means $g - \nabla u^T \nabla u >0$. We can make wrinkles. Wrinkling is written as a spiral $$\tilde{u} (x) = u(x) + \frac{a(x)}{\lambda} (\sin (\lambda x \cdot \xi) \zeta (x) + \cos (\lambda x \cdot \xi) \eta (x)),$$ where $\zeta, \eta$ are unit normal to $u(\omega)$. This looks like a “telephone cord”. What happens in orthogonal directions? How does this affect the metric? This is something you can calculate: $$ \nabla \tilde{u}^T \nabla \tilde{u} = \nabla u^T \nabla u + a^2 (x) \zeta \otimes \zeta + O(\frac{1}{\lambda}). $$ This means I have a lot of freedom to change the metric in a fixed given direction. This allows me to write $g - \nabla u^T \nabla u $ as a sum of terms $\sum a_j^2 (x) \xi^j \otimes \xi^j.$ It is important here that the $\xi$ does not depend upon $x$. This allows me to achieve a reduction in the difference

$$| g - \nabla u^T \nabla u |_0 = O(\frac{1}{\lambda}),$$

$$| u - \tilde{u} |_0 = O(\frac{1}{\lambda}), $$

$$| u - \tilde{u} |1 \thicksim | a |0 \thicksim | g - \nabla u^T \nabla u |_0^{1/2}.$$

Lipschitz isometries

$\Omega \rightarrow R^n$

Kirchlein Baire Category Method.

Consider the space $X = [ u \in Lip (\Omega): \nabla u^T \nabla u \leq Id]$ endowed with the supremum norm. The observation of Kirchlein is that $\nabla \cdot X \rightarrow L^1$ is Baire-1. Consider 1-Libschitz maps converging to the zero function. With this same argument, I can take any function and add corrugations.

Mather: What is Baire-1? Answer: It is a pointwise limit of continuous maps.

A corollary of Baire Category theorem. The points of continuity is dense. Despite the troubles with the corrugation possibility, most maps in this space are points of continuity. The only places where you can’t improve is where the gradient is already maximizing. As a consequence, most maps in this space are isometric.

This is an argument which can be generalized quite a bit and can be applied to the Euler equations.

Example. The original system (O). (Tartar-DiPerna ideas)

$$\sum_{i=1}^n A_i \partial_i z = 0, ~in D’$$ $$ z(x) \in K, ~a.e. ~x$$

and we want to move to same relaxed condition (R) but with $z(x) \in K^{\Lambda}$, which he refers to as the convex hull of $K$. The principle is that most (in the sense of Baire Category) solutions of the relaxed setting R are solutions of the original problem O.

Question: What is the set $K^\Lambda$? This is the “wave cone”. Let $z: R^n \rightarrow R^d$ here. $\Lambda = [ \hat{z}: \exists \xi \in S^{n-1} ~s.t.~ \sum_i A_i \xi_i \hat{z} = 0]$. So, these are the directions in which we can oscillate while maintaining the conservation law and keeping the constitutive relations intact. So, that is $\Lambda$ and then the $\Lambda$-convex hull is like this. $z \notin K^\Lambda$ if $\exists ~ f ~ \Lambda$-convex so that $f(z) > 0, f|_K \leq 0$. You can separate. This general point of view was developed by Tartar, DiPerna.

Klainerman: what is the “h”? Answer: In this setting “h” stands for homotopy but that is not so present in this discussion. My view of the weak version of the h-principle is that there are a lot of solutions which have less regularity. De Lellis: Gromov woud say that you can take your short map and homotopize it while maintaining that it is an isometry, except at the endpoint.

Application to Euler

Now, you can write the Euler equations in this form by renaming the nonlinearity as a new variable. He shows how to do this by renaming some variables, interprets the associated $K$ and $K^\Lambda$.

Theorem (DL-Sz): Let $\overline{e}$ be a given function on $T^n \times [0,T]$ and let $(\overline{v}, \overline{u}, \overline{q})$ be smooth strict subsolution. Then $\exists ~ v_k \in L^\infty$, a sequence of weak solutions of Euler such that $v_k \rightharpoonup \overline{v}$ in $L^\infty$ and $\frac{|v_k|^2}{2} = \overline{e}$ a.e. $(x,t)$.

This is the “local part” of the h-principle. Given one subsolution, I can construct a solution by adding these waves. More or less, what Scheffer-Shnirelman have done is to take 0 as the subsolution.

….I am very much running out of time….so just to state one more theorem which touches the initial data.

Admissibility. Those weak solutions for which the $L^2$ norm is nonincreasing. Under this type of assumption, you have the weak-strong uniqueness. Any such strong solution is unique within the larger class of weak solutions emerging from the same initial data.

Theorem: Let $n=2$. Let $v_0(x)$ be the shear flow (he indicates this graphically wiht an interface and an arrow to the right above and an arrow to left below). $\exists~$ infinitely many admissible weak solutions.

He draws the interface with a thickened interface of some growing-with-time size outside of which we have the share flow.

Among all the selection criteria you might be considering for restoring uniqueness among the weak solutions, you could ask for maximally dissipating, you could choose the shear flow itself. Or you could ask for the one which has the fastest interface thickening. It is not yet clear which is the physically relevant selection critereon.

Questions:

Sverak: If you take a sequence of smooth solutions onverging to your data, is there any relation to your solution? Answer: It depends how it converges. Sverak: The best you can with continuous. Answer: I’m not sure.

Camilo de Lellis: The h-principle for the Euler equations Part 2

(continuation of previous talk; chalk talk)

I’ll start by mentioning some related results in the literature.

Survey article: D-Sz posted on web in 2011 contains all this literature.

Wiedeman: Global existence of weak solutions for any $L^2$ initial data in $R^3$. This would have been a fantastic theorem if it had not been too many solutions. This is the global analog of what was outlined before. Kappeler: ARe they adminssible? Answer: No. We are not anywhere near the blowup problem.
Sz-Wiedemann: You can approximate any measure-valued solutions (a la DiPerna-Majda) with exact solutions.
Sz-Wiedemann: The set of “bad” initial data is $L^2$-dense.

These techniques can also be applied to other equations.

Incompressible porous medium equations. Some class of active scalar equations. Caddoba-Faoco-Grancedo, Shydkoy, Sz.
Compressible Euler. D-Sz, Chiararoli, Chiadaroli-D. (Higher dimensional conservation laws have a striking nonuniqueness, contrast with the entropy conditions in 1d)

The fact that there exist $C^1$ isometric embeddings uniformly approximating any short smooth embedding was surprising.

Theorem (D-Sz 2011): For any given $e: [0,1] \rightarrow R^+$ smooth. Then there exists a $C^0$ solution of incompressible Euler in $T^3 \times [0,1]$ such that $$ e(t) = \frac{1}{2} \int \frac{|v|^2}{2} (x,t) dx. $$

We are moving towards the lower part of the conjecture of Onsager. I can’t claim this is saying anything about turbulence, but it does speak to the issues of dynamics of solutions viewed on Fourier coefficients.

Remark: It seems we can reach some Holder regularity, something explicit like $\frac{1}{500}$.

Shnirelman: Can you say something about modulus of regularity about this solution? Answer: You can work out something. It would be painful to work out. Finding a Holder exponent is achieved through an iteration process.

The process uses smoothness properties of $e(\cdot)$.

….change gears.

Borisov ‘50: If $v \in C^{1,\frac{2}{3} + \epsilon}$ is an isometric embedding of a positively curved connected 2d surface in $R^3$ then the image is convex.

This gives you rigidity. This is a local theorem. The global theorem would say that the sphere has an isometric rigidity. Borisov also had an announcement…..never published his proof.

Borisov (1965—>2004): h-principle for 2d analytic surfaces in $R^3$ if $u \in C^{0, \frac{1}{13} - \epsilon}$.

Conti; D-Sz: Rigidity and h-principle in general dimension (with better exponenets). For 2d the exponent is $\frac{1}{7}$.

He describes a double iteration scheme. He emphasizes that the parameter $\xi_0$ appearing inside the $\sin$ and $\cos$ is independent of $x$.

There are always successive one dimensional layers in any convex integration, in any h-principle appliation. To improve these constructions and obtain the $C^0$ statement we need to replace these constructions by higher dimensional generalizations. This is a direction suggested also by Gromov in his book.

…slowing down on typing….I’m just going to watch this.

Vladimir Sverak: On the long-time dynamics of some infinite-dimensional Hamiltonian systems

2 main examples

$w_t + u \nabla w =0, ~ w = \curl u, \nabla \cdot u = 0, x \in T^2
$NLS_3 (T^2)$

Thanks to Shnirelman, Kuksin, Choffrut, Keel, Polacik (slide changed fast might be incomplete).

Speculative picutre in Fourier space.

There is an interesting possibilty that on the macroscopc scale the dynamics might be simpler than in finite dimensional cases. Why? If you look at the dynamics on the Fourier space, at time $t=0$, we impose some nice initial data and as tiem evolves, some part of the solution moves toward infinite frequency. Everything in this direction is very hard to establish. There are some obstructions to this behavior. For example, KAM and complete integrability block this phenomena. We consider here “generic solutions”. Even though the initial data might be very complicated, we might end up with complexity moving into the high Fourier modes and what remains will be dictated by the conservation laws. This should be contrasted with finite-d systems. If one truncates the PDE in a naive way, we will initially see the same type of picture (motion toward infinite frequencies) and then we will hit the frequency cutoff and there will eventually be a thermalization. The “complexity” has “nowhere to go”.

Let’s look at defocusing $NLS_3$. We have, for this equation, three basic conserved quantities:

Energy
Momentum
Mass

Variational principal. We can think of the equation as a variatonal principal: minimize E subject to the mass=m and momentum=p constraints. We can compare this situation to 2D Euler. (In 3D there is the possibility that everything goes to infinity.) The $L^2$ norm and momentum provide constraints and the minimization principle gives you some nontrivial solutions even in the linear case.

The most optimisitic scenario for the transfer to high frequencies is that for a generic solution over a long period of time, the solution will spend most of its time (in some weak topology) near this manifold of minimizers of $E$ subject to the constraints given by parameters $p,m$. The natural “phase space” for solutions is $H^1$ and we view the manifold $M(p,m) subset H^1$. Ergodicity.

The variational principle may be viewed from the statistical mechanics point of view. Consider a finite-d truncation via Dirichlet prjection. If we believe in stat mech in this scenario, we can use the microcanonical ensemble and look at the set of all points in our phase space where our energy lies between $E$ and $E + \delta$. We have a natural volume measure on this space. We can similarly $\delta$-thicken around the momentum and mass level sets. We can then hope that the solution will concentrate onto a measure living on these subspaces. This is provably rigorous in the linear case. In the limit as the truncation parameter goes to infinity, the energy $E$ is “forgotten” while the mass and momentum constraints are “remembered”. Another way to look at it is to follow the rule-of-thumb that the energy is equidistributed across possible states. Our initial energy is finite and we are equidistributing it across more and more states. Therefore, the temperature (energy per state) and the whole solution will weakly converge to zero. So, as $N \rightarrow \infty$, we concentrate on low frequencies consistent with the constraints and the rest of the solution goes to zero temperature.

Comparison with Gibbs measure. (Lebowitz, Bourgain,…) In this construction, one exponentiates the Hamiltonian and interprets this as a density with respect to the Wiener measure. For the Gibbs measure, $\langle E \rangle = + \infty$ and $\beta > 0$. These functions live on function spaces with infinite energy. The measure is concentrated on functions with low regularity and infinite energy. Remarkably, the dynamics is still well-defined by the PDE (Bourgain).

A rigorous result. NLS defines a dynamical system on the space $(X, w)$, where $X = [ \psi \in H^1, | \psi |_{H^1}\leq C]$. Here “w” denotes the weak topology. This is OK because Bourgain has shown well-posedness on $H^s, s<1$ (Bourgain).

Theorem: The $\Omega$-limit set (wrt the weak topology) contains solutions for which no movement to high frequencies goes to infinity. These are precompact in $H^1$.

These are the “end states” which solutions approach in the weak topology.

Question: Does every $\psi \in \Omega_+ (\psi_0)$ have this property?

Heuristics: If $\psi \in \Omega_+ (\psi_0)$ then the high frequency part has already separated.

Interpretation:

The solutions do not quite approach $M(p,m)$. there is probably some escape….ack slide changed.

2D incompressible Euler

the situation is quite similar. The difference between 2D Euler and NLS is that, in some sense, 2D Euler is a Poisson system instead of a Hamiltonian system. The system is a Hamiltonian system on symplectic leaves. Formally, 2D Euler should be viewed as a family of Hamiltonian systems and the orbit takes place on the leaf.

Fourier representation. Stream function. Energy. Conserved quantities associated with vorticity transport.

Natural “phase space”…. $L^\infty (T^2)$ with a weak * topology. Yudovich has shown the evolution is well-posed in $L^\infty (T^2)$. We have the necessary well-posedness pieces in place. Importantly for us, in this context of developing fine structures in teh flow, we have the stability result: If the data converges weak start then the same is true at later times. this follows from the proof of Yudovich.

Analogy with NLS. He draws a table.

Phase space: $H^1$ …. $L^\infty$
Weakly continuous: Momentum, mass …. $ E = \int -\frac{1}{2} \omega.$
Lower semicontinues: Energy….. ack slide changed.

Fourier picture. Variational principle, related to the notion of “mixing” introduced by A. Shnirelman). We minimize the energy subject to the constraints associated to the weakly continuous quantities. This produces a steady state solution (whih depends on $f_0$).

More geometric picture. We know that a good topology is the weak * topology on $L^\infty$. We can therefore study the weak * closure of the orbit.

Example: Data that looks like $\omega_0 = \chi_A - \chi_B.$ Here $B = T^2 \backslash A, ~ |A| = |B|$.

Onsager 1947, Montgomery-Johce, 1970s, Miller, Robert 1990s, Turkington 1990s, closely related to Shnirelman’s notion of “mixing”.

Looks similar to Ising model, except that the interaction is long-range. “Most-probable” configuration for a given energy $E$? But we specify here the number of configuration cells that have sign +1, and how many have -1. We can then do the usual statistical mechanics calculations.

He draws another analogy diagram.

NLS …. 2D Euler
$C^N$…..Ising config
Classical Maxwell Boltzmann microcannical ensemble picture……fermions (generalized) with long range interaction.

Full ergodicity seems….ack.

A more geometric picture for Euler (a sketch).

Geometric finite-d approximations inside the $SDiff (T^2)$ viewpoint. Remarkable fact learned from Khesin book showing that we have finite-d geometric approximations using $SU(N)$.

Determining the “end states” for Euler.

Calcuating the “entropy” etc., one gets different answers depending upon how one counts.

In Euler on the torus, the temperature is not zero but is instead negative as was observed by Onsager. All these predictions suggest that hte “end states” consist of shear flows.

An example where transfer to high frequencies was proved: Landau damping.

The only situation where this was rigorously proved wiath the Vlasov-Poisson system.

Landau 1946
Caglioti-Maffei
Hwang-Velazquez 2008
Mouhot-Villani 2009

Is there an analog of Landau dampoing possible for 2D Euler?

Linear Landau damping: Yes (Sverak)
Nonlinear case: probably yes, but seems more difficult than the Vlasov-Poisson case.

Is there an analog for this in the 2d NLS case? This seems more difficult in the NLS case. Transfer to high frequencies for dispersive equations:

2d NLS: “I-team”
Various model situations: Bourgain,…

Questions: 1. Can you formulate a stability statement related to your question about the “end states” in the Schrodinger setting? Answer: I expect there should be stability statements around the energy minimizers subject to mass and momentum constraints. 2. Do you expect corresponding statements for the focusing problem (when you don’t expect blowup)? Yes, but there will be a weak convergence to soliton instead of the “breather” energy minimizers.

In discussion after the talk, Sverak suggested that these minimizers are not localized onto single Fourier modes and instead appear to be some kind of “breather” solution. These objects should have variational stability properties resembling corresponding statements about solitons and built along the motif of Arnold Stability results as appearing in the book by Khesin.

T. Oh reported to me that some studies like those suggested in this talk appear in work of Chatterjee-Kirkpatrick.

Antoine Choffrut: On the local structure of the set of stationary flows to the 2D incompressible Euler equations

arXiv

(chalk talk)

The Euler flow evolves on symplectic leaves. If you start on one of these leaves, then the Euler flow stays on the leaf. The result I want to prevent is the following. Suppose you have a stationary solution on one leaf, then the other stationary solutions are located on a curve that passes through the leaves. There is a 1-1 correspondence between the stationary solutions and the leaves.

This talk will be a bit more elementary than the other talks. Some aspects were forecasted by Preston and Sverak in earlier talks in this workshop.

Consider a (potato shaped) domain $\Omega$. The Euler equation $ \partial_t u + (u \cdot \nabla) u _ \nabla p =0, \nabla \cdot u =0, u \cdot N =0 (\partial \Omega).$

In 2D, we introduce $u = (u^1, u^2)$ and the vorticity $w =\partial_x u^2 - \partial_y u^1$ and we can derive the vorticity equation $$ \partial_t w + u \cdot \nabla w = 0.$$

How do you recover $u$ from $w$? and vice versa? Introduce, in 2D, the stream function $u = \nabla^\perp \psi$.

$\psi|_{\partial \Omega} =$ locally constant. We have $\Delta \psi = w$.

Kelvin: A curve $c_0$ at time $t =0$ evolves along the flow to a curve $c_t$ at a later time. We obtain that $\int_{\Gamma_i} \frac{\partial \psi}{\partial N} = \gamma_i$. $\Delta \psi = w$ and some boundary conditions….

I can rewrite the vorticity equation as $\partial_t w + {\psi, w } = 0$ where the bracket is defined as the area of the parelellogram determined by $\nabla w$ and $\nabla \psi$.

Transport interpretation.

Coadjoint orbit.

Theorem (Choffrut-Sverak; GAFA 2012): Let $\overline{w}$ be a “non-degenerate” steady state. Then there exists a $C^\infty$-nhbhd $W$ of $\overline{w}$ such that any coadjoint orbit intersecting $W$ contains exactly one steady state in $W$.

The proof is by an inverse function theorem. I need to say how I am going to implement this function theorem. How do I describe my orbits? How do I describe my steady states? I want to show these are in one-to-one correspondence.

….lots of discussion…..smooth dependence….lots of chatter…..speaker needs to be able to describe more.

Characterization of steady states. The transport equation tells me that there is no dependence of $w$ on $t$ so that $w (\eta_t (x)) = w(x)$, $w = F(\psi)$ provided $\nabla \psi \neq 0$. I impose that $\nabla \psi \neq 0$ in $\Omega$ and I assume that $\Omega$ is an annular domain (with exactly one hole). Steady states are exactly parametrized by $F$.

Characterization of coadjoint orbits.

Transport $\implies A(0) = | [x \cdot w (x) < c ]| = |[x: w \circ \zeta (x) < c]|$

Correspondence.

Steady states ……… F
Orbits …………. A

Steady states can be characterized as conditional critical points of $$ E(w) = \frac{1}{2} \int_\Omega | \nabla \psi |^2 $$ (restricted to an orbit.)

Recall $\Delta \phi = w$. Calculating the first variation leads us to $\delta E = \int \nabla \psi \cdot \nabla \phi = - \int \psi { \alpha, w} = 0 $ (using a permutation property.) Since this is true for all $\alpha$, we find that ${\psi, w} = 0$. Similar ideas come up in the Arnold stability theorem.

Euler as a geodesic configuration space. Lagrangian least action principle. Marchioro-Pulvarenti, Chemin.

Clairaut, Noether, Lie group with (left) invariant metric.

That was the Lagrangian formulation on the tangent space. The real action takes place in the cotangent bundle where we have a Hamiltonian formalism.

Thomas Kappeler: Symplectic techniques for integrable PDEs

(pdf slides)

Aim: Survey of recent results on integraple PDEs obtained by symplectic techniques. The model equation is the $NLS_3^{\pm}(T)$.

Topics:

Phase portrait/ space of orbits; construction of normal coordinates
Asymptotic properties of solutions
KAM theorem

NLS as a Hamiltonian system.

Phase space: $L^2_C$. Pairs of functions. More generally, a function space.

There is a canonical Poisson bracket

$$ {F,G}(\phi) = -i \int_0^1 \partial_1 F \partial_2 G - \partial_2 F \partial_1 G) \phi dx. $$

Defocusing NLS. Focusing NLS.

Defocusing NLS as an integrable PDE on T. Grebert-Kappeler-Poschel “The defocusing NLS equation and its normal form” to appear in EMS.

Focusing NLS as integrable PDE: only few results.

Part 1. Review of NF for defocusing NLS.

Theorem (GKP): There exists a canonical map (closely related to the Fourier transform) which reveals that defocusing NLS may be viewed as a system of infintely many coupled oscillators.

Steps of proof:

Local part. We have to construct these coordinates $x_n (\phi)$ and $y_n (\phi)$. How to build these coordinates?

Global part. We have a global chart.

Zakharov-Shabat operator (ZS)

Lax pair for NLS
Periodic spectrum of $L(\phi)$ on [0,2].

Counting Lemma: He describes the spectrum with a picture on the board. Floquet theory.

characteristic function
two-sheeted spectral curve

von Neumann-Wigner 1929

The eigenvalues come in pairs. Asymptotically, they are like $n\pi + l_n^2$.

Construction of actions/angles

Choose cycles $a_n$
Cycles induce (i) actions and (ii) 1-forms.
1-forms induce angles.

Birkhoff coordinates

Euclidean versions of these action angle coordinates.

Important features of construction

same cycles $a_n$ were used to define $I_n$ and the 1-forms $\beta_n$.
Cycles/1-forms are defined on the spectral curve and not on phase space.

Part 2. Normal Form for fNLS

There do not exist global Birkhoff coordinates.
The associated Zakharov-Shabat operator $L$ is not necessarily self-adjoint.
Symmetries of $spec_p (L(\phi))$.
$spec_p (L(\phi))$ can be described.

We believe that one can characterize the (local) existence of Birkhoff coordinates near $\phi \iff$ spectral properties of the Zakharov-Shabat operator.

Standard Potentials

Describes how to carry out the construction. The focusing case requires local constructions.

The discussion here is very precise and I don’t think I can convey more than is available on the slides.

…

Part 4. KAM for defocusing NLS

Defocusing NLS is integrable on all of $L^2$.
Question: KAM on $L^2$, not only near equilibrium point 0?
Defocusing NLS frequencies can be expressed using the Birkhoff coordinates.

Proposition: Can check the Kolmogorov and Melnikov conditions and then conclude by analyticity.

Near resonances

$\omega_j - \omega_{-j} = O(1)$ for $ j \rightarrow \pm \infty$ jeopardizes measure estimate of standard KAM theorem for integrable PDE.
Ways to overcome difficulties:
Craig-Wayne-Bourgain method. Bourgain, IMRN 95
Kuksin-Poschel, Berti
Restrict the perturbations Geng-You CMP 06, JDE 05

Kappeler-Liang JDE 12

IAS Workshop on Symplectic Dynamics 2: Monday

Tue, 13 Mar 2012 18:39:20 +0000

I’m participating in a workshop at the Institute for Advanced Study on Symplectic Dynamics. This is part of the special concentration this year organized by Helmut Hofer. I’ll try to write notes on the talks I hear. Apologies to the speakers for typos and misquotations…. Comments (pending approval) are open for suggestions and edits.

IAS School of Mathematics

Workshop web page

10:15 - 11:15 Peter Constantin, Princeton University, “Long time, vanishing viscosity limits” abstract
11:30 - 12:30 Steve Preston, University of Colorado, “The inextensible string as a toy model of fluids” abstract
2:30 - 3:30 Emanuele Caglioti, University of Rome, “Long time behavior of solutions of Vlasov-like equations” abstract
4:30 - 5:30 Roberto Camassa, University of North Carolina, “Large amplitude internal waves and their stability” abstract

Peter Constantin: Long time, vanishing viscosity limits

(pdf slides)

Navier-Stokes

Navier-Stokes equations. $\nu$ multiplies the damping term. Studying it in $R^d$ or $T^d$, $d=2,3$. We are interested in $T \rightarrow \infty$ and $\nu \rightarrow 0$. (Reynolds number $\frac{UL}{\nu}$.)

Limits: selected stationary statistical solutions.

$$\lim_{Re \rightarrow \infty} \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T \phi( S^{NS} (t) ) dt.$$

Anomalous dissipation of energy:

$$ \lim_{\nu \rightarrow 0} \lim_{T \rightarrow \infty} \frac{\nu}{T} \int_0^T \int_{R^3} |\nabla u (x,t)|^2 dx dt = \epsilon > 0.$$

(K41)

2d Navier Stokes

No anomalous dissipation of energy. Enstrophy balance. Anomalous dissipation of enstrophy? Kraichnan 68: Yes. Bernard 00: add (extra…to be described…discussion….friction from the bottom) damping and then no.

Constantin-Ramos 07: Bernard was right. Heuristics.

We are talking about weak solutions. 2 reasons to talk about these. The singularities really describe the phenomena, e.g. shocks in hyperbolic conservation laws. Deduce from the few conservation laws that solutions exist and they are nice and smooth but we can’t so we content ourselves with what we can build.

Active Scalars

$$\partial_t \theta + u \cdot \nabla \theta = 0$$

Examples: 2d Euler, SQG.

$$ u = \Lambda^\gamma R^\perp \theta. $$

(Here $R$ is the Riesz transform, $\Lambda = \sqrt{-\Delta}$.)

Weak solutions for SQG. Resnick 95. Almost Lipschitz on $L^2$. Euler and NS does not have this weak continuity property. We don’t have uniqueness. If you have unique weak solutions, then you are golden. I’m going to talk about statistical solutions eventually.

Theorem (Chae-C-Cordoba-Gancedo-Wu): Let $\theta_0 \in L^2 (T^2)$. There exists a global $L^2$-weak solutions of the generalized SQG.

….too fast to type. Generalizes result of Resnick. Remarks…fast. Friedlander-Vicol. Moffat.

A commutator estimate

Summarizes the ideas of the proof. Stream function representation. Cancellations….end up with a commutator….end up with a weak vs. strong.

Damped Driven Navier-Stokes Equation

Standard Navier-Stokes with forcing and an added term $+ \gamma u$, which represents the damping.

You get some a priori bounds which are independent of the viscosity by following the usual arguments.

Stationary Solutions (a warm up)

$$ \gamma \omega + u \cdot \nabla \omega - \nu \Delta \omega - g =0.$$

Absence of anomalous dissipation

Theorem: $$ \lim_{\nu \rightarrow 0} \nu \int_{R^2} |\nabla \omega^\nu |^2 dx = 0.$$

The key idea is to extract extra structure by taking limits.

Statistical Stationary Solutions

Definition: A stational statisitcal solution (SSS) of the damped, driven NS equation on the phase space of vorticity is a probability meausre $\mu^\nu$ on $L^2 (R^d$ such that )…..some conditions indicating that the equation holds in a weak sense against “cylindrical test functions”.

Ideas of proof….pretty fast….hard to keep up while typing. He emphasizes the role of a nonlinear function $\beta$ which he uses to map from “$L^2$ to $L^\infty$”

One Key idea. Use $J(u \omega) - J(u) J(\omega)$ when you are considering a mollifier $J$. This is a useul trick. “Quadratic flux formula”

The hard part of the proof is to prove the enstrophy balance.

The delicate dance of the $\beta$ parameter reminds me of the role of the smoothing operator $I_N$ in the $I$-method. I’d like to understand this better….

The punch line is that you only need this “old language” for the proof. At the end, we obtain a proof of the absence of anomalous dissipation. I’d like to prove corresponding results for quasigeostrophic.

Remarks:

The existence of weak solutions for damped, driven Euler equations: Barcilon-Constantin-Titi
For absence of anomalous dissipation: Renormalized weak solutions. Enstrophy balance.
Results for gSQG. Mentions Caffarelli-Vasseur, Nazorov-Volberg-? on this equation.

Questions/Discussion

A suggested direction: Stability implies regularity. If you assume that NS solutions are $L^2$ stable then you should be able to prove regularity. Conjecture: Fix viscosity. Imagine you have a time horizon and you have a constant. Lipschitz regularity of the flow map implies regularity.

Steve Preston: The inextensible string as a toy model of fluids

(pdf slides talk; joint work with Ralph Saxton)

The original idea that the inextensiblestring might have soemthing to do with fluids was due to V. Yudovich. This problem was introduced to the speaker by A. Shnirelman. The part of fluids I want to have a toy model for is the geometric viewpoint.

Geometric aspects of fluid mechanics

At a point in a fluid, you have a velocity felid and we minagine this vector pushes a fluid element along that direction. You put an $L^2$-Riemannian metric on the space of maps $C^\infty (M, M)$ and you define geodesics wrt this metric.

Volumorphisms. He views this as a “submanifold” in the space of all maps. A lot of things break down here but formally we think of this curve of volume preserving maps inside the larger flat space of all maps. Hodge decomposition allows us to decompose an arbitrary map into one along the volumorphism submanifold and another involving a divergence.

Misiolek-Himonas

Ebin-Marsden

Riemannian exponential map.

Eulerian viewpoint is easy to formulate, but the Lagrangian form is more convenient for geometry. He emphasized the loss of dierivatives.

“Smoothness of the exponential map is the most basic requirement for doing infinite-dimensional Riemannian geometry rigorously.”

There is a gap between the topology and geometry of volumorphisms.

Topology. We want the volumorphisms to be a smooth submanifold of the space of smooth maps from $M$ to $M$. We can make this rigorous if we enlarge to Sobolev $H^s$ diffeomorphisms with $s > \frac{1}{2} {\mbox{dim}}(M) + 1$ to ensure that $\eta \in C^1$. For smaller $s$, we don’t get a smooth submanifold.
Geometry. The metric is defined only in terms of $L^2$ distance. So geometrycially, we should consider measureable maps $\eta: M \rightarrow M$ which preserve the measure. These may not even be bijections so that manifold structure fails (not all tangent spaces are isomorphic).

If $M$ is three-dimensional, the $L^2$ closure of smooth volumorhpisms is the space of all measure-preserving measureable maps (Shnirelman). Fluids should not behave like that! Or maybe they could come close?

In 2d, we don’t understand this so well.

Lagrangian Averaged Euler equations

We introduce a parameter $\alpha$ and define an alternative Riemannian metric which has an $H^1$ inner product (time $\alpha$) and we consider this only on the Volumorphism “submanifold”. The associated geodesic equation leads to the LAE-$\alpha$. “The idea is to average over small scales of a fluid; as $\alpha \rightarrow 0$ we expect the solutions to approach the usual Euler equation solutions.” When these were introduced, it was expected that 3d global existence would be easier than for Euler but this has not turned out to be the case.

To better understand what is going on, we want simpler one dimensional examples.

Camassa-Holm: $ u_t - u_{txx} + 3 u u_x - 2 u_x u_{xx} - u u_{xxx} = 0$
Constantin-Lax-Majda: $\omega_t = \omega H \omega, ~ u_x = H \omega$$

In the CLM equation $H$ is the Hilbert transform. Modified: $\omega_t - \frac{1}{2} u \omega_x = \omega u_x, ~ u_x = H \omega$. This is a geodesic equation on the space of diffeomorphisms on $S^1$ with a right-invariant $H^{1/2}$ metric.

….quick transition to the inextensible curves…..moving faster….

Unit speed parametrization. This constraint forces the curve the generate its own tension to satisfy the constraint. This is similar to the pressure which results from the zero divergence condition.

Geodesic equation for $L^2$ string

Orthogonal acceleration $\implies$

$$ \eta_{tt} = \partial_x (\sigma \eta_x)$$

where $$ \sigma_{xx} - |\eta_{xx}|^2 \sigma = - |\eta_{xt}|^2. $$ “Here the tension $\sigma$ is analogous to the pressure in the Euler equation, determined nonlocally by a purely spatial differential equation.

WE can think of this as an approximation of the nonlinear wave quation, in the same way as the incompressible Euler equation is an approximation of the compressible Euler equation.”

(This is an interesting nonlocal equation so it should be mentioned on the developing nonlocal equations wiki.)

Finite-d approximate model through a system of rigid rods oscillating like coupled pendula.

In $R^2$, we can write $\eta_x = (\cos \theta, \sin \theta)$ (which enforces the constraint) and rewrite the equations. Some discussion of boundary conditions. The boundary conditions then determine nonlinear constraints on the parameter $\theta$.

Motion of whips and chains. Preston JDE 2011
Bootstrap Chao Ma, PhD thesis forthcoming.

Movie:

$L^2$ whip. The whip cracks. The curvature becomes singular. Thess-Zikanov-Nepomnyashchy The loops make the crack!

So, I’d like to revisit the Lagrangian averaging in the setting of the inextensible string. He again adds the $H^1$ inner product to the Riemannain metric on the space of curves. This produces a new geodesic equation, which does not resemble the earlier wave equation. It does still have the nonlocal $\sigma$,…technical slide….

Preston-Saxton, DCDS-A (to appear) Some GWP results.

Movie:

Same initial condition and the loops don’t get pinched.

Emanuele Caglioti: Long time behavior of solutions of Vlasov-like equations

(pdf slide talk)

Outline

one slide, overview of the talk.

Vlasov-Poisson and 3d Euler

The Vlasov equation $$\partial_t f + v \partial_x f + F_f \partial_v f = 0$$

Here $f(x,v): S^1 \times R$ is the phase space density: $$ \rho (x) = \int dv f(x,v) $$ is the space density $$ F_f (x) = \int dy \rho(y) \mathcal{F} (x-y) $$ is the force.

Vlasov-Poisson Equation (VPE)

$\mathcal{F} = \partial_x \mathcal{V}, ~ \mathcal{V} = \partial_{xx}^{-1} \delta$. For $x \in [0, 2 \pi)$….ack slide change….

2d Euler equation is another example. The vorticity $\omega$ is transported along the flow.

The density $f(x,v,t)$ is transported along the trajectories of an Hamiltonian system: $$ \dot{x} = v, \dot{v} = F_f (x). $$

The hamiltonian is a functional of the density $f$ itself: self-consistent force field. Therefore the area of the level sets of $f$ is conserved. Same for 2D Euler the vorticity is tranported along the trajectories of an Hamiltonian system.

Mather: What does that mean “self-consistent force field”? Answer: Some discussion about the mean field limit and statistical mechanics. Here is a link to related discussion.

Possible behaviors

Stable stationary solutions of VPE.

Marchioro-Pulvirenti (1986)

$f = g(v) $ is an example.

BGK waves

Bernstein, Greene, Kruskal (1957)

$ f = f_0 (x - u_0 t, v - u_0).$

These solutions satisfy some conditions related to the Hamiltonian.

I try to make a parallel with 2D Euler.

Any radial vorticity: $ \omega = g(\rho), ~ \rho = \sqrt{x^2 + y^2}$ is a stationa soution of 2D Euler.

Kirchoff (1876) showed that elliptiacl patches are rotating solutions of 2D Euler. The patch is stable if $a < 3b$ (parameters refer to the geometry of the ellipse.)

Landau Damping

Landau, on the basis of the analysis of the VPE linearized around an equilibrium conjectured that for initial data close to equilibrium $$ f_0 = f(v) + \epsilon g(x,v) $$ asymptotically the electric field will vanish and the phase space density will become homogeneous. The linear case has ben fully characterized (Maslov- and Fedoryuk.)

Existence of a class of damped analytic solutions has been proved with a scattering approach by Caglioti-Maffei (1998). Hwuang-Velasquez (2009) extended this result to situations close to equilibrium solutions. Initial data cannot be characterized.

Mouhot-Villani (2009) proved that close to equilibrium initil data are exponentially damped (Landau Damping). The result is proved in an analytic framework (also Gevray type regularity).

Lin-Zeng (Recently) have shown that BGK exists for small regularity: $W^{s,p}, ~ s< 1 + \frac{1}{p}$. Therefore, there is NO LANDAU DAMPING for small regularity. This is interesting to me….a long time behavior which is dependent upon regularity.

Matthaeus (1991) simulation.

….Constantin: this is probably hyperviscosity. ok, but the point he wants to convey is not dependent on this issue. Shnirelman: this is a very robust picture. When you simulate NS on 2D torus, it always emerges that there are two vortices like this.

Possible limiting behaviors.

Stationary (stable) solution for VPE and for 2D Euler
time periodic solutions for BGK and Kirkhoff Ellipses for 2D Euler
Landau damping
Is it possible to prove damping to BGK solutions? Many people are working on this.

(This theory seems to be quite analogous to the state of the art in nonlinear Schrodinger and wave equations.)

What can we say in general?

Shnirelman ICM 2010 Mixing operators in $L^2$. Shnirelman’s construction. Bistochastic operators.

Partial ordering, minimal flows.

It is possible to prove that minimal flows are stationary stable solutions of 2D Eler.

A first conjecture: The set of minimal flows is an attractor for the 2D Euler flow (essentially Landau damping conjecture)

Motivation: If the fluid does not go to stationary solutions level lines of vorticity are stretched and stretched and therefore the solution reaches a minimal element.

The conjecture is probably wrong because more complicated behaviors are expected from simulations.

Brenier: True with probability 1. Caglioti: probably not.

Generalized minimal flows (Shnirelman):

The Navier-Stokes equation with random forcing in the null viscosity limit has an attractor which is concentrated on generalized minimal flows. We reformulate this conjecture in the language of Landau Damping from Vlasov.

A strictly related conjecture.

Given $f_0$, let us define $$ \Omega (f_0) = [ \mbox{weak limit points of} f(x,v,t): t \rightarrow + \infty ] $$

Then it is reasonable that generically:

$S(g) \geq S(f_0)$
If $g_1$ and $g_2$….ack slide changed.

Construction of periodic solutions for the HMF model

Morita-Kaneco PRL (2006)

See also Antoniazzi-Fanelli-Barre-Chavanis-Dauxois-Ruffo, PRE (2007)

work in progress with D. Benedetto and P. Butta.

Conclusions:

We might reasonably expect that asymptitocally the dynamics will become a simple motion. It might even be chaotic but with a few degrees of freedom involved.

Roberto Camassa: Large amplitude internal waves and their stability

Article about Carolina Wave tank

(Collaboration with A. Almgren, S. Chen, R. Tiron, C. Viotti.)

Somewhat soft…..filled with movies…..suitable for the end of the day.

Outline

Motivation: practical (quantitative) vs. “paradigm” (qualitative) significance of simple models of wave motion in fluids?

Three examples:

Two layer Euler vs. strongly nonlinear models
Wave induced instabilities
Integral equations:

Introduction. Fluids lab experiments. Cool movie.

Stratified incompressible Euler equations. 2 layer model. As you saw in the movie, diffusion can be ignored on the time scale of the movie.

Grue et. al JFM 1999 (Norway experimental group)

Stanton-Ostrovsky 1998 (Oregon Coast), 150m

ASIAEX 2004 (340m)

Helfrisch-Melville 2006

These papers reveal that large amplitude internal waves exist.

Miyata 1998, Choi-Camassa JFM 1999

Shallow water strongly nonlinear models.

Pictures….pictures….movies.

Richardson Number.

Taylor-Goldstein Eigenvalue problem controls stability.

Academic Trendiness at AAAS

Tue, 14 Feb 2012 19:37:12 +0000

The American Association for the Advancement of Science is gathering for their Annual Meeting for 2012. One way to measure academic trendiness is to look at search frequencies of keywords using the search tool for the AAAS conference.

“Basic Research” shows up 18 times; “Applied Research” shows up 93 times.

“Africa” appears as frequently as “mathematics” and both appear less frequently than “warming” and “online”.

“NSERC” appears as frequently as “Wellcome” and nearly 10 times less frequently than “NSF”.

The key words “environment”, “university” and “engineering” appear often. “Toronto” appears more frequently than “Alberta” and “Ontario” combined.

"Alberta" search results: (17) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|ontario|method|and|pge|1
"Africa" search results: (36) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|Africa|method|and|pge|1
"anthropology" search results: (12) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|anthropology|method|and|pge|1
"applied research" search results: (93) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|%22applied%20research%22|method|and|pge|1
"astronomy" search results: (15) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|astronomy|method|and|pge|1
"basic research" search results: (18) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|%22basic%20research%22|method|and|pge|1
"biology" search results: (97) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|biology|method|and|pge|1
"biotechnology search results: (14) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|biotechnology|method|and|pge|1
"chemistry" search results: (47) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|chemistry|method|and|pge|1
"digital" search results: (26) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|digital|method|and|pge|1
"engineering" search results: (110) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|engineering|method|and|pge|1
"environment" search results: (220) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|environment|method|and|pge|1
"experiment" search results: (69) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|experiment|method|and|pge|1
"evolution" search results: (77) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|evolution|method|and|pge|1
"industrial" search results: (27) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|industrial|method|and|pge|1
"math" search results: (11) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|math|method|and|pge|1 (all outreach related.)
"mathematics" search results: (36) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|mathematics|method|and|pge|1
"medicine" search results: (93) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|medicine|method|and|pge|1
"NSERC" search results: (3) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|NSERC|method|and|pge|1
"NSF" search results: (27) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|NSF|method|and|pge|1
"nanotechnology" search results: (13) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|nanotechnology|method|and|pge|1
"Nobel" search results: (5) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|nobel|method|and|pge|1
"online" search results: (39) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|online|method|and|pge|1
"Ontario" search results: (13) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|ontario|method|and|pge|1
"open access" search results: (24): http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|%22open%20access%22|method|and|pge|1
"peer review" search results: (17) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|%22peer%20review%22|method|and|pge|1
"philosophy" search results: (14) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|philosophy|method|and|pge|1
"physics" search results: (55) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|physics|method|and|pge|1
"sociology" search results: (6) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|sociology|method|and|pge|1
"solar" search results: (25) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|solar|method|and|pge|1
"statistics" search results: (20) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|statistics|method|and|pge|1
"terrorism" search results: (8) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|terrorism|method|and|pge|1
"Toronto" search results: (33) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|toronto|method|and|pge|1
"universities" search results: (833) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|universities|method|and|pge|1
"warming" search results: (44) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|warming|method|and|pge|1
"Wellcome" search results: (3) http://aaas.confex.com/aaas/2012/webprogram/start.html#srch=words|Wellcome|method|and|pge|1

Web Course Platform Prototyped by Toronto Mathematician

Thu, 09 Feb 2012 19:35:49 +0000

Suppose you’ve recorded a one hour lecture onto video and taken digital photographs of each blackboard used during the presentation. The video and photograph files contain metadata with time stamps: we know the time when each frame of video and photograph was captured. How should these files be blended to create a browser-based environment for studying the ideas in the lecture? My colleague Dror Bar-Natan has prototyped a remarkable software platform combining this data to support online courses and seminars.

The system is built atop MediaWiki (the software that runs Wikipedia) and allows metadata, such as comments and photographs, to be correlated in time to an embedded video stream. Comments can be attached to each frame of video. The system enables remote audience members to participate in the seminar frame-by-frame using their power to comment. In this way, the wClips system goes beyond playback by empowering the remote audience of the future to contribute to the past presentation.

The system was recently used to supplement a paper Dror is co-authoring with Zsuzsanna Dansco in the wClips seminar. Each section of the paper was discussed in detail in a seminar lecture captured on video. Each blackboard in the seminar series was photographed. Time links near the photographs move the video stream to the video time when that blackboard image was created. The sizes of the different elements can be changed by sliding the red line. In the final version of the paper, there will appear hyperlinks in each section pointing at the associated seminar presentation. Clicking on the photographs reveals another layer of comments highlighting the elements appearing on the blackboard.

Dror gave a video introduction describing how to operate the system and produced a one page guide. (The lighting is not so good.) Zsuzsanna’s lecture is also a nice demonstration of wClips.

The wClips system runs on a linux server using open source software so is inexpensive to deploy. Since it is based on MediaWiki, it has the potential to scale to support a massive audience. Getting the video content out of the camera and into the online course system involves some work but could be streamlined with the right scripts.

(Guest Post) Neil Turok Responds to Posts on Perimeter Institute

Wed, 08 Feb 2012 19:33:51 +0000

Dear James,

Thank you for this opportunity to respond to your recent blog posting on Perimeter Institute and offer some clarifications.

First, the comparisons of support for different institutions and programs presented in this posting mix total funding over time (including endowment contributions) with annual budgets and ignore substantial differences between their operations. For example, Perimeter Institute currently houses over 140 full-time trainees and researchers (from Masters students to senior faculty) whereas to my knowledge the Fields Institute has none.
Second, this blog post juxtaposes received funding with cultural events at Perimeter and implies that public monies are used to support ancillary activities, which is not accurate. Cultural and social events are supported through paid ticketing as well as private donations. Public funds are strictly allocated to the purpose for which they are intended: to support the research, training and educational outreach programs of the Institute.

You and your blog readers may be interested to know that a recent third party evaluation of Perimeter Institute’s cost-effectiveness was conducted by KPMG, as part of a comprehensive audit required as a condition of our government funding. The final report, available on our website, states “PI has designed and implemented practices and processes that promote economy and efficiency in the use of resources and that are effective in supporting the achievement of PI objectives and expected results.”

In closing, let me emphasize that Perimeter Institute takes very seriously its commitment to promoting basic scientific research in a collaborative manner with universities and other institutions across Ontario and Canada. In this context, Perimeter serves as an example of a successful public-private partnership which is helping to energize and strengthen the entire scientific community.

Sincerely,

Neil Turok

Director, Perimeter Institute for Theoretical Physics

The Lucky Few of Waterloo Part 2: Perimeter Institute Buys Culture

Sat, 21 Jan 2012 19:31:51 +0000

Efforts to understand the foundational issues of theoretical physics have been made by scientists over millennia. Human beings are naturally curious: we want to deeply understand nature.

With pioneering insights by Aristarchus and Archimedes, Copernicus and Kepler and, more recently, Dirac and Feynman, humans have made spectacular advances. The benefits of basic research investigations cascade into fundamental improvements for humans living on earth. Visionary investors, like the Duke of Braunschweig Charles William Ferdinand in the eighteenth century and Mike Lazaridis of today, have recognized the virtues of investing in basic research. Leaders like Alexander von Humboldt, John Charles Fields, and Vannevar Bush helped define frameworks for governments to support basic research.

Half a Billion Dollars

According to its mission statement, The Perimeter Institute is devoted toward the important goal of researching fundamental issues in theoretical physics. Perimeter's web page discloses that a stunning pile of public and private monies has been assembled to help the Institute's leaders advance humankind's understanding at the research frontier. Here is an extraction concerning public monies:

Government of Canada \$25 million grant through NSERC (2002) \$5.6 million grant from the Canada Foundation for Innovation (CFI) (2002) \$1.7 million grant from CFI Infrastructure Operations Fund (CFI-IOF) (2004) \$59,900 grant from Promoscience for ISSYP program (2005) \$50,700 grant from Promoscience for EinsteinPlus Program (2006) \$50 million Government of Canada announcement (2007) \$10 million commitment from Canada Foundation for Innovation (CFI) for building expansion (2009)
Government of Ontario

$15 million grant through MEDT (2002) $5.95 million grant from the Ontario Research and Development Challenge Fund (ORDCF), shared equally with the Institute for Quantum Computing (2002) $5.6 million grant from the Ontario Innovation Trust (OIT) (2002) $20,000 for 2003 Summer Institute (2003) $150,000 grant through MEDT for outreach programming (2005) $120,000 grant from the provincially administered Research Performance Fund (RPF) (2005) $50 million through Ministry of Research and Innovation (2006) $10 million commitment from Ministry of Research and Innovation for building expansion (2009)

Adding up the bold figures above reveals a public investment of more than $178.85 million. The 2011 Federal budget commitment of $50 million and the additional 2011 Ontario budget commitment of another $50 million to the Institute are not yet listed (nearly a year after the gifts). I wrote about these gifts last year in Part 1 of this series of posts. The total Canadian public investment in the Perimeter Institute over the past ten years exceeds $278.85 million.

Here is an extraction concerning private donations supporting the Perimeter Institute:

Mike Lazaridis, President & Co-CEO Research In Motion: \$100 million (2000), \$50 million (2008), and \$20 million (2009) for total donation of \$170 million Doug Fregin, Vice President (Operations) Research In Motion: \$10 million (2000) and \$20 million (2009) for total donation of \$30 million Jim Balsillie, Chairman & Co-CEO Research In Motion: \$10 million (2000)

These highlighted private gifts total to \$210 million producing a rather impressive (but incomplete) total sum of gifts in excess \$488.85 million. To gain some sense of the scale of this amount of money, consider the following facts:

The annual budget for the Fields Institute (Ontario's treasure supporting mathematical research) is less than \$5 million.
The Connaught Fund for research at the University of Toronto has a total value of \$77 million. These funds were raised by the sale of Connaught Labs, the first lab to commercially produce insulin following the discovery by University of Toronto researchers F. Banting and C. Best. That fund generates \$3 to \$4 million dollars to support research activities by UofT researchers.
Investment in Perimeter exceeds the five year budget request by TRIUMF labs which employs 340 full-time scientists and engineers, plus a lot of specialized equipment for experiments. (Perimeter lists 14 faculty and around 45 other faculty whose permanent positions are at other, mostly non-Canadian, institutions.)
NSERC's Discovery Grants Program is the principal source for funding basic scientific and engineering research activities performed by faculty at Canada's colleges and universities. The annual budget for the Discovery Grants (supporting about 10,000 investigators across all fields from Biology, Chemistry, Physics, all flavors of Engineering, Computer Science, Mathematics, ...) program is about \$360 million.

Perimeter Institute Buys Culture

The Perimeter Institute has invested some of its funds to create a "lively and dynamic atmosphere for research." The "Pushing the Perimeter" series has hosted musical events artists Brian Eno and Kronos Quartet. Last Thursday, the Perimeter Institute hosted musical and experimental performance concert by Laurie Anderson. Based on the official Twitter feed from the Perimeter Institute, it looks like the concert was a lot of fun, culminating in drinks at the Black Hole Bistro. Another upcoming series of events aimed at enlivening the research atmosphere in Waterloo will take place when the Perimeter Institute hosts Indulgence$^2$ (matched selections of red wine and chocolate) and Bask in the Cask (selections of fine ales brewed especially for the Perimeter Institute), as part of their Gastronomy series.

Celebrated Discovery Continues in Toronto

Meanwhile, and despite the relative paucity of their research funding, my colleagues at the University of Toronto continue to advance the boundary of human knowledge:

The top global breakthrough in physics (according to the editors of Physics World) was made by Aephrim Steinberg's group probing fundamental aspects of quantum measurement.
James Graham of the Dunlap Institute at Toronto and Nicholas McConnell of U.C. Berkeley discovered monstrously large black holes ($> 10^9$ solar masses).
Adrian Nachman is leading a special program at the Fields Institute on inverse problems and medical imaging, advancing on ideas leading to new methods for diagnosis and treatment of diseases.
Sajeev John's discovery (with Eli Yablonovitch) of photonic band gap materials is in the running for the Nobel Prize.

With cultural attractions like the National Ballet of Canada, the Canadian Opera Company, the North by Northeast Music Festival (Devo at Yonge and Dundas square last year!), excellent restaurants and more, Toronto investigators are fortunate: our research money can be spent on research instead of on investments aimed at building an atmosphere conducive to research. Confronting the unfolding impact from this month's cut of \$42 million from the Ontario Research Fund, tri-council research mission drift, and persistent troubles with peer review, the next generations of Canadians will look back and wonder what might have happened if Perimeter's half-a-billion dollars had been invested differently.

Fingertip Search Using University Library Resource

Fri, 06 Jan 2012 19:30:27 +0000

UTORvpn

Members of the University of Toronto have access to a fantastic library. There are great resources like the Oxford English Dictionary, MathSciNet and Web of Knowledge and a whole lot more. To access these resources while away from campus, members of the UofT community can use the virtual private network service UTORvpn. That site tells you how to install the VPN service on your computer. When you turn on the VPN, your computer can access restricted library resources even though it is not located on the University network. On my computer, the active and inactive VPN service is indicated like this:

[caption id=“attachment_711” align=“alignleft” width=“241” caption=“Active VPN”][/caption]

[caption id=“attachment_712” align=“alignleft” width=“236” caption=“Inactive VPN”][/caption]

Standard Search

Suppose you wanted to look up the definition of the word syzygy. You could use google to get some information or you could drill down through the University of Toronto library looking for the Oxford English Dictionary to eventually find a more definitive result.

Researchers are busy. So, many will merely use google since the drilling down through the library takes too much time.

Customized Search at your fingertips

UofT’s restricted databases can be made available at your fingertips. Here is an example using Firefox. Right click (or control-mouse) the search field on the OED to reveal the option Add a Keyword for this Search

I chose to use the keyword oed.

Now, I can type into the URL bar, oed WORDTOLOOKUP, and I get the OED search results instantly.

The same thing can be done with AMS MathSciNet author search and other search engines.

NSERC Rethink: Engage Grants Illustrates Mission Drift

Tue, 03 Jan 2012 19:28:13 +0000

The Globe and Mail recently posted a story entitled “Building partnerships between businesses and universities” which highlights NSERC’s Engage Grants Program. Following the article, there appears an attribution I don’t usually see in The Globe:

Content in this section is provided in partnership with the Business Development Bank of Canada. BDC provides entrepreneurs with financing, venture capital and consulting services. To find out more go to BDC.ca.

I found this a bit strange so I followed the link and found the mission statement of BDC.ca:

Our mission Help create and develop Canadian businesses through financing, venture capital and consulting services, with a focus on small and medium-sized enterprises (SMEs).
Our vision Accelerate entrepreneurs' success.

In fact, we’re solely dedicated to Canadian entrepreneurs. We have a nationwide team helping more than 29,000 businesses reach their full potential.

The Engage program is now three years old. NSERC President Suzanne Fortier has characterized the program as providing funds for academic-industrial “first dates”. The Globe article contains an anecdote on one (reportedly among ~~240~~ to date [Correction: there are now over 1000 Engage grants.]) academic researcher who is having fun “making lane changes”. NSERC Vice President for Partnership Programs Janet Walden’s assertion that the program is “win-win” for business and academic sectors is not justified with any statistics or summary facts about the program as a whole. Not everyone sees it this way but the businesses helped by BDC probably like the Engage program a lot. Who wins?

[caption id=“attachment_670” align=“alignright” width=“261” caption=“NSERC President Suzanne Fortier”][/caption]

Consider the details of the Engage program:

NSERC provides $25K of taxpayer funds to pay for a six-month research and development project between a university researcher and a company already involved in research and development.
The company is not required to invest any money on the project.
Any intellectual property developed by the project is owned by the company. There is no direct return back to taxpayers on their investment.
NSERC has not revealed conversion rates of Engage grants into Collaborative Research and Development program grants. Why?

Meanwhile, ~~240~~ 1000 Engage grants, each costing $25K, bleed 6 25 Million dollars away from other NSERC programs like the Discovery Grants which support basic research by Canada Research Chairs and other university researchers. NSERC’s foray into business development should be contrasted with the Council’s original mission:

"...encourage excellence in research; provide a base of advanced knowledge in the universities; assist in the selective concentration of research activities; aim for a regional balance in scientific capability; maintain a basic capacity for research training; encourage curiosity-oriented research; and encourage research with a potential contribution to national objectives. ... these objectives are intended ... to ensure long-term coherence in the federal system of university research granting." (Honourable Hugh Faulkner, then Minister of State for Science and Technology, during the opening comments of the second reading of Bill C-26.)

[caption id="attachment_637" align="alignleft" width="300" caption="Panel Chair Tom Jenkins, Minister of State (Science and Technology) Gary Goodyear, Arvind Gupta, Monique F. Leroux and Nobina Robinson (Not shown are panel members David Naylor and Bev Dahlby)"][/caption]

With Engage and other programs aimed at building academic-industrial partnerships, the current leadership has drifted away from NSERC’s core mission. This “mission drift” at NSERC was highlighted (see page 128 in the PDF version or follow this link) in the Review of Federal Support to Research and Development produced by the federal R&D panel chaired by Tom Jenkins (who was recently appointed an Officer of the Order of Canada):

The granting councils have played a pivotal role in developing both talent and ideas for Canada’s innovation agenda. Their core raison d’être has been and remains investigator-initiated research of both a basic and applied nature, and each needs to continue to be generously supported. However, there has been mission drift for the granting councils, as they have responded to pressure from government to be more business facing.

By excogitating the recommendations of the expert panel report, Minister Goodyear and Prime Minister Harper are poised to lead Canada’s research and development enterprise back to the launch pad: basic research; dream big.

2011 Winter CMS Town Hall Notes

Tue, 13 Dec 2011 19:26:51 +0000

The 2011 Winter CMS Meeting took place this past weekend. On Sunday (2011-12-11), there was a CMS Town Hall meeting. The discussion was led by a panel consisting of Long Range Plan Committee Chair Nancy Reid, CMS President Jacques Hurtubise, and CMS Director Johan Rudnick (who was partially obscured by a poinsetta). Attendees to the town hall meeting were fed a free lunch.

Long Range Plan Update

Nancy gave a status update on the long range plan. She reported that the committee met in October and is in the process of writing. They hope to have a draft version of the report available in late February with a target goal of final publication in June 2012. Nancy also reported that she and Math-NSERC Liaison Committee Chair Walter Craig had sent a letter of recommendations to NSERC for this year's Discovery Grants competition. The letter is posted here and also on the LRP web space.

Nancy reported that the overall federal funding envelope for math/stats through NSERC (Discovery grants, Institute, …) is around $21M/y. To function effectively, the Institutes require additional funds from the Provinces and other partners. To avoid the departures of talented mathematicians from Canada as forecasted last year, there is a need for more funding. I asked if the LRP report would contrast the circumstances faced by Canada’s financially threatened math/stats community with the recent <a href="/tag/perimeter/">$100M ( $50M federal, $50M Ontario) gift to the Perimeter Institute. Nancy replied that the LRP report will only address funds granted through NSERC and will not comment on grants given through other sources.

Persistent Concerns about NSERC and Discovery Grants

The open discussion revealed that there remain serious concerns within the Canadian mathematical community about NSERC and the Discovery Grants program. Nancy reported that consultations with NSERC have “not always been easy.” Brett Stevens expressed the view that the weight given to the training of highly qualified personnel in evaluating the merit of proposals was problematic. Nancy relayed that these issues had been raised with Isabelle Blain. Based on those conversations, Nancy predicted that no substantial review of the new system would take place until 2014.

According to Nancy, NSERC staff claims that no discipline other than mathematics has complained about the new peer review system. I relayed that there have been reports by computer scientists and physicists of troubles with the outcomes of their recent competitions. NSERC might not be aware of the troubles in other disciplines but that doesn’t mean they don’t exist. Through the Liaison Committee, the LRP process, the Institutes, meetings of chairs, trans-Canada research collaborations, and meetings of the CMS and SSC, the math/stats community in Canada has strong communications channels. These channels allowed us to see the anomalies in the 2011 outcome through a national lens. I wonder if other disciplines would see problems with the conference model if they too had a wide enough vantage point on the outcome of their recent Discovery Grants competitions.

There was some discussion about how NSERC has moved funds away from the Discovery Grants program into a potpourri of programs supporting commercialization and academic-industrial partnerships. As mentioned in the 2007 report of the International Review panel (which led to the new peer review system), pure mathematics is unfairly punished when basic research funds are redirected toward short term commercialization goals.

Eddie Campbell isolated an issue that our community must confront. Should NSERC fund lots of smaller grants or fewer larger grants? There is a fixed amount of money in the federal budget for mathematics. How should those funds be spread out? Jacques mentioned that “spreading peanut butter” is a frequent metaphor used in discussions around this issue.

Transparency

I highlighted Minister Tony Clement’s call for federal government transparency. With this background, I asked Nancy if the LRP could arrange for the release of NSERC President Suzanne Fortier’s slides from her public presentation at the Summer 2011 CMS meeting. Nancy replied that the LRP has the slides but has been instructed not to circulate them. Nancy also reported that the LRP had requested data regarding the appeals numbers and success rate. She said the success rate is running around 25% but the LRP had not yet received the requested data. In light of Mr. Clement’s call for federal government transparency, I wonder why the data presented publicly by NSERC’s President to the Canadian Mathematical Society and requested by the LRP remains concealed from public view.

Immigration Policy Concerns

David Pike expressed frustration that Canada’s immigration rules prevent international graduate students from seeking permanent residency. I didn’t quite understand the details. David reported that the policy is seriously affecting his finishing PhD student who wishes to stay in Canada but will probably be forced to leave. Canada and the provinces invest heavily in the training of international graduate students. David reported that current immigration policy prevents Canada and the provinces from benefiting from this investment since these highly qualified graduates are often required to leave after earning their PhD.

The CMS town hall meeting was a great opportunity for discussion among members of the Canadian math/stats community. I look forward to the LRP report and am grateful for all the hard work that Nancy Reid and the committee have done.

2011 Winter CMS Meeting Notes

Tue, 13 Dec 2011 19:25:24 +0000

This past weekend, I attended the 2011 Winter meeting of the Canadian Mathematical Society. I have posted below some (often rough) notes on the talks that I attended.

Robert McCann

(joint work with J. Denzler and H. Koch)

ROBERT MCCANN, University of Toronto Higher-order asymptotics of fast diffusion in Euclidean space: a dynamical systems approach.

With Denzler and Koch, we quantify the speed of convergence and higher-order asymptotics of fast diffusion dynamics on Rn to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution semigroup becomes differentiable with respect to H"older initial data on the cigar. The linearization of the dynamics is given by the Laplace-Beltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity. We provide a detailed study of the linear and nonlinear problems in H"older spaces on the cigar, including a sharp boundedness estimate for the semigroup, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution, and higher order asymptotics. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

Nonlinear diffusion Equation

$$ \rho_t = \frac{1}{m}\Delta (\rho^m) = \nabla \cdot (\rho^{m-1} \nabla \rho) $$

Heat Equation: ($m=1$) Spreads out at rate $\tau^{1/2}$.
Porous Medium regime: ($m>1$). Rate of diffusion $\rho^{m-1}$ varies directly with density $\rho$ of diffusing materiel.
Fast Diffusion regime: ($m<1$). He rewrites $m = m_p = 1 - \frac{2}{n+p}$ with $ p>0$. This leads to tails of algebraic thinckness which decay fast enough to stay in $L^1$. The size of the support grows like $\tau^{\frac{1}{2}[1 + \frac{n}{p}]} = \tau^\beta$.

We are interested in the convergence to a characteristic shape called the Barenblatt solution.

McCann-Slepcev 06: An $L^1$ convergence result for general conditions in the case where $p \geq n$. The rate of convergence is $\tau^{-\beta}$ with constant related to the initial first moments.

Main Goal: Slide change…ack. It involves an estimate showing that $\frac{\rho}{\rho_B}$ goes where it should.

Friedman-Kamin 80, CarilloToscani 99, Dolbeault-del Pino 00, Otto 01.

Blanchet, Bonforted, Dolbeault, Grillo, Vazquez 07, ….

There are other norms that can be studied besides $L^1$. A nice choice is the so called relative $L^\infty$.

Vazquez 03

Kim-McCann 05

Denzler-Koch-McCann 11 Higher order asymptotics.

Using an infinite dimensional analog of a finte dimensional dynamical systems approach.

A time-dependent rescaling of space. This is explicit because they know the rate of spreading of the solution.

Otto 01. Gradient flow structure and linearization. Rescaled dynamics is the steepest descent of the energy $E(u)$ with respect to an infinite-d heurisitic Riemannian structure whose intrinsic distance is the $L^2$-Wasserstein distance.

Theorem pretty long….can’t type it fast enough. Seems quite strong. It captures the convergence up to exponetially small errors and they can improve the exponent by adding more terms to their formula for the asymptotic corrections.

This is different than solitary wave resolution. The conclusion reports that there is one big Barenblatt that describes the asymptotic behavior. The spreading is faster than any translation and eventually it absorbs everything in the wings. In contrast, soliton equations have solutions with long time asymptotic descriptions involving widely separated non-interacting bump functions.

Mostapha Fazly

MOSTAPHA FAZLY, UBC Liouville-type theorems for some elliptic equations and systems

In this talk, we consider the problem of non-existence of solutions for some basic elliptic equations and systems with weights. Starting with Henon-Lane-Emden system, we present a Liouville-type theorem for bounded solutions in dimension N=3 as well as the statement for the full Henon-Lane-Emden conjecture in higher dimensions. Since systems are normally much more complicated than equations, in higher dimensions we back to single equations (both second order and fourth order) to prove such theorems under some additional assumptions on solutions. This work has been done under supervision of N. Ghoussoub.

arXiv1

arXiv2

Liouville’s Theorem: Any bounded entire function is a constant.

For PDEs, any bounded harmonic funciton is a constant.

Lane-Emden equation. $-\Delta u = u^p$ on $R^N$.

Gidas-Spruck 1981: Assume $N \geq 3$. Let $u$ be a nonnegative solution of L-E with $ 1 < p < \frac{N+2}{N-2}$. Then $u=0$.

For the critical exponent, there is an explicit solution.

C.S. Lin 1998, similar result for a 4th order generalization. In the critical case, there is an explicit solution.

Lane-Emden system: On $R^N$, consider $$ -\Delta u = v^p, -\Delta v = u^q$$

Lane-Emden conjecture: Suppose $(p,q)$ lies under the critical hyperbola $$ \frac{N}{p+1} + \frac{N}{q+1} > N-2 .$$ Then there is no positive solution for the above system.

Mitidieri 1996, Serrin-Zou 1996, Serrin-Zou, Polacik-Quittner-Souplet 2007, Souplet 2009.

Known to be sharp for radial solutions. $N=3$ True for polymomially bounded solutions. Condition removed. $N=4$ done. $N \geq 5$ is open.

Henon Equation: On $R^N$, we consider

$$-\Delta u = |x|^a u^p$$

Conjecture: Assume $ N \geq 3$. Let $u$ be a nonnegative solution of the above euqtion with $1 < p < \frac{N+2 + 2a}{N-2}$. Then $u=0$.

Some partial results listed….

Fourth order Henon equation.

Also open for $N \geq 5$. There is a stability condition in this context. Methods are linked to ideas used to prove the De Girogi conjecture.

Henon-Lane-Emden System and conjecture

….some details.

Mary Pugh

MARY PUGH, University of Toronto A new result in blow-up for long-wave unstable thin film equations

This talk will provide an introduction to long-wave unstable thin film equations of the form $$u_t = −(u^n u_xxx)x−B(u^m ux)_x $$ The exponents n and m determine whether or not finite-time blow-up of the solution might occur. In this talk, we present new results for the critical $(m=n+2)$ and supercritical cases $(m>n+2)$ on the line. This is joint work with Marina Chugunova (University of Toronto) and Roman Taranets (University of Nottingham).

(joint work with Chunganova and Taranets)

arXiv

$$ u_t = - (u^n u_{xxx})x - B(u^m ux)_x $$

This is a long wave unstable thin film equation.

Consider data $u_0 (x) = \overline{u} + \epsilon u_1 (x,0) + \epsilon \cos (\xi x + \phi)$.

The small perturbation $u_1$ will approximately satisfy a related equation and the cosine perturbation generates a low frequency instability.

Critical regimes:

Compare height scales and length scales. Cannot have finite-mass blowup if $m < n+2$.

Another path is to plug in a self-similar ansatz and look. Eventually, you find this relationship.

Bertozzi-Pugh (CPAM) 1998:

$m< n+2$ subcritical
$m = n+2$ critical
$m> n+2$ supercritical.

Dissipated energy. Can it somehow be related to a useful norm like $H^1$? Could it ever go to $-\infty$?.

(Analogous somehow to the Weinstein threshold.)

Noviick-Cohen+Shishkov 2010.

Critical regime involves an $L^4$ potential energy. The main difference is that the $L^1$ norm here plays the role of the $L^2$ norm in the NLS setting. Therefore, 4 turns out to be the critical potential energy power whereas in NLS criticality occurs when potential energy is $u^6$. There is an optimal mass threshold. For mass less than this value, there is a spreading of the dsolution. At equality, there is a blowup.

Witelski+Bernoff_Bertozzi did a simulation with two disjoint subcritical mass droplets whose joint mass is supercritical. Computationally, they see a focusing one point blowup effect.

In the critical and supercritical regimes (Bertozzi+Pugh 2000, building on a formal argument by Bernoff only for $n=1$). There is a variance equality, analogous to the Glassey identity, which eventually proves the existence of finite time blowup solutions.

“This is not a hugely strong statement.” It does not prove what happens but does show that blowup occurs.

In the case $ n \in (0,2)$, Chugunova+Pugh+Taranets managed to find an obstructive argument implying blowup.

OK, so it seems they have found a flexible virial-type structure governing the blowup phenomena for this problem. In principle, there should be next steps towards a definitive understanding of the blowup. For example, one should be able to prove an $L^1$ concentration result with lower bound of on concentrated mass in terms of the optimal constant in their interpolation inequality. It seems that all the pieces are in place to prove the analog of the Merle-Tsutsumi concentration result. I find the exponential weights intriguing. What is the meaning of these in this setting?

Alexei F. Cheviakov

ALEXEI F. CHEVIAKOV, University of Saskatchewan Conservation Laws of Surfactant Transport Equations

We present interfacial convection and convection-diffusion equations which model the transport of surfactants in an incompressible two-phase flow. The model employs the level set formulation of the interface. In both convection and convection-diffusion settings, in three dimensions, we derive infinite families of conservation laws for these equations. Using these conservation laws, surfactant transport equations can be written in a fully conserved form. This is a joint work with C. Kallendorf, M. Oberlack, and Y. Wang (TU Darmstadt).

What is a surfactant?

Surfactant is a furface active agent. They are molecules with a hydrophobic tail and a hydrophili head. Examples include shampoos and fatty oils. These molecules have various interesting properties. They have many interesting industrial and medical applications. Bilayer sheets arise in soap films. Micelle. Liposome. Soap bubbles.

Wang-Oberlack 2011

Governing equations. Euler, Navier-Stokes, incompressibility condition and surfactant transport equation involivng the surface laplacian.

To do numerical modeling, we require that our equations are discretized into a fully conserved form. Can the surfactant transport equation be written in conservation law form.

Conservation laws

Divergence expression equal to zero.

Simple PDE example. 1d wave equation. Conservation of momentum and conservation of energy. Both can be derived from a multiplier calculation.

Noether’s thoerem. Direct construction method is sometimes more flexible. Introduces the notion of an Euler operator with respect to $U^j$. (This discussion seems to involve the same notions as the geometric description of Noether’s equation involving killing and conformal killing fileds.)

Completeness.

For the majority of physical DE systems all conservation laws follow from linear combos of equations derived in this way.

This provides a path to build conservation laws that is different from Noether’s theorem, placing the equation as the primary object instead of the Lagrangian. This is all reminiscent of the Friedrichs ABC method as described by Morawetz. What about monotone formulae? Does thhis general theory lead us to any new monotone quantities?

Differential Grobner basis….other methods allow us to reduce the number of equations encountered here.

Applications.

Using this direct method machinery, he derives some conservation laws. The method emplys mulipliers and Euler operators. The method is implemented in a symbolic package GeM for Maple. There emerges an infinite faily of $c$-dependent conservation laws for cases with and without diffusion. Surfactant dynamics can be written ina fully conserved form.

Open problems.

Higher order conservation laws? Can we get more by including fluid dynamics equations?

References.

Anco-Bluman 1997

Anco-Bluman 2002

Bluman-Cheviakov-Anco 2010

Kallendorf-Cheviakov-Oberlack-Wang 2011

Gideon Simpson

GIDEON SIMPSON, University of Minnesota On the Well and Ill-Posedness of Degenerately Dispersive Equations

In some physical problems, such as granular media, sedimentation, and magma dynamics, the leading order continuum model is a degenerately dispersive equation. A rigorous analysis of equations of this type has only recently begun and remains incomplete. Though some cases are locally, and globally, well-posed, others may be ill-posed. In this talk, we consider the Rosenau-Hyman compacton equations. Inspired by a proof of ill-posedness for a surrogate equation, we present robust numerical evidence that the $K(2,2)$ compacton equation is ill-posed for data about the zero background state. The mechanism of ill-posedness is an observed loss of continuity of the solution operator; arbitrarily small data may become arbitrarily large at a fixed time $T>0$. We also explore the equation about a nonzero background state, and examine the limit as this reference value goes to zero.

This work is in collaboration with D.M. Ambrose, J.D. Wright and D.G. Yang (Drexel University).

(joint work with D. Ambrose, J.D. Wright, D.G. Yang)

Degenerate Dispersion

The magma equations. When $\phi$ vanishes, the dispersion turns off.

The Rosenau-Hyman compacton equations. Again, we have a nonlinear prefactor appearing inside the third order derivative term.

In both cases, we can observe the nonlinear term influencing the dispersion relation.

Danger of Heuristics.

Camassa-Holm, 1993. CH has a term $u u_{xxx}$ which looks scary. But, this is a completely integrable system so we should not be scared about the degenerate nonlinear dispersive equations.

Origins of the Magma equations.

Origins of the $K(2,2)$ compacton equation. Solitons have exponential tails. In nature, we don’t have exponential tails. We need to have localized pulses.

Granular chains have similar properties.

Rigorous progress on well-posedness

Analytical alternatives in studying degenerate dispersion.

Self regularizing well-posedness.

Assume the initical conditions are nondegenerate; the data is strictly positive. Prove that the flow preserves nondegeneracy for some time; the solution remains striclty positive.

Accept degeneracy.

Develop a motion law for the boundary of the support….ack slide change.

For the magma equation, assume that we have data which looks like 1 + $H^1$. Then the solution exists uniquely on a positive time interval and remains bounded away from zero.

Ambrose-Wright 2010: $C^3$ solutions of the Compacton equations preserve their support.

compuatational results on wp

To justify the simulation, the equation should be well-posed. We don’t know that yet. So, we regularize by adding on a 4th order regularizing term. Inverting that smoothing operator helps tame the nonlinearity. He calls the regularized equation $K_\delta (2,2)$ and studies this in the limit as $\delta \rightarrow 0$.

He forecasts a wp result for positive data for $K(2,2)$.

Rigorous results on ill-posedness

Hints of ill-posedness. Backwards diffusion effect?

Surrogate equation. $u_t = 2 u u_{xxx}$. Differentiating this wrt x leads to a backwards diffusion term. The backwards diffusion effect inside the surrogate equation might give us a hint toward proving ill-posedness.

Describes a sequence of steps leading to the conclusion that the surrogate equation has a norm inflation effect driven by the backwards diffusion effect. These steps suggest that one might expect an ill-posedness result.

compuational results on ill-posedness

Interesting simulations involving the collision of two compactons for small diffusive regularization parameter $\delta$. It appears to be very sensitively dependent upon $\delta$ and creates a wildly oscillatory effect when $\delta$ is small.

Fridolin Ting

FRIDOLIN TING, Lakehead University Nonradial solutions to magnetic Ginzburg-Landau equations on the whole plane

We show that there exists non-radial, degree-changing, finite-energy solutions to the magnetic Ginzburg-Landau equations on the whole plane. These solutions are polygonal type configurations with $2\pi k$ symmetry for $k\geq 7$. This is joint work with J. Wei.

Magnetic vortices.

Plohr 78, Berger-Chen 89.

Stability, Gustafson-Sigal 2000.

Effective dynamics, Gustafson-Sigal 2006.

Bethuel, Brezis, Helein 1993.

Sandier-Serfaty 2007

Gustafson-Sital-Tzaneteas 2010

Ovchinnikov-Sigal 2004.

Main theorem. Construction of a bunch of new vortex configurations! The construction is almost in equilibrium.

Kapouleas 1991. Idea came from here. Finding compact constant mean scurvature surfaces.

Musso-Pacard-Wei 2010. Finite energy sign changing solutions of the NLS with dihedral symmetry.

Ehsan Kamalinejad

EHSAN KAMALINEJAD, University of Toronto Gradient flow methods for thin-film and related higher order equations

We will discuss recent results on a class of higher-order evolution equations that can be viewed as gradient flows on the space of probability measures with respect to the Wasserstein metric. The simplest of these equations is the thin-film equation ∂tu=∂x(u∂3xu), which corresponds to the Dirichlet energy. We will consider questions of existence and uniqueness of these gradient flows. A key probem in the analysis is the lack of convexity of the relevant energy functionals.

(Joint work with Almut Burchard).

Existence and uniqueness of fourth and higher order nonlinear evolution equation

Powerpoint slides actually slide in this talk. General overview of gradient flow on the space of probability measures wrt the Wasserstein metric.

Ask for energy to be convex along geodesics. This is the notion of displacement convexity.

Kumar Murty

KUMAR MURTY, University of Toronto The Tau of Ramanujan

The mathematical work of Srinivasa Ramanujan is striking for its originality and insight. But these very qualities also make it hard to understand. In this talk, we shall look at one aspect of Ramanujan’s legacy and use it as a backdrop to more generally discuss creativity and discovery in mathematics.

This was an interesting public lecture held at Ryerson University. The talk addressed the ideas of creativity and genius in mathematics.

Dmitry Pelinovsky

DMITRY PELINOVSKY, McMaster University Rigorous justification of the short-pulse equation

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals. This is the joint work with Guido Schneider, University of Stuttgart.

(joint work with Guido Schneider)

$$ u_{xt} = u + \frac{1}{6} (u^3)_{xx} $$

Schafer-Wayne 2004

Stefanov et. al. 2010

Pelinovsky-Sakovich 2010

Liu-Pelinovsky-Sakovich 2010

This model is integrable. The short pulse equation is closely related, via coordinate trnasformations, to the sine-Gordon equation. This relationship was observed by Sakovich-Sakovich 2005, 2006. The transformation is not globally well-defined. Not all solutions can be generated in this way. For example, short pulse equation does not have a solution in anlogy with the loop soliton solution of sine-Gordon. sine-Gordon breathers spawn 2-soliton solutions of short pulse.

Alterman-Rauch 2003

Barrailh-Lannes 2002; Colin-Galice-Laurioux 2005

Colin-Lannes 2009; Lannes 2011.

For the short pulse equation, only linearized equations were justifed from Maxwell equations using oscillatory integrals and Fourier analysis. Chung-Jones-Schafer-Wayne 2005.

Toy problem—quasilinear Klein-Gordon equation:

$$u_{tt} - u_{xx} + u + (u^3)_{xx} = 0.$$

Justification theorem. A good ($H^s$, $s>7/2$) solution of the short pulse equation valid on a time interval of size 1 with small data is close to a solution of the quasilinear K-G equation on the same time interval. (theorem statement is actually more involved but this appears to capture the jist of it.)

Brunelli 2005

Using integrable structure, small $H^2$ data for short pulse equation evolves globally in time.

Blowup alternative.

I am a bit confused. Here we have a completely integrable pde which forms singularities in finite time? After the talk, Dmitry pointed out that the integrable structure involved here is revealed via changes of variable to convert the short pulse equation into the sine-Gordon equation. The changes of variable become singular at blowup.

Anton Sakovich

(Image no longer available: Anton)

ANTON SAKOVICH, McMaster University Wave breaking in the short-pulse equation

In this talk, we discuss sufficient conditions for wave breaking in the short-pulse equation describing wave packets of few cycles on the ultra-short pulse scale. Our analysis relies on the method of characteristics and conserved quantities of the short-pulse equation and holds both on an infinite line and in a periodic domain. We provide numerical illustrations of the finite-time wave breaking in a periodic domain. This is a joint work with Yue Liu and Dmitry Pelinovsky.

Magdalena Czubak

MAGDALENA CZUBAK, SUNY Binghamton On some properties of the Navier-Stokes equation on the hyperbolic space.

Contrary to what is known in the Euclidean case, finite energy and finite dissipation solutions to the Navier-Stokes equation on a two dimensional hyperbolic space are nonunique. We review the nonuniqueness result and discuss possible ways to arrive at uniqueness of solutions in the hyperbolic setting. This is based on joint works with Chi Hin Chan and Pawel Konieczny.

(joint work withChi Hin Chan, Pawel Konieczny)

arXiv

L-H solutions

NS on $R^n$. Vector valued unknown. $P$ is the pressure. $u$ is divergence free.

Theorem: Leray-Hopf solutions on $R^n$. Energy inequality.

Goal: investigate how geometry (of the underlying space) affects the uniqueness of the L-H solutions of NS equation.

NS on manifolds

What is the equation on a Riemannian manifold? What happens with the Laplacian?

Ebin-Marsden 1970. Laplace beltrami works well on functions be we have a vector valued unknown. Hodge laplacian? E-M say no…the right thing to use is the deformation tensor. We get the hodge laplacian witha correction. Basically, you soulnd also introduce a term associated with the Ricci curvature. In this setting, it is more natural to work with one-forms than with vector fields. This can be done easily by raising and lowering indices using the metric.

On the hyperbolic space, we can identify the Ricci term very nicely. It produces $Ric (U^) = - a^2 (n-1) U^.$ On a general manifold, the Ricci formula is not as nice.

$NS_M$ is then expressed as an equation on one forms with an additional terma ssocatiated with the Ricci curvature. T

Ebin-Marsden 1970

Priebe 1994

Mazzucato 2003

Dindos-Mitrea 2004

Q.S. Zhaing 2006: also gets nonuniqueness using a connected sum of two copies of $R^3$.

(without the Ricci term)

Avez-Bamberger 1978

Filatov-Il’in 1989

Ilin 1991

Temam-Wang 1993

Nonuniqueness of the L-H solutions

Theorem (Chan-Czubak 2010):

There exists infinitely many smooth solutions of Navier-Stokes on the hyperbolic 2d space which satisfy finite energy, bounded $L^2$, …

On a general negatively curved manifold, the modified NS equation (drop the Ricci term), with a certain nonpinching condition on the manifold, there is nonuniqueness.

Anderson 1983, Sullivan 1983. Anderson-Schoen 1985.

There is an abundance of nontrivial harmonic functions on negatively curved manifolds. (Actual statement is more precise.)

The idea of using harmonic functions to build nontrivial solutions of NS goes back to Serrin but, because of the Liouville theorem, these solutions are not interesting in the $R^n$ case.

Solution Pairs.

Using the nontrivial harmonic functions, we can build solution pairs $(U^*, P)$.

Hodge was a Scottish mathematician who defined the structures required to define the Laplacian on one forms.

S.-T. Yau 1975

description of gap preventing extension to 3d hyperbolic space.

Martin Boundary

Lack of Liousville theorems in a hyperbolic setting

Koch-Nadirashvili-Seregin-Sverak 2009.

Let $u$ be a bounded weak solution of NS on $R^2$….ack slide changed…..liouville theorem.

Uniqueness?

Work in progress.

Heywood 1976 shows that there is an interesting separation of natural versions of $H^1$ zero divergence vector fields on manifolds.

Xiao Liu

XIAO LIU, University of Toronto Numerical simulation for Derivative Nonlinear Schrodinger Equation

We present the numerical simulation of generalized Derivative Nonlinear Schrodinger Equation (gDNLS): $$i\phi_t + \phi {xx}+ i|\phi|^{2\sigma} \phix=0.$$ In the case of σ=1, it describes the long wavelength dynamics of dispersive Alfven waves propagation. We observe that for σ>1, solutions may develop a singularity after a finite time, and we give a precise form of the blow up rate and the solution profile.

Blowup for derivative NLS. Very nice talk with simulations probing the generalized DNLS. Current studies investigating what happens as the parameter $\sigma \searrow 1$.

Followup discussion considered the idea to prove LWP for the quintic DNLS. Fractional powers would involve another layer of difficulty, but the quintic case might be accessible using $X^{s,b}$-type spaces and multilinear tools.

Walter Craig

WALTER CRAIG, McMaster University On the size of the Navier - Stokes singular set

Consider the hypothetical situation in which a weak solution $u(t,x)$ of the Navier-Stokes equations in three dimensions develops a singularity at some singular time $t = T$. It could do this by a failure of regularity, or more seriously, it could also fail to be continuous in the strong $L^2$ topology. The famous Caffarelli Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of the singular set $S(T)$. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle $T^* (\mathbb{R}^3)$ above this set. Our first result is that, if the singular set is nonempty, then there is a lower bound on the size of the wave front set $WF(u(T,.))$, namely, singularities can only occur on subsets of $T*(\mathbb{R}^3)$ which are sufficiently large. Furthermore, if the solution is discontinuous in $L^2$ we identify a closed subset $S^{L^2}(T)$ of $S(T)$ on which the $L^2$ norm concentrates at this time $T$. We then give a lower bound on the microlocal manifestation of this $L^2$ concentration set, which is larger than the general one above. An element of the proof of these two bounds is a global estimate on weak solutions of the Navier-Stokes equations which have sufficiently smooth initial data.

(joint work with Maxim Arnold [UIUC] and Andrei Biryuk [U. Krasnodar])

Link to MathSciNet entry about first paper along these lines.

IVP for NS

NS initial value problem. This may be viewed as Newton’s laws for a fluid. Initial finite energy ($L^2$) data: how does it evolve? We can consider the spatial domain to be $R^3$ or $T^3$. (Real physical problems have boundaries.) The problem on $R^3$ with smooth boundary remains open.

Do solutions exist? Yes, in some sense.

If no singularities form, then yes. Solutions exists, they are unique and the theory is satisfactory.

If singularities form, then weak solutions exist but they aren’t known to be unique…

Leray weak solutions.

$(u,p)$ solves the problem in the sense of distributions AND
Integrability conditions and Sohr-vonWahl 1986 pressure condition.

Leray 1934.

I am studying every solution that satisfies these properties.

$L^s_t L^p_x$

Singular sets

The singular set $S(u)$ is the set of space-time points at which $u(t,x)$ is not locally bounded. That is $(t_0, x_0) \notin S(u)$ if there is a neighborhood such that $u(t,x)$ is bounded on $Q$.

Serrin theory says this is good enough to define the singular set. We can upgrade to Holder contiuity.

These results have been improved along a sequence of steps culuminating in Escauriaza-Seregin-Sverak 2003.

Upper bounds on the singular set.

Singular times.

Leray said that singular times have zero 1/2 dimensional Hausdorff measure.

Caffarelli-Kohn-Nirenberg 1982.

The one dimensional parabolic Hausdorff measure of $S(u)$ is zero.

Restrict to a time slice.

CKN says that the 1d Hausdorff measure is zero.

How to define the Hausdorff dimension? Cover the singular set by balls and sum up over the radii to the power beta and then optimize over the cover.

Theorem 1.

If $T$ is a singular time for $u$ then $”dim” (WF(u)) \geq \frac{1}{2}$.

In each fiber it is at least half dimensional.

Three estimates on Leray weak solutions

Energy inequality (an axiom)
Fois-Guilope-Temam 1981, Chemin 2004. $$ \int \| u(\cdot, t) \|_{\infty} dt < + \infty, ~ \forall ~T \in R^+. $$
Biryuk-Craig 2009. A (future) invariant set for NS flow….

Energy concentration set $S^{L^2}$. Suppose $T$ is a singular time when the $L^2$ norm drops down.

Theorem 2 (Arnold-Craig 2010): If $T$ is a singular time and is a time of energy discontinuity then the dimension of the WF is $\geq 1$.

Identify a geometry subset of $T^* (R^3)$.

Probe the solution with the Weyl calculus of pseduodifferential operators.

Identify the $L^2$ discontinuities with defect measures and $WF^{L^2} (u)$ with their support.

Define the “dimension” of the sets.

Argue by contradiction.

Phase space volume: lower bounds on the size of wave front sets

Ideas of proof: microlocal analysis and defect measures

Unpredictable Societal Benefits of Basic Research Illustrated

Sat, 26 Nov 2011 00:00:00 +0000

The American Association of Universities has a trove of documents illustrating the societal benefits of basic research. A wide portfolio of scientific investments selected strategically via peer review by scientists produces unexpected benefits. Here are some examples, courtesy of the AAU:

Innovation and the Disrespect of Scientific Invention

Wed, 23 Nov 2011 19:23:22 +0000

“Innovation is the creation of better or more effective products, processes, technologies, or ideas that are accepted by markets, governments, and society. Innovation differs from invention in that innovation refers to the use of a new idea or method, whereas invention refers more directly to the creation of the idea or method itself.”

Dean Roger Martin of the Rotman School of Business, an iThinker pondering Canada’s innovation gap, writes in a recent op-ed piece that

“The first lesson is that commercial success and impact is more about innovation than about invention. Invention is the creation of some new-to-the-world technology, molecule, material, or formula. It is typically the product of the curiosity of a scientist. It can be pretty earth-shattering when it is electricity or insulin. But it can be pretty irrelevant when it is a technology in search of a user.”

The laser, created by scientist inventors A. Schawlow and C. Townes at Bell Labs, was dismissed as “a solution looking for a problem.” Eventually, this seemingly irrelevant technology found applications in fiber optic communications systems, in semiconductor device fabrication and in powerpoint presentations at leading business schools worldwide. Bell Labs invested heavily in curiosity driven basic research by scientists. Scientist inventors, free to pursue their potentially irrelevant curiosities, eventually produced important technologies:

The Transistor (invented by scientists John Bardeen, Walter Brattain, and William Shockley)
Unix Operating System (invented by scientists Ken Thompson, the late Dennis Ritchie, Brian Kernighan a U. Toronto alum and others)
Electromagnetism (understood by James Clerk Maxwell and Richard Feynman)
Global Positioning System (built on Albert Einstein’s Theory of Relativity)

These inventions by scientists are the foundational elements Mr. Jobs “cobbled together to make the Macintosh, iPod, iPhone and iPod”. Without them, Apple’s innovative product line would simply not exist.

Basic scientific research is the soil in which innovation grows.

Addendum to Arnold Memorial Workshop: Khesin on Pinzari Talk

Mon, 24 Oct 2011 21:02:50 +0000

The following message is a guest post by Boris Khesin. Boris summarizes the wonderful talk given by Gabriella Pinzari at the workshop. –Jim Colliander

Gabriella Pinzari (30min talk) described her joint result with her advisor Luigi Chierchia on a recently found fix for the famous KAM theorem, or rather for its application to the stability of the Solar system.

Namely, the original KAM theorem in Arnold’s 1963 paper claimed the persistence of the Liouville tori for perturbations of integrable systems under some nondegeneracy assumption - some determinant must be nonzero. This was a perfectly correct statement proved in the paper. But Arnold applied it to the Solar system without properly checking that for that system the determinant is indeed nonzero. (More precisely, Arnold checked the non-degeneracy condition for the first nontrivial case, the planar three-body problem, and claimed that this could extend to the general case: spatial, arbitrary $n$.) However, it turned out to be identically zero in the spatial case.

So later M.Herman developed a theory for how to deal with such a degeneracy in the KAM theory. Essentially it shows how to use a slighter nondegeneracy “in the next term”, which worked for the application to the Solar system. (It was published by J.Fejoz.)

But now Gabriella Pinzari explained that the source of the degeneracy in the application of the KAM theorem to the Solar system was the necessity to mod out rotations!! Once one mods out rotational symmetry in the Laplace plane, the system becomes nondegenerate (i.e. Arnold’s determinant is nondegenerate) on the quotient! This is somewhat similar to the relation of Morse and Morse-Bott functions, as far as I understand.

Quoting Pinzari, “Arnold gave only some ideas on how to construct (by series) such a reduction, but did not develop these ideas. What we have done was in essence to construct explicitly such a reduction. The miracle is that you can do it without singularities in the transformation (in a sense this is needed to control the convergence of the series that Arnold had in mind).”

[Note that since the determinant is identically zero, no higher order terms are nonzero. So M.Herman introduced a modification of the Hamiltonian which broke the rotation invariance of the modified system and computed a modified torsion with nondegeneracy in higher orders and this worked for the application to the Solar system. On the contrary, keeping the rotation invariance allows one to stay in the nondegenerate setting on the quotient.]

I am really shocked that this unavoidable degeneracy had such a simple explanation, which came unnoticed for 40 years. So all one needed was to develop a rotation-invariant version of the KAM theory. (As far I as understand, an equivariant KAM still does not exist beyond this rotation case.) And Arnold, who developed both KAM and group actions was in the best position to marry these two domains, but somehow it did not happen then! –Boris Khesin

Here are some references from G.Pinzari:

Edinburgh Arnold Memorial Workshop Notes

Mon, 03 Oct 2011 21:00:10 +0000

I am at an interesting workshop in Edinburgh entitled Dynamical systems and classical mechanics: a conference in celebration of Vladimir Arnold 1937 - 2010. Boris Khesin and Serge Tabachnikov have coordinated a Tribute to Vladimir Arnold which will soon appear in consecutive issues of the Notices of the AMS. These tributes were shared at the workshop and are also available here:

I will post below my notes from the talks. I apologize, especially to the speakers and readers (if any), for errors and typos.

Welcome

Remarks at introduction of workshop by S. Kuksin:

Kuksin highlights the openness of Arnold to discussions with students. A main message from Arnold “Mathematics must be interesting.”

Small program changes: Lai-Sang Yang speaks on Thursday at 1630. Laurent Stolovich speaks on Friday at 930.

Wednesday will have some short afternoon talks. Young people can either speak to me or Hakan Eliasson. We will make a page with a list of the talks.

Program

Monday 03 October

Alexander Shnirelman, Concordia University,Some problems of the fluid mechanics

I am grateful and touched because of the invitation to speak at this conference. I had the influence of Arnold for several decades. I remember that the pace of sdiscoveries in his seminar was so fast. It reminds me a bit of the situation like 500 years ago. There was not enough time to examine in detail the discoveries as they developed. It is a bit similar to the discovery of continents.

1. Arnold’s presntation of basics of the fluid dynamics.

A lagrangian system whose configuariton space is a Lie group $G$ with unit element $e$. Take the kinetic energy defined by a right-invariant Riemanniain metric via $$ E(u) = \frac{1}{2} \langle u , u \rangle = \frac{1}{2} (Au, u)$$ where $A: H \rightarrow H^*$ is the inertia operator. ….oh my this is too fast to follow while typing.

This is a survey of the Arnold approach to fluids. The critical points of the energy on each orbit are steady solutions.

Example: $SO(3)$. Surfaces $S$ are ellipsoids. levels lines. stable points.

We collapse to $n=2$. Take the quadratic form to be the $L^2$ norm. Discussing 2d Euler equation. Arnold called the surfaces isorotated velocity fields. There exists a unique stream function.

The second variation of $E$ on $S$ at the point $u$ is given by the quadratic form $$ \delta^2 E (\phi) = \int_M (\nabla \phi)^2 + \frac{\nabla \psi}{(\nabla \Delta \psi)} (\Delta \phi)^2 dx. $$ The solution $u$ is called Arnold Stable if this form is either poisitive or negative definite. What is $\phi$ here? It is a perturbation of $\psi$ and he writes on the board $\delta \psi = [ \psi, \phi ]$ (but with curly brackets).

2. Difficulties of the Arnold’s Approach

The group $D$ is infinite-dimensional; its topology is usually stronger than topology defined by the Reimannian metric. Hence the existence and uniqueness of geodesics are not certain.
The surfaces $S$ may be nonsmooth, and the partition of $H$ into these surfaces may be locally nontrivial.
It is unclear whether the energy functional $E$ attains a maximum or a minimum on a given orbit.

3. Group as a Banach Manifold

Theorem (Lichtenstein, Giunter, …):

For any initial velocity $u_0 \in X$, where $X$ is one of the above spaces (e.g. Holder, Sobolev, …), there exists $T>0$ and a unqique solution $u(x,t) \in X$ of the Euler equations with initial velocity $u_0$ defined for $|t| < T$.
If $n=2, T = \infty$. (Volibner, Yudovic, Kato, ….)

Mentions Ebin-Marsden.

4. Mixing operators.

If a 2d domain $M$, the vorticity $\omega$ is transported by the flow; it is distorted and effectively missed. He considers a class of operators on $L^2$ given as integral kernel operators with positive kernel (so positive measures) and with marginals which are equal to 1.

He defines a partial order in $L^2$: $f \ll g$ if $f = Kg$ for some $K \in {\bf{K}}$. Now we write $u \ll v$ for two vector fields $ u, v \in V$ if $ curl u \ll curl v$.

By Zorn’s lemman, there exist a minimal flow wrt this ordering.

Theorem Minimal flows are Arnold stable.

Minimal flow is called energy excessive if $F’ \leq 0$; energy deficient if $F’ \geq 0$.

5. Long-time behavior of the flow; mixing of vorticity

Natural conjecture: minimal flows form an attractor. However, this is wrong.

Movie.

Shows a 2d torus. Vorticity on square patches. After some transition period, the solution becomes more or less periodic plus a constant velocity drift. In this experiment, the flow converges to time periodic flow whih is not a stable configuration. More detailed experiments show that the final flow is quasiperiodic with more details. Even possible to find almost periodic. These are not stable flows. There exists a wider class of flows which are attracting.

6. The evidence of irreversibility: Liapunov function.

Defines a Liapunov function. Existence of Liapunov function always shows that there is some kind of irreversibililty. Shows an elemtary example. For a free particle, we can consider $L(x, \dot{x}) = x \cdot \dot{x}$. (This is reminiscent of the Morawetz estimate.)

The first Liapunov functional for the fluid was found by V. Yudovic (1973). The LF is given by $L = \omega \omega_x \omega_y$. It turns out this combination satisfies that its time derivative is given by a square $(\omega^2 \omega_x^2)|_\pi \geq 0$.

7. Generalized minimal flow

Suppose $u_0 \longmapsto u$. Let’s close the orbit of the evolution in $L^2$.

Definition: A flow $u(t)$ with initial veloicty field $u_0$ is called a generalized minimal flow (GMF) if ….curl condition.

slide changes fast.

8. Construction of GMF by pseudoevolution

A process is decribed which produces GMF’s from a given seed data using the curl ordering.

Conjectures:

The set $N$ is an attractor for the Euler equations in the ordinary sense.
GMG are either statiornay or qaiperiodic with at most countable set of periouds.
Stircktly speaking, we have not proven that the set $N$ is a a proper subset of $V$. e.g. there might exist flows which are not GMF. (This is analogous to the Landau damoping recently proved for the Vlasov-Poisson equation by Villani.)

9. Local regularity of partition into isovortical surfaces

The equation was addressed recently by V. Sverak and A. Choffrut (2010). Consider the distribution function for the vorticity $$ \lambda (s) = {\mbox{mes}} [ x \in M : \omega (x) \leq s]. $$

Theorem (Sverak-Choffrut): Steady solutions close to a fixed Arnold stable one are in a smooth 1-1 correspondence with distribution functions $\lambda (s)$.

The proof is difficult and based on Nash-Schwarz implicit function theorem.

10. The structure of the exponential map.

Ebin-Marsden 1970
Ebin-Misiolek-Preston 2008
Misiolek 1993

John Mather, Princeton University,Near a double resonance

(blackboard talk)

I feel honored to speak here, especially since something I have been working on for a long time is called Arnold Diffusion. Some people speak about the problem. What is interesting to me is the program. Arnold posed many problems. There are two aspects that should be highlighted.

They are very interesting.
There is a chance that we could do them.

Generic quasiergodicity of Hamiltonian systems….a famous problem attributed to Boltzmann. This problem seems incredibly hard and seems to be inaccessible. In contrast, there appears to be hope for Arnold diffusion.

I apologize to those of you have heard this talk before. I want to speak about something that I announced in 2003 in Russian and it appeared in English in 2004. I announced some results at that time but, in the meantime I found some mistakes in the proof. I want to speak about corrections to those proofs.

This is a question about small perturbations of integrable systems. Usually, people discuss this in the setting of the Hamiltonian form. I prefer to approach it using the Lagrangian (equivalent form) since my method of approach is variational.

Lagrangian: $$L(\theta, \dot{\theta}, t) = l_0 (\dot{\theta}) + \epsilon P(\theta, \dot{\theta}, t)$$

We consider here the case where $P(\theta, \dot{\theta}, t+1 ) = P(\theta, \dot{\theta}, t)$.

$\theta \in T^n$
$\dot{\theta} \in B^n \subset R^n $

We are looking for solutions of the Euler-Lagrange equation $$ \frac{d}{dt} ( L_{\dot{\theta}} ) = L_\theta. $$ Let’s assume that $d^2 l_0 >0$. (This is a strong restriction; we’d rather like to do it under the assumption that the determinant of the Hessian is nonzero. A great deal of the theory is developed under that assumption. For the methods I use, I need this stronger condition.)

The goal is to somehow show that the solutions go everywhere. That is too strong, but I can prove something along those lines in a special case.

The Lagrangian I am looking at is a small perturbation of an integrable system. For the integrable system, the E-L equation is $$ \frac{d}{dt} ( L_{\dot{\theta}} ) = 0. $$

Arnold Question: Are the orbits confined or do some go everywhere? (This is a vague question; certainly Arnold was more precise.)

In the case when $n=1$, the orbits are confined. This is a consequence of KAM theory. The KAM tori persist. There are Birkhoff regions of instability and the orbits are confined by the KAM tori.

In the case $n>1$, the expectation is that the orbits are not confined. There are results along these lines which show, in some cases, that this is the case. For the program we have in mind, we want to build methods which show this phenomena is somehow generic.

In the case $n=2$ (this is what I had announced in 2003): Generically (assuming the positive Hessian part), the orbits are not confined.

The methods that I use are variational. I want to say a little bit about those tools.

Consider $U_1, U_2, \dots, U_k$ open non-void sets in $B^2$. The construction guides the orbit to only be allowed to move in certain ways. The trick is to choose the conditions in such a ways so that when you minimize within those conditions, the solution stays inside the open set and doesn’t get pushed off to the boundary. This is a method that I introduced in the past in studying twist maps. The conditions are really complicated so you need a guide to tell you what the conditions are. The basic guide involves something called Aubry sets. These are sets contained inside $T^2_\theta \times T_t$. The phase space consists of $T^2 \times B^2 \times T$ and the state space is $T^2 \times T$. The Aubry sets are defined by a global minimizing condition. What shall I say about them? First of all, it is useful to consider invariant probability measures for the Lagrangian system. Let $\mu$ be a probability measure on $T^2 \times B^2 \times T$. You can then define a cohomoology class $c \in H^1 (T^2)$. This allows you to define an average action: $$ A_c (\mu) = \int_{T^2} L d\mu - c $$ …..I don’t ususally write it this way….scratches it out and writes instead $$ A_c (\mu) = \int_{T^2 \times B^2 \times T} (L (\theta, \dot{\theta}, t) - c \dot{\theta}) d\mu (\theta , \dot{\theta}, t). $$ We can then define $$M_c = [~{\mbox{invariant probability measures that minimize}}~ A_c].$$ The support of $M_C \subset T^2 \times B^2 \times T$ and the map turns out to be injective.

Aubry set….defined by a minimizing condition.

Theorem: $M_c \subset Au_c$.

The conditions I build for the minimization process. First you have to know something about the Aubry sets, you can then state what the conditions are. The process was carried out successfully in the case of twist maps. I was able to prove that there could be wandering in the Birkhoff zones of instability. This procedure was also used by Cheng-Yan in the case of a priori unstable systems. They were able to prove a version of Arnold diffusion in these systems. One considers a rotator and a pendulum and take the product of the two. The diffusion is related to the unstable fixed point.

…trouble typing…..discussion of double resonance, with strong so that the resonant linear combination arises with control on the integer prefactors by a constant.

Massimiliano Berti, University of Naples Federico II,Quasi periodic solutions of Hamiltonian PDEs

(Similar to what I saw in France at Ile de Berder, so I will watch rather than type…)

There was some discussion afterwards between me, Massimiliano and Walter Craig. I suggested that Massimiliano’s improvement of Walter’s pseudodifferential result might be reconsidered in the setting of the $MMT_{\alpha, \beta}$ models introduced by Majda-McLaughlin-Tabak. These models are slightly more general and consider parametrized $\alpha$-power dispersion relation with a $\beta$-smoothing operator inside the cubic nonlinearity. This might provide a generalized framework for investigating the relationship between dispersive smoothing and the derivative properties appearing in the nonlinearity. Admittedly, these are not directly physical models but the mathematical motivations for their study seem to keep appearing….

Andrei Agrachev, SISSA Trieste, The long-time behaviour of dissipative systems

A natural mechanical system on a Riemannian manifold. Traectories are curves on this manifold. The kinetic energy is the usual Riemannian length. The potential energy is a function on the manifold.

Hamiltonian $ = \frac{1}{2}|p|^2 + V(q)$ where $p \in T_q^* M$ and $|p| = \max [ \langle p, \xi \rangle: \xi \in T_q M, |\xi | =1 ]$

WE consider this system but with an isotropic dissipation: $$ \dot{p} = - H_q - \alpha p, ~\alpha > 0$$ $$ \dot{q} = H_p. $$

Toy example: $M=R, V(q) = b q$. All solutions converge to one particular solution. Eventually, the particle moves with a fixed velocity. If we perturb the V a little bit, the phase portrait will be very similar. There will be a limiting profile and the structure will be very similar.

Toy example: Pendulum. $V(q) = b \cos q, ~ \frac{\alpha^2}{4} < |b|.$ Then, we don’t have limiting behavior like that beffore. We have instead a vortex. The dissipation brings the trajectory down to the minimum of the potential energy. However, when we have $ \frac{\alpha^2}{4} > |b|,$ we have a different limiting configuration. We obtain a limiting potential “gradient” flow on the circle.

There is strong dissipation in life. The limitig dynamics of systems we observe, like a ship on the ocean, has a transitional period but eventually there is a balance.

We try to view things using the Eulerian viewpoint.

Definition: Potential stationary flow is a gradient vector field $\nabla u$ where $u \in C^2 (M)$ and $[d_q u: q \in M] \subset T^* M$ is an invariant submanifold of our system.

In particular, $\dot{\gamma}(t) = \nabla_{\gamma(t)} u$ implies that $t \longmapsto (d_{\gamma(t)}u, \gamma(t))$ is a solution.

Definition: The curvature of the Hamiltonian $H$ at $p \in T_q^* M$ is a self-adjoint linear operator from the cotangent bundle to the cotangent bundle defined by the formula $$ R^H_{(p,q)} \xi \cal{R} (\xi, p)p + (\nabla_q^2 V) \xi, ~ \xi \in T_q^* M, $$ where $\nabla$ is the covariant derivative and $\cal{R}$ is the Riemannian curvature.

(This is a natural extension fromt he standard symplectic setting to the dissipative systems. Some interesting discussion…what is the connection….natural…only this kind of isotropic dissipative systems.)

Assume that $M$ is complete, $\cal{R}$ and $\nabla^2 V$ are uniformly bounded. (This follows if $M$ is compact.) Let $\Phi_t$ be the flow on the cotangent bundle. We consider a strip defined by $\Omega_c = [ (p,q) \in T^*M: |p| \leq c]$.

Theorem:

If $R^H_{(p,q)} < \frac{\alpha^2}{4} I, ~ \forall (p,q)$ such that $H(p,q) \leq \max V$, then $\exists$ a potential stationary flow $\nabla u$ such that $$ \Phi_t (\Omega_c) \rightarrow [ d_q u : q \in M] ~{\mbox{as}}~ t \rightarrow + \infty$ $$ with an expoenetntial rate, $\forall c>0$.
$[d_q u: q \in M]$ is a normally stable submanifold of $\Phi_t$.
If $M$ is compact and $R^H_{(p,q)} < \frac{\kappa -1)\alpha^2}{\kappa^2} I$ then $ u \in C^k (M)$.
The map $(H,\alpha) \longmapsto u$ is continuous in the $C^2$-topology.

Smaller dissipation: When dissipation is smaller, we have some hopeful hints. Discussion is moving a bit fast for me to type….Markov process…not an invariant measure but the measures can be propagated…

Interesting discussion about the use of measures in the presence of small dissipation limits as a device to probe the structure of the original Hamiltonian systems.

Slides stop….he still has about 10 minutes. He tries to explain the proof. This discussion is reminiscent of a principal theme I will try to convey in my talk. Infinite dimensional systems might be viewed as the envelope system of limits of finite-d systems. When we study the infinite-d system, one strategy of attack is to identify convenient finite-d systems which limit on the infinite-d system. One possible source of these convenient systems might be through appropriate choices of isotropic dissipative systems, which truncate high frequencies.

Some discussion striving to describe the “curvature” of a Hamiltonian system….family of vertical and horizontal Lagrangian distributions. Curvature of dissipative systems is easily accessed…..cheating a bit, but look in the paper for the details. Levi-Civita connection is tangent to the zero section.

Walter Craig, McMaster University,The water wave problem as a Hamiltonian system

I thought I would speak a bit about Arnold’s influence on my mathematical life. We were given his Mathematical Methods book in graduate school. His perspective has pervaded our approach to problems.

(joint work with Catherine Sulem; along with Alessandro Selvitella and Yun Wang)

Outline:

Two ODEs
Euler’s equations
Zakharov’s Hamiltonian
Partial Differential equations as Hamiltonian systems
Birkhoff Normal Forms
Implications of the normal form
The KdV scaling limit

Two ODEs

$$ \dot{z} = z^2, z(0) = \epsilon $$ versus $$ \dot{w} = w^3, w(0) = \epsilon $$ …..ack slide changed and I missed the point.

Euler’s equations

Newton’s laws, Eulerian coordinates, incompressible fluid. We work on a pre-Columbian model of the earth.

Free surface water waves. We imagine that the velocity field is irrotational (oceanographers do this) so we can recast this as a potential flow. The bottom is not a sponge, so no penetration and we assume that the fluid velocity has no normal component at the bottom.

Free surface conditons: Kinetic BC at the top; Bernoulli condition.

Hamiltonian systems: Zakharov 1968. This was a poorly understood paper which has emerged as being very important.

Goals: explain this fact; use it to understand the PDEs.

Partial Differential equations as Hamiltonian systems

Zakharov’s Hamiltonian

The energy functional $H= K + P$ so it should be $$ H = \int \int_{-h}^{\eta(x)} \frac{1}{2} |\nabla \phi |^2 dy dx + \int_x \frac{g}{2} \eta^2 dx. $$

This is pretty clear but the difficulty is what are the choices of variables?

Zakharov’s choice: $$ z = (\eta(x), \xi(x) = \phi(x, \eta(x))).$$ That is $\phi = \phi[\eta, \xi] (x,y).$

In these coordinates, we can realize the PDE for Euler flow for the free surface as a Hamiltonian system in Darboux coordinates. The subtlety is how to differentiate the Hamiltonian wrt the canonical coordinates.

Other Hamiltonian PDEs:

Boussinesq system; KdV equation; NLS; …

Dirichlet-Neumann oeprator:

Laplace’s equation on the fluid domain $-h < y < \eta(x)$. $$ \xi(x) \longmapsto \phi(x, y) \longmapsto N \cdot \nabla \phi (1 + |\nabla_x \eta |^2)^{1/2} = G(\eta) \xi (x). $$
In Zakharov’s coordinates, we can express the Hamiltonian in terms of the D-N operator $G$ as $$ H(\eta, \xi) = \int \frac{1}{2} \xi G(\eta) \xi \frac{g}{2} \eta^2 dx.$$
The water wave system rewritten: $$ \partial_t \eta = G(\eta) \xi ,$$ $$ \partial_t \xi = -g \eta - {\mbox{grad}}_\eta K. $$ (This discussion is closely related to a variational formula of Hadamard from 1911, 1916)

Lemma (Properties of the Dirichlet-Neumann operator): A singular integral operator $G(\eta)$ related to the Green’s function.

Hermitian symmetric.
$G(\eta) \geq 0$ and $G(\eta) 1 = 0$.
$G(\eta): H^1_\xi \rightarrow L^2_\xi$ is analytic in $\eta$ for $\eta \in C^1$ [using a theorem of Christ-Journé (1987)]: $$ G(\eta) \xi = G^{(0)} \xi + G^{(1)}…$$

ack….slide changed.

Conservation Laws:

Mass is conserved. Mass is $\int \eta dx.$
Momentum is conserved. Momentum is $\int \eta \partial_x \xi dx.$
Energy is conserved.

(Poisson bracket calculations are quite direct.)

Taylor Expansion of the Hamiltonian:

From analyticity, we can expand around the stationary zero solution using the Taylor expansion of $G$.

Flow of the Harmonic oscillator. He is considering here the linearized problem and showing that we can explictly solve this problem using Fourier/superposition methods. We encounter a Fourier series with rotating phases. The typical solution is almost periodic.

Basic facts:

The flow preserves the (linearized) energy.
Actions are preserved. (This is the moment map.) Therefore, all Sobolev norms are preserved.
Phases involve linearly in time.

Basic Questions:

Add in the perturbations. Do any of those orbits persist? This turns out to be quite hard. In fact, it is challenging to show that any of them persist. The progress on these questions have been made using KAM theory. We know that there exist periodic solutions.

Do there exist quasiperiodic or almost periodic solutions?
Given a point $z^0$ in some phase space. Does the flow exist in M? This is hard.
Does it exist globally in time? This is basically open, although there are some recent advances. Do we have stability? Do we have a Nekhoroshev stability property?
Can you make the actions grow? Weak turbulence. Growth of Sobolev norms?

Birkhoff Normal Forms

Fix the dimension to $d=2$, so we have $x \in R$. We restrict to the periodic-in-$x$ case. We want to perform canonical transformations to move the Hamiltonian into a normal form at least in some neighborhood of the origin.

Conditions:

Make the transformation canonical.
Make the new Hamiltonian have the same linearization plus only resonant terms up to some order with a new truncation/residual error.
If $Z^{(3)} = 0$, we will have a chance to get longer existence intervals based on the analogy of the first ODEs at the beginning of the talk.

This transformation process is called the reduction to Birkhoff normal form.

Theorem (Craig-Sulem 2009):

Let $d=2$ (and $h = + \infty$) and fix $r>3/2$. Then, there exists a neighborhood of the ball at the origin in $H^r$ on which we have a Birkhoff normal form which kills off the quadratic nonlinear terms resulting a cubic equation.

Note: This transformation mixes the variables $\eta$ and $\xi$.

Outline of the proof: flying slides…..cohomological equation turns out to be a linear equation. THere are no nonzero $m=3$ resonances. It turns out to be rather challenging to show that the flow of the vector field exists.

Implications of the normal form

Long time existence theorem. Work in progress.

We should be able to build solutions that last for time intervals on the order of $\epsilon^{-2}$. I want this time so that I can study the NLS limit of the water wave equation. On this time scale, we would like to have a nice justification. This justification requires the desired long time existence result.

Wu 2009: Small Sobolev data lasts for exponentially long times.

Germain-Masmoudi-Shatah 2009:

For $d=3$, small Sobolev data exist globally in time.

The difference is that I am on a compact domain. Wu is in a dispersive situation.

Nathan Totz & Sijue Wu have done the NLS limit in the non-periodic case. Schneider-Wayne also have results in this direction.

The KdV scaling limit

pretty fast slide switching….but nice moves. He shows how the KdV Hamiltonian can emerge from the water wave Hamiltonian. Now perform the above sequence of transformations on the Birkhoff normal form for water waves. In the limit as the small parameter goes to zero, the water wave Hamiltonian collapses to the KdV Hamiltonian.

Tuesday 04 October

Yann Brenier, University of Nice,From incompressible fluids to dust

It is a great honor for me to be here. I would like to discuss two issues that were familiar to V.I. Arnold. We saw fluids discussed yesterday in Shnirelman’s talk. Dust is a singularity of the Hamilton-Jacobi equation. (Nepecmponcka perestroika)

First part: euler equations and minimizing geodesics for volume preserving maps

Euler equations of incompressible fluid mechanics
Leas action principles
Geometric analysis issues
Minimizing geodesics: existence and uniqueness results for the pressure gradient

Euler’s Equation: Geometric Definition.

The fluid is moving inside a box denoted by $D$. We consider incompressible motion. This is viewed as a time dependent map $M_t: D \rightarrow D$. Points in $D$ are called $a$. $M_t$ is viewed as a map in the Hilbert space $H = L^2 (D, R^d)$, valued in the subset $VPM(D)$ of all Lebesgue measure-preserving maps.

Solutions of theEuler equatiosn, introduced in 1755, correspond to those curves $t \rightarrow M_t \in VPM (D)$ for which there exists a time dependent scalar function $p_t$ called the “pressure field” defined on D such that $$ \frac{d^2}{dt^2}M_t + (\nabla p_t) \circ M_t =0 $$ where $\nabla$ is the gradient operator on $R^d$ (wrt Euclidean norm).

The Principle of Least Action.

Theorem: Assume $D$ convex. The $(M_t, p_t)$ be asolution of the E equations, with constant $\lambda$ such that $$ \sum \frac{\partial^2 p_t}{\partial_i \partial_j} \xi_i \xi_j \leq \lambda |\xi|^2 $$ Then $M_t$ is the unique minimizer, among all curves along $VPM(D)$ that conincide with $M_t$ at the endpoints $t = t_0, t=t_1$ of the following action $$ \frac{1}{2} \int_{t_0}^{t_1} \int_D | \frac{dM_t (x)}{dt}|^2 dx dt. $$

In other words, such a curve is nothing but a (constant speed) geodesic along $VPM(D)$ wrt metric induced by $H = L^2 (D, R^d)$.

Arnold 1966, Ebin-Marsden 1970, Arnold-Khesin book 1998.

The Dual Action

Minimizing the actrion can be written as a saddle point problem, just by using a time-dependent Lagrange multiplier to relzs $$ \inf_M \sup_p \int_{t_0}^{t_1} \int_D [\frac{1}{2} | \frac{dM_t (x)}{dt}|^2 - p_t (M_t (x)) + p_t (x) ] dx dt. $$ This is trivially bounded fro below by $$ \sup_p \inf_M (same) $$ which naturally leads to a dual least action priciple.

Theorem: Using exactly the same conditions ($D$ convex and $(t_1 - t_0)^2 \lambda < \pi^2$, the pressure $p$ is the unique maximizer of the concave dual action $$ I[p] = \int_D J_p (M_{t_0} (x), M_{t_1} (x)) dx + \int_{t_0}^{t_1} \int_D p_t (x) dx dt, $$ where $$ J_p (y,z) = \inf \int_{t} ( \frac{1}{2} |\frac{d\xi_t}{dt}|^2 - p_t (\xi_t)) dt $$ where the infimum is taken over all curves $\xi_t \in D$ such that $\xi_{t_0} = y \in D, \xi_{t_1} = y \in D$.

The proof is elementary and follows from 1d Poincaré inequality.

Geometric Analysis Issues 1

Density of diffeomorphisms in $VPM(D)$. $SDiff(D)$ is the set of volume preserving orientation preserving diffeomorphisms. This is more refined than the $VPM(D)$ condition. For $d \geq 2$, it turns out that $VPM$ is the $L^2$ closure of $SDiff$. The identification of the closure of $SDiff (D)$ for the a prior finer geoesic distance induced by $L^2$ is a much more difficult issue. For simple (say contractile) domains $D$, this closure is still $VPM(D)$ for $d \geq 3$ (but defintely not for $d=2$) as show by Shnirelman in his land mark paper (Math USSR Sb 1985). These results have striking consequences: in particular maps of form $$ M(x) = (h(x_1), x_2, x_3) $$ where $h$ is any Lebesgue-measure preserving map of the unit interval, are in the closure of $SDiff([0,1]^3)$. (Thus, even though we are interested in volume preserving maps, we have to open our eyes to all Lebesgue measure preserving map of the unit interval. This is a much much richer class than $SDiff$!)
Density of permutations in $VPM(D)$. Another interesting subset of $VPM([0,1]^3)$ is made of all “permutations” of all dyadic divisions of the unit cube in sub-cubes of equal volumes. You divide the buce into dyadic sub-cubes, like a Rubick’s cube. To every permutation, you define a permutation which shuffles the cubes. It turns out that the union of these permutations taken over all scales defines a dense set of maps in $VPM$! He shows some remarkable gifs where dust appears related to the orientation reversal.
Geodesic completeness. Big issue….global well-posedness of E.
Minimizing Geodesics. (Shnirelman Math USSR Sb 1986) The 3d case turns out to be “easy” with a crucial use of the convex structure of the dual problem. The case $d=2$ is clearly linked to symplectic geometry and seems extremely difficult: a fascinating strategy has been developed by Shnirelman, by adding braid constraints to the minimization problem, which certainly deserves further investigations.

Approximate Minimizing Geodesics:

fast slide…The existence of such approximations is in no way trivial and is a consequence of a key density result due to Shnirelman (GAFA 1994).

Main Theorem: Let us assume $D$ to be convex, with $ d \geq 3$, fix $t_0 = 1, ~ t_1 = 1$ and consider maps $M_0, M_1 \in VPM (D)$. Then there is aunique pressure gradient $\nabla p_t$ such that for all $\epsilon$-minimizing geodesics, we have in the sense of distributions $$ \frac{d^2 M_t^\epsilon}{dt^2} \circ (M^\epsilon_t)^{-1} + \nabla p_t \rightarrow 0, \epsilon \rightarrow 0.$$

Yann insists that this has “nothing to do with geodesic completeness”. I am confused…..

Minimizing Geodesics: Final Comments

Uniqueness of the pressure gradient. This is a remarkable feature of the theory. There is no equivalent result for finite-d configuration spaces such as $SO(3)$,on which geodesic curves (for appropriate metrics) correspond to the motion of solid bodies in classical mechanics. WE believe this strange phenomenon to be the consequence of the “hidden convexity” of the problem in dimension 3 and more.
Limited regularity of the pressure gradient.

Some references.

Second Part: From incompressible fluids to dust

Gravitating particles as a natural approximation of Euler incompressible fluids ….fast slides…..jet lag….fascinating….Yann is a fast thinker…

Penalization of the Euler Action

Use a penalty method to try to approximate “geodesics” on discrete sets using permutations.

Monge-Ampere (instead of Poisson) nonlinear correction to the classical Newton gravitation.

Rafael de la Llave, Georgia Institute of Technology,Arnold diffusion in a-priori unstable Hamiltonian systems of high dimension

(joint work with Delshams, de la Llave, T.M. Seara)

(Related collaborators: Elisaget Canalias, Marian Gidea, Gemma Huguet, Vadim Kaloshin, …)

Instability for a priori unstable Hamiltonian systems

We consider a periodic in tim perturbation of $n$ pendula and a $d$-dimensional rotor described by non-autonomous Hamitonian, $$ H(p,q, I, \phi, t , \epsilon) = P(p,q) + h(I) + \epsilon Q (p,q, I, phi, t, \epsilon) $$ with $$P(p,q) = \sum P_j (p_j, q_j), ~ P_j = \pm (\frac{1}{2} p_j^2 + V_j (q_j)). $$

Elemntary and regularity assumptions.

H1: Assume that the functions $h, V_j, Q$ are $C^r$ in their corresponing domains with $ r \geq r_0$ sufficiently large.
H2: Assume that the potentials $V_j$ have non-degenerate local maxima, say at $q_j = 0$, each of which gives rise to a homoclinc orbit of the pendulum $P_j$: They are penduli, they have critical points and have homoclinc connections.
“The enemies to this problem are the KAM and the Nekoroshev. Of course, they are my friends in other talks…”
H3: The mapping $I \rightarrow \omega(I) = …$ ack slide change.
H4: The function $Q$ is assumed to be a trigonometric polynomial. (This can be removed)

Remark: [Delshams-Llave-S06], [Delshams-Huguet09], [Gidea-Llave06].

Melnikov Potential:

Poncare-Arnold-Melnikov. This basically measures the effect of the perturbation on a homoclinic orbit at first order. (Big integral….too long to type this fast.)

H5: Assume that the system of equations $$ \frac{\partial}{\partial \tau} L(\tau, I, \phi, s) = 0 $$ admits a nondegenerate solution. This allows us to eliminate the $\tau$ in terms of the other variables.

Poincaré reduced function

H6:
H7: $Q$ satisfies some nondegeneracy assumptions.
H8: You don’t want the resonances to be flat.

….I can’t really keep up….so I will just watch.

Tenyson 83 Probed many mechanisms of diffusion by observing numerically.

Chirikov 82

Vadim Kaloshin, University of Maryland,Hausdorff dimension of oscillatory motions for three body problems

(blackboard talk)

I have a deep admiration of V.I. Arnold. He was a “god of mathematics and still is.” Here is a list of topics that have occupied my interest for research. They are all explicitly linked with ideas of Arnold.

My undergraduate thesis was on a topic called prevalence, a notion of probability one in infinite dimensional spaces. My project emerged from Arnold’s note that one could understand genericity through this notion.
Hilbert-Arnold Problem. This was my first problem I studied with Ilyashenko. This was motivated by the 2nd part of Hilbert 16th problem. You look at a family of vector fields on $x \in S^2, ~ \epsilon \in B^k$. You consider $\dot{x} = v(x, \epsilon)$. Generic fanily of $C^\infty$ v. fields has $LC(\epsilon) < \infty.$
Growth of the number of periodic points. $M$ is a compact manifold. You look at $f: M \rightarrow M; f \in Diff (M)$. You look at $P_n (f) = {\mbox{Number}} [x: f^n x = x]$. How quickly $P_n(f)$ generically?
Arnold Diffusion.

As you can see, most of my research is either inspired by or directly leads from questions suggested by Arnold as interesting directions.

Qualitative analysis of 3-body problem. Let $q_i \in R^d, ~ d =2,3$ are point masses. Each point has mass $m_i$. The Newton law then gives the dynamical law $$ m_i \frac{d^2}{dt^2} q = - \sum m_i m_j \frac{q_i - q_j}{|q_i - q_j |^3}. $$

Kepler motions:

2 Body problem (2BP). $m_0 q_0 + m_1 q_1 = 1$. $$ H(q, \dot{q}) = \frac{\dot{q}^2}{2} - \frac{1}{|q|} $$ Three cases for the 2BP:

$H<0$ either circular or elliptic.
$H=0$ parabolic; escapes to infinity with zero velocity.
$H<0$ hyperbolic; escapes to infinity with nonzero velocity.

Three Body Problem: (Sun-Jupiter-Comet)

Four types of motion:

B: $\sup_{\pm t>0} |q_i (t)| < K < + \infty$
Parabolic: escapes to infinity with zero velocity at infinity.
Hyperbolic: escapes to infinity with nonzero velocity at infinity.
Oscillatory: $\limsup_{t \rightarrow \infty} |q_i (t)| = \infty; ~ \liminf_{t \rightarrow \infty} |q_i (t)| < + \infty $.

What kind of behavior is possible in the future? What kind of behavior is possible in the past?

A famous result of (missed the names…) showed that there are solutions with any of these four behaviors in either direction of time infinity.

Shows a table. He reports that every one of the boxes (except one) in the table have been shown to have positive measure. He will focus on the case whether the situation with oscillatory motions in the past and in the future. We want to know whether this event has positive measure or not. There was a conjecture of Kolmogorov: He conjectured that this was expected to have measure zero.

Shows two papers by Alexeev. The French version has no attribution to Kolmogorov. The English version has attribution to Kolmogorov. Katok says you should attribute this to Kolmogorov. The English version was published after his death.

(joint work with A. Gonodetski; preprint)

Kolmogorov conjecture: $Mes OS = 0$.

Main Result 1. Often 2 degree of freedom 3 body problem have Hausdorff Dimension maximal possible.

Leading Idea: Build a “fat” Cantor set $\Lambda \ni \infty$ with ergodic dynamics.

If one could produce an ergodic component with positive measure, one could perhaps prove a counterxample to Kolmogorov’s conjecture.

Second version of the main result:

Remark: 2 degrees of freedom Hamiltonian dynamics locally reduces to a 2 dimensional area preserving map. My analysis will concern those maps since they have some advantages, for example I can draw pictures.

Newhouse domains in dissipative setting. Suppose $f:M^2 \rightarrow M^2$. Suppose $f$ has a homoclinic tangency (HT). If $f$ has a saddle point then $f^k p = p$ such that unstalbe and stable manifolds satisfy….. RETURN HERE….Newhouse domains…..

Main Result 2: 2 dof 3BPs have Newhouse domains.

Remark: Duarte proved a conservative Newhouse phenomenon. (20 year interval between Newhouse and Duarte.)

Newhouse domains give rise to striking dynamical examples.

1st Model (Sitnikov): There is a beautiful book by Moser which gives an example of oscillatory motions. You have two bodies $q_0, q_1$ in elliptic orbits with eccentricity $e$. The masses $m_0 = m_1 = 1.$ The third body $q_2 = (0,0,z)$ lies on the $z$ axis. $$ H(t, z, \dot{z}) = \frac{\dot{z}^2}{2} - \frac{1}{\sqrt{z^2 + r_e^2 (t)}}. $$

Theorem 1 (GK): $\exists$ open nonvoid $\cal{N} \subset (0,1)$ such that for a generic $e \in \cal{N}$ we have $HD(OS) = 3$. $\cal{N}$ is a subset of a Newhouse domain.

2nd Model (Restricted planar circular 3BP): $m_2 = 0$ (restricted, comet). $q_0, q_1$ move in circular orbits. Motions are planar. In a rotating frame, these masses are fixed. Set $m_0 + m_1 =1$ and choose $m_0 = \mu, ~ m_1 = 1 - \mu$. The parameter $\mu$ is called the mass ratio. Form the so called Jacobi constant $J(x,y, \dot{x}, \dot{y}) = \frac{\dot{x}^2 + \dot{y}^2}{2} - [\frac{ {x}^2 + {y}^2}{2} + \frac{1-\mu}{d_0} + \frac{\mu}{d_1}]$.

(Here $d_0$ represents the distance from $q_0$ to $q_2$ measured in the rotating frame. $d_1$ relative to $q_1$. )

Theorem 2 (GK): $\exists ~ J_0$ such that $\forall ~ J > J_0$ then $\exists ~ \cal{N}J \subset (0,1)$ with property generic $\mu \in \cal{N}J$ and $HD(OS \cap J(…) = J^*) = 3.$

Meta Theorem. Let $[H_\delta]$ be a 1-parameter family of Hamiltonian systems of 2 dof (or 1.5 dof) satisfy Hypothesis:

H1: As $\delta \rightarrow 0$, the limiting Hamiltonian $H_0$ is integrable with a separatrix loop.
H2: For $\delta \neq 0$, we want the separatrix to split tranversally.
H3: Melnikov function satisfies a certain open condition.

Then for a generic $\delta$ in an open nonempty set, $H_\delta$ has a hyperbolic (nonzero Lyapunov exponent) set of HD = 3.

Ideas from proof (Sitnikov):

$e = 0, ~ H = \frac{\dot{z}^2}{2} - \frac{1}{\sqrt{z^2 + 0.25}}$. He draws a picture in the $(z, \dot{z})$-plane and identifies the region $H<0$ and the region $H>0$ and highlights the “separatrix loop”. He glues the points at infinity at $ z = \pm \infty$ to highlight this as a separatrix loop. Constructing oscillatory motions corresponds to building orbits that come arbitrarily close to this point.

$C^2 - \lambda$ - Lemma…. why do we need this? We need quadratic tangency. This is an explicit system so we can’t use genericity.

Marc Chaperon, Université Paris 7,Generalised Hopf bifurcations

It’s a great honor to be here. I admired Vladimir Arnold very much. We liked each other. What I will speak about appears in the MMF v11(3) in memory of Arnold. The reason I became a mathematician was because of Thom but the reaosn why I persisted was probably because of ARnold. It was amazing how much energy he had. When he came to Paris, he knew more about it than I did, more than most natives. He was some kid of wunderkind and remained so his whole life.

Motivation: interest in the coupling of oscillators.

Chenciner-Iooss 1979

….I’m a bit tired so stopped typing.

Anton Zorich, University of Rennes,Lyapunov exponents of the Hodge bundle

(joint work with Alex Eskin and Maxim Kontsevich)

I am jealous towards my colleagues. I can’t claim this work was motivated by work of Arnold. But I can report that he was constantly interested in this topic. I enormously regret that, now that the story is complete, I can not tell it to Arnold.

Motivations. Consider a billiard in the plane with $Z^2$-periodic rectangular obstacles.

Theorem (Delcroiz, Hubert, Lelivre 2011): For almost all parameters of the problem, the billiard trajectory ecapes to infinity with a rate of $t^{2/3}$.

How can we capture this $2/3$? The obstacles that can appear in this story must involve rectangles.

Exponents like this have appeared in work by Giovanni Forni.

Geometric interpretation of multiplicative ergodic theorem:

Consider a vector bundle endowed with a flat connecton over a manifold $X^n$. Having a flow on the base, we can take a fiber of the vector bundle and transport it along a trajectory of the flow. When the trajectory comes close to the starting poitn, we identify the fibers using the connection and we get a linear transformation of the fiber. The multiplicative ergodic theorem says that when the flow is ergodic a “matrix of mean monodromy” along the flow.

Moduli spaces of Abelian differentials.

Abstract version and a concrete version…..slides are pretty dense and mving a bit fast.

Teichmuller discs.

Teichmuller geodesic flow:

Teichmuller geodesic flow acts in the modulie space of pairs (complex structure, holomorphic quadratic differential.) Away from the zeros of a quadratic differential $q$ one can find a local coordinate $z$ on the underlying Riemann surface in which $q = (dz)^2$. This distinguished local coordinates defines a flat metric $|dz|^2$, which has a canonical singularities at the points where the quadratic differential has zeroes. Teichmuller geodesic flow acts as a uniform contraction in teh vertical direction and unform expansion in the horizongal direction.

(This is like Asteroids on a much richer surface than the 2-torus!)

The unraveled quotient space can be viewed as a polygon with parallel sides identified. There are some rich combinatorics available by cutting and regluing.

Hodge bundle and Gauss-Manin connection:

This reduces things down to $g-1$ Lyapunov exponents. To compute these exponents appear to be out of reach in most every dynamical system.

Siegel-Veech constant:

Closed regular geodesics on flat surfaces appear in families of parallel closed geodesics sharing the same lenght. Every such family fills a mximal cylinder having conical points on each of the boundary components. Denote by $N_{area} (S, L)$ the sum of areas of all cylinders spanned by geodesics of length at most $L$.

Theorem (Veech-Vorobets): For every $SL(2, R)$-invariant finite ergodic measure the following ratio is constant (ie.e does not depend on the value of a positive parameter L): $$ \frac{1}{\pi L^2} \int N_{area} (S, L) d\nu_1 = c_{area} (d\nu_1 ). $$ (The integration here is over the entire family of flat surfaces. Here $\nu_1$ is the invariant measure on the space of these surfaces.)

The constant $c_{area}$ is called the Siegel-Veech constant.

Eskin-Masur have a similar theorem for fixed $S$ where you take the limit $L \rightarrow \infty$.

What happens for the torus? For most of the tori, you can’t find closed small geodesics. However, inside the family of flat tori, there are some with very narrow cylinders with closed geodesics.

Eskin-Masur-AZ

Eskin-Okounkov computed the volumes explicitly.

Main Theorem: The sum of the Lyapunov exponents can be expressed as a sum of two (explicitly computable) constants.

V. Knizhnik

This stuff is amazing, truly beautiful. But, I can’t keep up with the typing….

Big advance by Eskin-Mirzakhani is in redaction.

Wednesday 05 October

Jacques Féjoz, Université Paris- Dauphine & Observatoire de Paris,Diffusion along mean motion resonance in the restricted three-body problem

(blackboard talk)

(joint work w M. Guardia, V. Kaloshin and P.Roldan)

This work would not exist w/o the marvelous 1964 paper of Arnold. This talk is closely related to the talk that Rafael gave yesterday and also to Vadim’s talk. I will try to concentrate on other aspects.

In the solar system, there is one priviledged place where we should look for instabilities. It is called the Asteroid Belt. It is located between Mars and Jupiter. Dust particles in this part of the solar system never condensed to form additional planets. Instead, there remain nearly 2 million particles ranging from microscopic to larger asteroids, some having a size of a few hundred kilometers in diameter. If you look to the current distribution of the asteroids in this belt, it gives a quite precise idea about the stability and instability zones between Jupiter and Mars.

In 1857, an American mathematician and astronomer named Daniel Kirkwood observed that there are gaps inside the belt where there are basically no asteroids and other zones where there are lots of asteroids. These gaps correspond to orbital resonances with Jupiter. Since the orbital frequency can be read off from Kepler’s third law….he draws a graph with vertical axis as the number of asteroids and the horizontal axis is the semi-major axis between the Mars and Jupiter radii. He highlights a gap appearing at the zone located in 3:1 resonance with the Jupiter orbit. Why are there these gaps? The conjectural explanation is that an asteroid is in resonance with Jupiter (which has a mass of about 1/1000 of the mass of the sun) then the eccentricity will be unstable. If the eccentricity goes through large variations, then its perihelon will be at size $a(1 - e)$ from the sun. Therefore, the asteroid will get closer and closer to Mars. Due to this very close encounter with Mars, the dynamics will transform so that the principal force acting on it will be due to gravity from Mars rather than with the sun. In this talk, I would like to focus on the first step in this scenario. Namely, why should the eccentricity vary a lot when the asteroid is in resonance with the Jupiter orbit?

Planar restricted 3-body problem: Sun, Jupiter, Asteroid. Restricted means we take the limit when $m_{asteroid} = 0$. Practically speaking, this means that we imagine the Sun and Jupiter take place along the 2 body motion and they are not influenced by the motion of the asteroid. Let’s normalize two things. Set $\mu = mass_{jupiter}$ and the mass of the Sun is $1 - \mu$.

There are 3 “small” parameters.

The mass of Jupiter $\mu \rightarrow 0$. The asteroid motion collapses then to an integrable 2 body problem.
The eccentricity $e_0$ of Jupiter. This parameter is slightly more subtle. The limiting dynamics as $e_0 \rightarrow 0$ is not integrable. We then obtain the circular restricted problem in which the two primary objects orbit on a circle centered at the center of mass. The problem restricts from 2.5 dof down to 2 dof.
Semimajor axis $a$ of the Asteroid. When we let $a$ go to zero, or to infinity, we encounter 2 body problems. In one limit, the Sun dominates the asteroid motion and Jupiter is irrelevant. In the other limit, the asteroid essentially sees the gravity of a combined mass of the Jupiter and Sun.

In the problem inside our solar system:

$ \mu = \frac{1}{1000}$
$a = (\frac{p}{q})^{2/3}$. This implies that the periods of the asteroid and Jupiter satisfy $ \frac{T}{T_j} = \frac{p}{q}$.
We will restrict attention to $0 < e_0 \ll 1$. This allows us to view the problem as a singular perturbation of the restricted circular problem. There is also a computational reason for doing this. Part of the proof will require some numerical computations. Our strategy was to make these calculations as simple and convincing as we possibly could. All the numerical computations are done on the circular problem and boil down to checking for zeros of a one variable function. A final reason is that the real eccentricity of Jupiter is $\frac{1}{20}$. There is hope that we could in fact claim our theorem for the real value. This will require some quantifications which in principle we could extract.

Theorem: Set $\mu = \frac{1}{1000}$. Fix $\frac{p}{q} = 7$ (chosen for incidental reasons; we expect this can be relaxed; so this asteroid is outside of Jupiter corresponding more closely with Uranus….nice discussion). Assume $0 < e_0 \ll 1$. There exists a solution and a time T with $e(0) < e_{min} = 0.48$ and $e(T)> e_{max} = 0.67$ and all the while the asteroid radius $a(t) \thicksim (\frac{p}{q})^{2/3}$. Thus we have a $(p:q)$ orbital resonance with Jupiter.

What is the time scale of $T$? Conjecturally, we have $T \thicksim - \frac{\ln \mu e_0}{\mu^{3/2} {e_0}}$.

What are the units of time? Year of Jupiter.

Circular Problem:

$$ H = \frac{|p|^2}{2} - \frac{1}{|q|} + \frac{1}{|q|} - [ \frac{1-\mu}{|q + \mu|} - \frac{\mu}{|q - (1 -\mu)|}]. $$ This formulation views the principal force as provided by a fictitious mass at the origin perturbed by the separation of the Jupiter and Sun masses.

Delaunay Coordinates (written in notation of Poincaré): $(L, l, G, g) $

$L = \sqrt{a}$
$G = \sqrt{a}\sqrt{1 - e^2}$ (angular momentum)
$g$ is an angle to Jupiter.
$l$ is the angle of the asteroid advanced past Jupiter.

If we set $\mu =0$, the perturbing term vanishes and we are left with a degenerate 2BP. Understanding this limit does not bring much light to the problem.

Assume $\mu >0$. The first natural idea is to average out the fast orbital angle remaining inside this Hamiltonian. This involves first making a change of variable. The averaging process leads to an integral. This is a transcendent process. However, when $e_0, e \ll 1$, you can use the Laplace coefficients to make some computations by hand. However, this computation is not much help because we are interested in proving diffusion in the eccentricities. This calculation does give some intuition which suggests that the perturbation looks generic and the degeneracy observed in teh $\mu =0$ limit is broken.

Fact (Numerical): There exists a normally hyperbolic cylinder foliated by periodic orbits $\gamma_e$, $0.48 < e < 0.67$. This is related to an idea of R. Moeckel from the late 90s. He draws some surfaces intersecting and explains that there are some small splitting issues requiring high precision arithmetics. We limited our attention to higher eccentricities to avoid these issues. Morally, the larger value of eccentricity the faster the diffusion. As we go above $0.67$ we get closer to the Euler relative equilibrium $L_{1,2}$.

In order to lower the number of dimensions, it is perhaps a good idea to consider a Poincaré return map to $[g = 0]$. This means we are looking at what happens every time the ellipse of the asteroid is aligned along the line connecting the Sun and Jupiter. He draws a helix of 7 levels high above an ellipse. Then he slices the helix with a vertical plane and identifies the intersection points as 7 normally hyperbolic invariant cylinders. It is now time to introduce the analogs of the “inner” and “outer” maps that Rafael introduced in the more general case yesterday.

OK, mostly pictures now….

Discussion: There is no steepness in this Hamiltonian so Nekoroshev’s theorem does not apply.

Boris Khesin, University of Toronto,Optimal transport and geodesics on diffeomorphism groups

(blackboard talk)

(joint work with J. Lennels, G. Misiolek, S. Preston)

Plan:

Euler equation on $SDiff$. Otto’s calculus.
$SDiff \subset Diff$ (numerous applications in optimal transportation)
$L^2$ and $H^1$ metrics.

I. Arnold’s approach to the Euler equation

This was discussed in Shnirelman and Brenier’s talks so I can perhaps be brief.

Definition: Consider $v$ to be a velocity field on a manifold $(M, (,))$ which is divergence free. The Euler equation is then given as $$ \partial_t v + (v \cdot \nabla) v = - \nabla p, ~ \nabla \cdot v = 0. $$ Here $\nabla$ is the covariant derivative. If there is a boundary, we assume that $v$ is parallel to $\partial M$.

Draws a picture and identifies the Lie algebra $g$ at the identity. He explains how to use Lie algebra transportation along a geodesic in $G$.

Theorem (Arnold, Bombshell in the 60s): The Euler equation may be viewed as a the geodesic equation on $G = SDiff (M)$ (the group of volume presernving differomorphisms) w.r.t right invariant energy $L^2$-metric given on $g = Lie(G)$ by $$ E(v) = \frac{1}{2} \int_M (v,v) \mu.

Remark: Other groups and energies give Euler top, Kirchoff equations for motion of rigid body in a fluid, KdV, Camassa-Holm, MHD, Landau-Lifschitz equation, …

II. Geometry of full diffeo group

This is the point of view rather common in optimal transport. This discussion unifies these two perspectives.

Consider $Diff$, the group of diffeomorphisms on $M$. Inside this group, we have the subgroup of volume preserving diffeomorphisms $SDiff$. The notion of $SDiff$ requires the volume form. Ebin-Marsden. He views $Diff$ as a space of fibers over the space of densities. Once you specify the volume form, this induces the fibration over the densities. Fibers $=F_\nu = [g \in Diff: g_* \mu = \nu]$.

$\exists$ “natural” $L^2$-type metric on $Diff$ for flat $M$: $$ l^2 [g(t, \cdot)] = \int_0^1 ( \int_M (\partial_t g, \partial_t g) \mu) dt. $$

Remark: $\forall ~M$,

$$ (v \circ g, v \circ g){L^2} = \intM (v \circ g, v \circ g)_{g(x)} \mu(x). $$

Properties:

Not right invariant on $Diff$
Is right invariant on $SDiff$, because the Jacobian term arising from the change of variables disappears.
“flat” for a flat $M$: $Diff \thicksim L^2 [ g(x)]$…pre-Hlibert.
geodesics in $Diff(M) \iff $ solutions of the Burgers equation: $$ \partial_t g (t,x) = v(t, g(t,x)); ~ \partial_t v + (v \cdot \nabla) v = 0. (*) $$ This differs from the Euler equation since the right side is zero. It also does not require the zero divergence condition.This formulation appears in the paper of Ebin-Marsden.
Geodesics which are orthogonal to $SDiff \iff $ potential solutions $v_0 = \nabla \phi$.

The proof of (*) follows from the chain rule and the fact we are considering geodesics. $$ 0 = \partial_t^2 g = \partial_t (v(t, g(t,x)))$$ $$ = (\partial_t v + (v \cdot \nabla v))(t, g(t,x))$$.

Remark: Geodesics on $SDiff$ are constrained within the larger family $Diff$ to remain on the subgroup. This requires imposing a force to keep the evolution within $SDiff$. This force is the pressure.

What I am describing right now is called Otto’s Calculus which arose in the study of optimal transportation.

Remark: It turns out there is a natural metric on the space of densities. We can introduce a measurement of the cost to move one density $\mu$ to another $\nu$. The natural metric is called the Wasserstein-Kantorovich $L^2$-metric on densities: $$ dist(\mu, \nu) = \inf_{g_* \mu = \nu } \int_M |x - g(x)|^2 \mu(x). $$ This is the cost of transporting $\mu$ to $\nu$.

Theorem (F. Otto): $$(Diff, L^2) \longmapsto (Densities, dist)$$ is a Riemannian submersion.

Corollary: Geodesics in the space of densities starting at $\mu$ are in 1:1 correspondence with horizontal geodesics in $Diff$ starting at the identity.

This is the picture behind the scenes driving the proofs of many theorems.

Applications:

Conjugate points along the base of densities correspond to focal points inside the space $Diff$. P. Lee, A. Agrachev and a former student (I missed the name…)
Asymptotic Directions $\iff$ geodesics with higher than ususal $(\epsilon^3)$-tangency. Theorem: Asymptotic directions to $SDiff$ must satisfy $$\nabla \cdot v = 0,$$ $$ \nabla \cdot (v \cdot \nabla) v = 0.$$ (These are called the Bao-Ratiu equations and arise naturally from this perspective.)

Theorem (K-Misiolek): For $M$ of dimension 2 (surface), ${\overline{K}} \neq 0, ~ \forall x \in M$ there do not exist asymptotic directions. (For $K > 0$ this result was called Palmer’s theorem.)

So asymptotic directions are “rare.”

What happens if we consider slightly more general metrics instead of $L^2$. Recently, there was interest in $H^1$ metrics so let me say a few words about that.

dimension	1	2	3
$SDiff$	Rot	$H(x,y)$	$[Jac = 1]$
Density	$[\hat{f}(x)]$	$[f(x,y)]$	$[f(x,y,z)]$

III. $H^1$-(right-invariant) metrics on $Diff$

article

$$ (v,v){H^1} = a | v |{L^2}^2 + b | \delta v^\flat |{L^2}^2 + c | d v^\flat |{L^2}^2 $$

where $ v \in Vect \rightarrow v^\flat \in \Omega^1 (M)$. So, we have terms involving $\nabla \cdot v$ and another involving $curl v$, etc.

Theorem: The Euler-Arnold equations are

$(n = 1)$

$b = 0 \implies$ Burgers (KdV for Virasaro)
$a = b = 1 \implies $ Camassa-Holm equation: $v_t - v_{txx} = -3 v v_x + 2 v_x v_{xx} + v v_{xxx} $\nabla \cdot
$a=0 \implies$ Hunter-Saxton equation: $u_{txx} = - 2 u_x u_{xx} - u u_{xxx}$

For other $n$, one can write the corresponding equation. Various other equations arise such as Euler-$\alpha$ and many others….

There exists one metric which has nicer properties than others.

IV $H^1$-metric on Densities

$(a = c = 0, b = \frac{1}{4} \neq 0)$. So, we are considering the metric $\| v \|_{\dot{H}^1}^2 = \frac{1}{4} \int |\nabla \cdot u |^2 \mu.$

Theorem: For any compact $M$ there exists an isometry $\Phi: Densities \rightarrow U \subset S_\rho^\infty$ (an infinite dimensional sphere) where $\rho = \sqrt{vol(M)}$.

Remark: The dimension 1 case was observed by Lennels. At first, we thought this was a special case but turns out to be general and produces some nice insights.

Geodesics on densities are great circles on the sphere.
They are solutions of a high dimensional Hunter-Saxton equation (completely integrable system)

Proof: $\Phi: \eta \in Diff \rightarrow f = \sqrt{Jac (\eta)}$. Then $$ \int_M f^2 \mu = \int_M (Jac (\eta)) \mu = \int_{\eta(M)} \mu = vol(M)$. $$ This very metric on the sphere arised earlier in probability theory and is known as the Hellinger distance, aka Fisher-Rao metric. All these objects come together from this point of view.

article

Bassam Fayad, IMJ CNRS,Smooth linearization of commuting circle diffeomorphisms

(blackboard talk)

My work was completely influenced by Arnold’s papers and ideas. I had very close friends who were his students. They were completely venerating hi. I like to think that I am representing them here. They are two. These students report that what they miss most is long discussions with Arnold who had patience and knowledge and he fills you with interest.

I work on small denominators.

Mixing tori. Impossible on $T^2$. He drew a cube representing a 3-torus. The statement was not so explicit.

He connects this to the original motivation of Kolmogorov which revolved around an interest in finding mixing.

Circle Diffeomorphisms:

$$ f(\theta) = \theta + \tilde{\alpha} + \phi(\theta) $$

$\rho (f) = \alpha$ is the (irrational) rotation number.

Denjoy theory. If $f \in C^2$ then $ f = h R_\alpha h^{-1}$ where $h$ is a homeomorphism of the circle.

What is the regularity of the homeomorphism?

Siegel. Arnold.

Arnold showed that if $\alpha$ is Diophantine and $f$ is close to $R_\alpha$ then $ f $ is analytic and $h$ is analytic.

What is diophantine?

$\alpha \in DC (\gamma, \tau)$ if $ |\alpha - \frac{p}{q}| > \frac{\gamma}{q^{2 +\tau}}$
Best approximations. $\| k \alpha \| = \inf_l | k\alpha - l|$. The sequence of best approximations is defined by $$ \| q_n \alpha \| < \| q \alpha \|$$ for all $ q < q_{n+1}, ~q \neq q_n$.

Linearized equation:

$\phi (x) = \psi (x + \alpha ) - \psi (x)$.

The global problem remained open and was conjected by Arnold. Even without the closeness condition, Arnold conjectured that the rotation number being diophantine was all that was required to ensure the analyticity of the homeomorphism.

Herman 1976 ($H$-class of numbers), Yoccoz 1981 (all Diophantine numbers)

If $f = h R_\alpha h^{-1}$ with $h$ a homeomorphism and $\alpha$ is irrational and you have $f \circ g = g \circ f$ then $g = h R_{\rho{g}} h^{-1}$. (My quotation of the Qualifiers might be wrong here….be CAREFUL….)

Why does commutation imply higher regularity, more rigidity? The idea emerges from a paper by Moser 1981 who proved KAM smooth linearization of $f,g$ commuting if $(\alpha, \beta) \in SDC$

$(\alpha, \beta)$ are SDC (Simultaneous Diophantine Condition) if $\max(|k \alpha|, | k \beta |) \geq \frac{\gamma}{k^\nu}$.

Applying these techniques, you can show: If $(\alpha, \beta) \notin SDC$ then $\exists ~ (f,g)$ commuting then $h$ is not absolutely continuous.

Theorem (K. Khanin, F): $(\alpha, \beta) \in SDC$ and $ f \circ g = g \circ f$ in $Diff^\infty \implies h \in Diff^\infty$. KF Paper: Annals 2009

Very clever pivots in a case-by-case analysis. Some pigeon holes. Make friends with your enemy.

Thursday 06 October

Chong-Qing Cheng, Nanjing University,One way to Cross Complete Resonance

Nice introductory discussion of Arnold Diffusion, placing the principal settings studied so far in context. Mentions that there is a “definition” of Arnold Diffusion in v3 of Arnold’s book.

Some nice pictures suggesting the mechanism.

A big issue to overcome is that there are uncountably many barrier functions. One way is to study the regularity. This will impliy the finiteness of the Hausdorff dimension.

Resonance path. KAM iteration at complete resonant point. Very nice pictures of the Aubry set!

Interesting discussion following the talk between Cheng and Mather, comparing their respective strategies.

James Ellis Colliander, University of Toronto,Hamiltonian PDEs

I spoke so I didn’t type.

Artur Avila, IMJ, CNRS, Global theory of one frequency Schrödinger operators

(blackboard talk)

This topic can be introduced in several ways. I try to present this work in a way that is connected to the work of Arnold.

KAM-persistence of quasiperiodic motion.

One theorem of Arnold: $f$ analytic diffeo of $T = R/Z$ orientation preserving has a rotation number $\rho$.

Theorem: If $\rho$ is Diophantine and $f$ is close to translation then $f$ is linearizable (analytically conjugated to tranlation).

He also makes a conjecture. This should be global. This means that the hypothesis “f close to translation” should not be necessary. This was proved by Herman in a breakthrough work introducing new techniques, and eventually completely resolved by Yoccoz. These results are now understood in a new framework called renormalization.

What is the situation in higher dimensions? Not every $T^2$ has a notion of translation number. Suppose we have a diffeo that has a rotation number and is close to translation. What can be said? In higher dimensions, the local theorem survives. Herman asked: which aspect of the global theorem will survive? It’s a paper of Herman with a very long title…

“Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnolʹd et de Moser sur le tore de dimension 2.”

Example:

$$(x,y) \longmapsto (x + \alpha, A(x) \cdot y)$$

$A(x)$ will be some projective action. In particular, I will imagine that $A(x) \in SL(2;R)$. So, I am viewing the second coordinate as an element of $PR^2$.

$A(x)$ is a matrix ($E - \lambda v(x), -1$) (top row) and (1,0) in bottom row. Here the parameters are $E \in R$, $v$ is a trig polynomial, $\lambda >0$.

Rotation number?

In the first slot, it is clear that the number is $\alpha$. In the second slot, it depends on all the parameters. It turns out that in this case, it is well-defined and is montonic wrt $E$. Diffeos of the circle have those regions called Arnold Tongues and there are similar structures here…rationality condition…draws a Cantor-like set. He draws a “vertical curve” in the $(E, \lambda)$ plane and along that curve we have $\rho = constant$.

Theorem (Herman):

$\lambda \ll 1$, KAM works
$\lambda \gg 1$, Lyapunov exponent $LE >0$. Independent of E

$(\alpha, A)^n = (n \alpha, A_n)$. $A_n (x) = A(x + (n-1) \alpha) … A(x)$.

$$L = \lim \frac{1}{n} \int \ln | A_n (x ) | dx \geq 0. $$

If $(\alpha, A)$ is conjugate to translation then $LE = 0$.

OK, so what is the obstruction to globalization? IS $LE$ the only obstruction to conjugacy?

$LE = 0$, continuity argument. Goldstein-Schlag, Bourgain-Jitormskaya

At the endpoint of the supremum of the good parameter, we can not have analytic conjugacy. What broke down? You might think it is just that we lose analyticity. But this turns out to not be the case because in this context topological conjugacy $\implies$ analytic conjugacy. So, it is possible to have $LE =0$ while losing even the topological conjugacy.

$A$ is analytic, extends to a neighborhood $[|\Im x | < \epsilon]$. $A_n (x)$ is defined on the same neighborhood. Maybe the LE changes a bit as we move off the real axis? You see that if $(\alpha, A)$ is analytic conjugate to translation then this kind of complexified LE $$ \lim \frac{1}{n} \int_{Im x = a} \ln | A_n (x) | dx = 0. $$ is still zero. Now, you have some kind of necessary condition that is implied by analytic conjugacy.

Theorem: $(\alpha, \rho)$ diophantine. $(\alpha, A)$ is analytically conjugate to translation $\iff ~ LE =0$ for $|\Im x|$ sufficiently small.

The method of proof of this theorem is quite interesting, but I won’t do that right now. You can’t just apply KAM theorem. It is closer to a different theorem. You need an a priori bound on the renormalization. In fact, you have that the complexified LE is behaving subexponentially…lots of work to be done to get conjugacy. What I want to do instead is to connect with the title of the talk.

$l^2 (Z)$

$ (Hu)n = u{n+1} + u_{n-1} + v(n \alpha ) u_n. $

$$ v: T \rightarrow R, ~ analytic. $$

$$ Hu = Eu. $$

This generates a one-parameter family of cocycles. $H \iff $ one-parameter family of cocycles. Those cocycles have been studied a lot for similar reasons why dynamicists should interested in cocycles. They are the simplest class with $LE>0$, are consistent with KAM theory, and have rich dynamical behavior (e.g transition from absolutely continuous and discrete spectra, etc.)

(spectrum should usually be thought to be a Cantor set, similar to the situation of those Arnold tongues.)

Local Theories:

When $v$ is small. This is often quantified by writing $\lambda v$ with $\lambda$ small. This was developed by Dinaburg-Sinai, Eliasson, Bourgain, Jitormiriskaya, Avila, Fayad, Krikorian, …. From the beginning, it started with $\lambda$ small so that you can apply KAM theory. An end conclusion here is that the spectral measures are absolutely continuous. sigh… What does this imply? You start with some state that lives in this lattice and you let it evolve. What happens here is that it spreads at the fastest possible transport. This is called ballistic motion, so it moves like the free problem.

Eliasson: not the same speed?….Avila:….in average over time, you can see that it is the same speed whenever there is continuous spectrum. Eliasson:….oh you time average, I see…. Craig:….seems you need some smoothnes. Avila:..(eagerly)….which kind of smoothness do you need? Craig:….need to integrate by parts. Avila:….I have some weak smoothness. Craig:….maybe can help. Avila:…I expect it will help but don’t know how to use it yet.

When $\lambda \gg 1$, there is a different theory. Sinai, Frohlic-Spencer, Eliasson, Bourgain, Goldstein, Schlag. This theory corresponds to the situation where $LE > 0$. (Typically…some almost every conditions), the spectral measure is pure point. This means that the infinite matrix is diagonalizable and the quantum dynamics is quasiperiodic. It is known that the eigenfunctions decay exponentially, so they are better localized than required by $L^2$. Two approaches to this: KAM and more recent interactive techniques like renormalization. With Bourgain, Goldstein, Schlag, new nonperturbative techniques developed based upon the assumption that $LE >0$. It might be possible to understand the dynamics across the entire parameter space.

He draws an egg. He draws a line along the egg representing strength of nonlinearity. When the nonlinearity is small, we KAM-like behavior. Both regions can be shown to be open. When the nonlinearity is big, we $LE >0$. The egg has an “region” in between. Does that intermediate region have a non-empty interior?

Natural global questions:

What is the behavior of typical one frequency Schrodinger operator?
In particular, is there an influence of other behaviors of cocycles?
You might be optimistic and hope to prove there is some kind of phase transition between the KAM-like and $LE >0$ regimes?
Describe the phase transition as “interface-like”? This would go in the direction of showing that there are not these other types of dynamics of cocycles.

This was basically blocked for some time. But, recently, well maybe not recently, it was in 2008, there emerged some new ideas to approach these questions. Large parts of the emerging program have been carried out.

Center your attention on $LE$ and its dependence upon parameters. It’s good to get some kind of target to focus your attention upon. There will be three regimes:

Supercritical: $LE >0$ (leads to localization; point spectrum)
Critical: otherwise
Subcritical: $LE = 0$ in a complex neighborhood $[|Im x| < \epsilon]$. (show that it is KAM-like; AC spectrum)

The main parts of the program. Study the critical “interface”. Study how it intersects one-parameter families and so on. This would be a kind of geometric approach to begin to understand the dynamics across the parameter space.

What parts are completed? Several parts…..here is a main theorem.

Theorem: $H$ is a typical one-frequency Schrödinger operator. $H = H_+ \oplus H_-$. $\sigma(H_+) \cap \sigma(H_+) = \emptyset$. $H_+ $ is supercritical, localization. $H_-$ is subcritical and KAM-like, AC spectrum.

What about the growth of critical energies? There are no critical energies. This is a bit surprising. As I said before, I had a picture moving up from the KAM region and we encounter a transition point. It would be natural to expect that you will face a critical energy. This turns out to not be the case. The critical interface has zero measure, a kind of Cantor set, inside a codimension 1 subspace. This means it won’t intersect a typical one-parameter family. All this takes place inside the geometric analysis of the parameter space.

$\alpha$ diophantine, subcritical is $KAM$-like. Even thought I don’t have the conjugacy, I still have good control on the dynamics. This is the concept of almost-reducibiilty.

What is the meaning of typical? $\alpha$ is almost every. We also have $v$. What’s a zero measure set in infinite dimensions? It involves some kind of concept called prevalence and Gaussian measures. My good set of $v$’s has the following property: $\forall ~ v_0, $ for almost every $\lambda_n \in [-1, 1]^Z$ then $v_0 + \sum \lambda_n^{\epsilon_n} e^{2 \pi i n \cdot x}$ where $\epsilon_n = \frac{1}{(n!)^2}$.

Mikhail Sevryuk, Russia Academy of Sciences, The reversible context 2 in KAM theory for lower dimensional tori

A review of KAM theory. “Meta-reason” for the ubiquity of invariant tori carrying conditionally periodic motions: any finite dimensional connected compact Abelian Lie group is a torus.

Various structures (contexts):

Hamiltonian
reversible
volume preserving
dissipative (no structure at all)

Key advances highlighted here.

Moser 1966
Herman 1988

Lai-Sang Young, New York University,Toward a smooth ergodic theory for infinite dimensional systems

(Scanned hand-written marker slides! Cool….we party like its 1999!)

Aim of larger project:

Extend finite-d nonuniform hyperbolic theory (=ergodic theory of chaotic systems) to $\infty$-d
New phenomena
include settings related to some PDEs. (principles should include nonempty set of PDEs.)

The body of finite-d stuff that I have in mind does not include Hamiltonian systems. Instead, we are looking at problems which include some dissipation. So the invariant sets we are looking at are like attractors, etc.

Today’s talk:

Reduction to finite-d via $W^c$-inf and $W^\epsilon$-foliations. Upshot: notion of “a.e.” initial conditions in $\infty $ dimensions.
Example of strange attractors from Hopf bifiurcation + forcing. Illustration of how to leverage finite-d techniques
Lyapunov exponents, periodic orbits and horseshoes for semiflows on Hilbert spaces (extend Katok’s results for finite-d diffeos.)

Some background info:

Givne an evolutionary PDE, view this as an “ode” on a function space. I want to see it as a dynamical system.

$$ \frac{du}{dt} + Au = F(u) $$ where $u \in X, ~ A $ is a linear operator, $F$ is the nonlinear part.

To define (smooth) dynamical system, need $(X, | \cdot |)$ such that

$u(0) \in X \implies u(t) \in X ~ \forall t \geq 0$,
$ t \longmapsto u(t), ~ t \geq 0$, continuous,
Smoothness of the time-t map $f^t : (X, \| \cdot \|) \longmapsto (X, \| \cdot \|)$ which maps $ u(0) \longmapsto u(t)$.

for nonunif hyperbolic theory, generally require $C^{1 + \alpha}$.

Model setting:

$X$ is a Banach space. $A$ is an operator on $X$. Assume $A$ is “sectorial” or self-adjoint w eigenvalues on $[a, \infty)$.

Q: Can I just think of $A = \Delta$? A: Yes. (OK, I’ll think that way….knowing that there are generalizations.)

A sample result:

Theorem (Henry ~80): …

Discussion….skip it….just know that we are not talking about an empty set.

“Solution” means mild solution.

I. Reduction via center manifolds

$W^c$ can be local, global, or “medium size”

Constantin-Foias-Nicolanenko 86, Chow, Sell, Mallet-Paret, Lu, …

Think of $f$ as the time one flow-map associated to this dynamical system on $X$.

(A1) Reference Splitting: $X = E^c \oplus E^s$, closed subspace, not invariant. (A2) Absorbing “slab”: $\forall ~ R ~ \exists ~ R’$ such that $f( E^c \times B^s (0,R)) \subset E^c \times B^s (o, R’)$. (A3) INvariant cones ….lots written on slide here, can’t keep up with that. Some nice pictures to explian what is going on.

$E^s$ is vertical, $E^c$ is horizontal.

….questions….is the center manifold infinite dimensional?…..this is just the setting. I’ll be precise about the theorems soon.

some spectral assumptions.

Theorem 1 (Existence of $W^c$): $\exists ~ ! ~ W^c = graph(h^c), ~ h^c : E^c \rightarrow E^s, ~C^{1+\alpha}$, invariant.

Theorem 2 (Existence of $W^s$ foliations): slide slid up…..

Theorem 3 (Absolute continuity of $W^s$-foliations): (Zeng Lian, Chongchun Zeng, LSY): Assume $dim(W^c) < \infty. Then $W^s$-foliation is absolutely continuous.

Strange Attractors arsising from periodically forced Hopf bifurcations.

(joint work w. Kening Lu and Qiudong Wang)

Result for ODE in 2D $\rightarrow $ Corresponding equation for evolution equation in Hilbert space $\rightarrow $ Application to a specific PDE.

We have an unforced system with a parameter which is undergoing a “generic” supercritical Hopf bifurcation at $\mu = 0$.

$\mu < 0$
$ \mu = 0$
$ \mu > 0$.

Normal form. Introduce the twist number, expressed in terms of coefficients appearing in the normal form.

Forced system. Periodically, we kick it and then let it relax. It doesn’t have to be a kick. It just needs to relax in between the applications of the forcing.

Theorem (LWY): …I read it rather than type it…. there is some number you can calculate that as to be pretty big. We have a big kick period. Then you have a strange attractor with complicated dynamics. The attractor has an SRB measure.

An SRB measure is an important concept in finite-d and is the first challenge to bring it to infinite dimensions. If you look at a Hamiltonian system with flowmap $\phi_t$. Let $m$ be the Liouville measure. Assume ergodic. Then $\forall$ cts $g$ we find $$ \frac{1}{T} \int_0^T g( \phi_t) x dt \rightarrow \int g dm $$ for $m-a.e.$ x. (Birkhoff Ergodic Theorem)

Now suppose you have an attractor. (Sinai-Ruelle-Bowen). An invariant measure $m$ is called SRB or physical measure if the same convergence takes place for Lebesgue-a.e. In this setting $m$ is completely singular compared to the ambient Lebesgue measure. This is considered to be the analog of the Liouville theorem for dissipative systems.

I am claiming that these attractors support these measures.

Example, nice pictures.

Kick can be quite general.

…

Lyapunov exponents and $W^u, ~ W^s$-manifolds

Ruelle, Mané, Thieullen 80s, Lian-Lu (later)

Cocycle set up. Biggest differences w. finite-d:

$\Phi (x)$ generally not onto (possibly 1:1)
“Essential spectrum” - Lyapunov exponent is not defined.

Kuratowski measure of noncompactness.

Extension of Katok’s results….moving a bit fast here.

Friday 06 October

(Alas, my flight departure time will force me to miss out on hearing these talks.)

Laurent Stolovitch, CNRS, Université de Nice, Smooth Gevrey normal forms of vector fields near a fixed point

Claude Viterbo, Ecole Normale Supérieure, Symplectic Homogenization

Anatoly Neishtadt, Loughborough UniversityAveraging, passages through resonances, and captures into resonance in dynamics of charged particles

(Image no longer available: Anatoly Neishtadt)

Anticipating the Report of Canada's Expert R&D Panel

Fri, 30 Sep 2011 20:58:36 +0000

I was happy today to learn that Canada Post issued a stamp honoring my Toronto colleague and Nobel Laureate John Polanyi. The stamp is issued as part of the celebration of the International Year of Chemistry. This bit of good news tempered the alarming developments across the ocean where actions by the EPSRC appear to be destroying the scientific fabric of the UK. Here in Canada, despite an anomalous 2011 Discovery Grants competition for math/stats and recent news that some of my colleagues’ appeals were rejected, I hope to soon hear good news from the Expert R&D Panel which will hopefully reset Canada’s priorities and shore up support for basic research. The original vision of NSERC (supporting research at universities) is quite different from today’s gallimaufry of industrial Research Partnership Programs which are taking away money from Discovery Grants and basic research in general.

The report of the six member R&D panel is forecasted to arrive sometime in October. UBC’s Nassif Ghoussoub reports that “all eyes are on” David Naylor, the President of the University of Toronto. Ghoussoub’s post encouraged me to spend some time reviewing some of President Naylor’s opinion pieces looking for insights into his perspectives on research policy. Here are some extractions I found encouraging:

Summer 2011: Meeting Global Challenges

“We cannot afford to fall behind. Higher education and advanced research in today’s world has a massive impact that extends into every other field of human endeavor. And Canada must have universities that can do two related things: conduct the advanced research that will help surmount the grand challenges that humanity now faces, and offer the best and brightest students an education that will help them build a more successful nation and a better world. No university in Canada is better positioned to meet those objectives.”

Spring 2008: The Topography of Innovation

“…governments should fund basic research more generously. From lasers to Teflon, countless economically important advances have piggybacked on basic research. And in regions where Nobel Prize winners congregate in great universities, knowledge-based industries flourish in a wonderfully synergistic relationship.”
“Canadians are efficient at turning dollars into research but inefficient at turning research into dollars. Commercialization is not the enemy of fundamental research; nor is the converse true. However, it is wrong headed to insist that granting councils and research agencies constantly look downstream to the marketplace when their sights are justifiably set upstream on knowledge generation. Instead, we need dedicated commercialization agencies and infrastructure.”

Summer 2009: Universities and the Innovation Economy

“Unfortunately, one still hears grumbling about overspending on “irrelevant” basic research. The last hundred years have shown us time and again that basic research, driven by curiosity and arbitrated by peer review, is absolutely essential to human progress – and its practical impacts are totally unpredictable.”
“We should also be clear about what universities don’t do. Commercialization happens in companies, not in universities. To be sure, universities can collaborate more often and more effectively with industrial partners. We can try to ensure a strong outflow of well-protected intellectual property with interesting potential. And we should promote a culture of civic engagement and entrepreneurship among our students and trainees. The University of Toronto is taking positive action on all those fronts. But the onus in commercialization rests squarely on the private sector.”

I also found a lot of discussion explaining that, on all metrics, Toronto is an outstanding University. President Naylor also presented detailed comparisons showing how American, British (this might have changed recently), and European grants cover the “indirect costs of research” at much higher levels than Canadian grants. Indirect costs are very important but appear to be harder to sell to policy makers than, say, research institutes in vulnerable ridings.

In addition to developing the method of infrared chemiluminescence, Professor Polanyi has made many insightful statements concerning science policy. These three are especially relevant right now:

(Concerning the allocation of research funds) “It is folly to use as one’s guide in the selection of fundamental science the criterion of utility. Not because (scientists)… despise utility. But because. .. useful outcomes are best identified after the making of discoveries, rather than before.”
— Speech to the Canadian Society for the Weizmann Institute of Science, Toronto (2 Jun 1996)
“Faced with the admitted difficulty of managing the creative process, we are doubling our efforts to do so. Is this because science has failed to deliver, having given us nothing more than nuclear power, penicillin, space travel, genetic engineering, transistors, and superconductors? Or is it because governments everywhere regard as a reproach activities they cannot advantageously control? They felt that way about the marketplace for goods, but trillions of wasted dollars later, they have come to recognize the efficiency of this self-regulating system. Not so, however, with the marketplace for ideas.”
— Quoted in Martin Moskovits (ed.), Science and Society, the John C. Polanyi Nobel Lareates Lectures (1995)
“At one time, it would have been thought mistaken to suggest that scientists meddle in politics. Today, it would be shameful to deny that they have this responsibility.”
— Hope lies in the scientific method, The Globe and Mail 2009-05-29

Île de Berder Workshop Notes

Fri, 09 Sep 2011 05:22:49 +0000

Île de Berder Workshop Notes

I am at a workshop on Île de Berder. The post below contains the notes I am taking during the talks. I apologize (especially to the speakers) for misquotations and typos but I hope the notes might be useful.

Tuesday 2011-09-06

Rafik Imekraz: Non resonant normal form for perturbed quantum harmonic oscillator

Study high sobolev norms of solutions $\psi$ solving a NLS with harmonic trap and a nice smooth bounded potential. There is also a compact operator $M$ which is introduced to avoid the resonance. The natural Sobolev spaces are those naturally associated to the linear operator (without the compact operator).

Almost global existence. $\exists ~M$ with small operator norm and for data of size $\epsilon$ in $H^s$ then the solution stays small in that norm for times on the order of $\epsilon^{-3}$.

Earlier results on NLS and NLW on $S^1$. Delort-Szeftel 2007, Bambusi-Delort-Grébert-Szeftel on Zoll Manifolds.

In his thesis he replaced the $x^2$ by $x^{2p}$, e.g. the quartic oscillator.

When $V=0$, we have the usual Hermite eigenfunctions.

This PDE can be given a Hamiltonian formulation. The operator $M$ is given as a Fourier multiplier on the eigenbasis with $m_j \rightarrow 0$.

Normal Form Procedure:

$H_0 + P $ transforms into $H_0 + Z + R$ with $Z$ in normal form and $R$ a negligible error term.

Spectral key points. $H_0$ has a nonresonance condition. We will say that the spectrum is nonresonant. Definition introduced initially by Bambusi, used by Delort, Brebert, Imkeraz, Paturel, Szeftel. The condition involves infinitely many eigenvalues.

Delort-Szeftel argument. The spectrum is not explicit when $V \neq 0$. However, there is a 2005 theorem by Klein-Korotyaev-Pokrovski: $$V \in BC^\infty(\mathbb{R}, \mathbb{R}) \cap L^1 (\mathbb{R}) \implies |\lambda_j - 2j -1| \leq \frac{C}{j^\delta}.$$ Same assumption on a Zoll manifold.

We have a multilinear estimate. Technical proof with a commutator lemma. Uses smoothness and $L^p$ estimates on eigenfunctions by Yajima-Zhang 2001. It is not necessary to know the eigenfunctions explicitly.

Conclusion: A normal form procedure is possible to deduce almost global existence for $$ i \partial_t \psi = ( - \partial_x^2 + x^2 + V(x) + M) \psi \pm |\psi|^2 \psi. $$

Is it possible to say something when $M=0$? Does there exist a $V$ which has a nonresonant spectrum? We give a partial answer. Yes, there exists such a $V$ but without the regularity we’d like to impose. The function $V$ will be continuous and bounded but we don’t know if it is possible to create a more regular $V$ which remains nonresonant. The lack of regularity seems to preclude the multilinear analysis.

Chelkak-Kargaev-Korotyaev 2004

Q: Does the proof of almost global existence imply global well-posedness with polynomial-in-time bounds on the high Sobolev norms?

There was some discussion but the answer was not clear. It turns out the question was naive because there are examples showing blowup.

Massimiliano Berti: Quasiperiodic solutions of Hamiltonian PDEs

Hamiltonian PDES.

Goal: existence of qp solutions of pdes. techniques, based on nash-moser implicit function theorem and KAM theory. Techniques apply to NLW, NLS and 1d derivative-NLW.

Perspectives: water waves?

Model case:

$$u_{tt} - \Delta u + V(x) u = \epsilon f(\omega t, x, u)$$

$x \in T^d$, periodic boundary conditions
$\epsilon$ small
$V(x) \in C^k (T^d; R)$
$f \in C^k$

NLW is a Lagrangian equation on an infinite dimensional phase space.

The problem is to construct qp solutions of the NLW for $\epsilon \neq 0$.

QP Definition: $u(\omega t, x)$ where $u(\phi, x): T^\nu \times T^d \rightarrow R$

Linear equation.

superposition principle….harmonic oscillator with frequence $\sqrt{\lambda_j}$ when $\lambda_j >0$. For $\lambda_j < 0$, we have a harmonic repulsor.

There are infinite dimensional spaces of qp solutions of the linear equation. We want to see if these persist for the nonlinear equation when $\epsilon >0$ is small.

The embedding $T^\nu \ni \phi \longmapsto u(\phi, x)$ solves the “NLW” $$ (\omega \cdot \partial_\phi)^2 u - \Delta u + V(x) u = \epsilon f(\phi, x, u). $$ in a Sobolev space $H^s (T^\nu \times T^d )$.

This can be approached as a bifurcation problem. Let $F(\epsilon, u)= (\omega \cdot \partial_\phi)^2 u - \Delta u + V(x) u - \epsilon f(\phi, x, u).$ We know there are zeros when $\epsilon =0$ and we want to branch off these via the implicit function theorem. The standard hypotheses of the implicit function theorem are not satisfied.

We need to make a diophantine assumption to proceed. We use Newton Method + “smoothing” following the Nash-Moser IFT.

Newton tangent method for zeros of $F(u) = 0 + “smoothing”$: $$ u_{n+1} = u_n - S_n (D_u F)^{-1} (u_n) F(u_n), $$ where $S_n$ is a regularizing operator.

Advantage: Quadratic scheme! $$| u_{n+1} - u_n |s \leq C(n) | u{n1} - u_{n-1} |_s^2. $$ This is convergent even when the constants $C(n)$ are exploding.

However, there are also disadvantages. We are studing a linearized equation on an approximate solution. Linear differential operator with non-constant coefficients. It is not diagonal in Fourier basis. We know the eigenfunctions exist and are orthonormal in $L^2$ but we don’t have much explicit control. This is a “singular” perturbation problem.

Literature:

Kuksin 89; Wayne 90: Dirichlet b.c., $f$ analytic, KAM theory. Eigenvalues of $- \partial_x^2$ are simple so the KAM nonresonance conditions are satified (so-called 2nd Melnikov conditions)

Craig-Wayne 93: Periodic case. Eigenvalues have multiplicity 2. Lyapunov-Schmidt, f analytic, Newton-Method, periodic solution. Extended to PDE nonresonant of Lyapunov center theorem. …breath mention by Bourgain.

Berti-Bolle DMJ 06, Advances 08. Berti (book) 08. Extend to PDE the Weinstein-Moser and Fadell-Rabinowitz theorems.

In the space dimension $d \geq 2$, main difficulties:

eigenvalues of $-\Delta + V(x)$ appear in clusters of increasing size. For example all the lattice points on spheres have the same linear frequency.
Feldman-Knonner-Trubowitz. The eignefunctions of $-Delta + V(x)$ are NOT localized with respect to exponentials. This means that there are strong interactions between the eigenmodes. In this frame, it is often convenient to work with pseudo-PDE involving Fourier multipliers. (Bourgain, Kuksin-Elliason)
Bourgain 95-98 f analytic.
Bourgain’s question 97: for differentialble nonlinearities? See Berti book.
Berti-Procesi DMJ 11, General Riemannian manifolds. General Lie group: products of eigenfuctions can be represented as a sum over eigenfunctions. Related to Birkhoff normal form results by Bambusi, Delort, Grébert, Szeftel.

QP solutions for $d \geq 2$:

Newton Method. Bourgain Annals 98 ($d=2$); Annals 05; Wang 11 Supercritical (completely resonant) NLS-NLW, no parameters.
KAM Method: Kuksin-Eliasson Annals 10. analytic NLS w Fourier multipliers.
Procesi-Xu 11, Procesi-Procesi 11, any dimension, Birkhoff normal form for completely resonant NLS.

New results for qp solutions in $d \geq 2$:

Berti-Biasco CMP 2011
Bambusi-Berti-Magistrelli JDE 2011.

Techniques:

Optimal Nash-Moser iterative scheme: different from the analytic Newton iteration.
For measure estimates, we use simpler techniques that Bourgain avoiding semi-algrebraic sets.

Very interesting technical discussion highlighting the favorable constellation effects when the frequencies are in tight resonance.

Granville: More “torsion” of a manifold there are less integers nearby. (This seems interesting…look up and discuss with Andrew.)

Wednesday 2011-09-07

Frédéric Bernicot: Bilinear Strichartz inequalities and space-time resonances

(joint work with P. Germain)

Linear Strichartz inequalities for Schrodinger. $L^2$ admissibility pairs. History: Strichartz 77, Ginibre-Velo 89, Keel-Tao 98. Extensions to compact manifolds with loss of derivatives.

What about bilinear Strichartz inequalities? Suppose $f \longmapsto e^{it \Delta} f = u, g \longmapsto v$. We’d like to know:

$$ | vw |{L^p L^q (R^{1+d})} \leq |f |2 | g |_2. $$

In previous works on this topic, these were typically studied with $p, q$ both equal to 2. We are interested in the cases where p and q are not equal to 2.

Applications: some large time behavior results; some stability results.

Time resonant set; space resonant set; their intersection is called the spacetime resonant set.

Take advantage of geometric properties of the resonant set to prove boundedness properties for solutions.

Some related works could be mentioned:

The detailed study discussed here also resonates with the paper of Bejenaru-Herr-Tataru 2009:A convolution estimate for two-dimensional hypersurfaces.

Emanuele Haus: Asymptotic stability of the synchronous resonance for an elastic satellite with internal friction

(joint work with D. Bambusi)

Synchronous resonance: the satellite always shows the same face to the planet. For example, the moon does this. Why does this happen? Tidal effect. The satellite is deformed and stretched towards the planet. If the satellite is not in a circular synchronous orbit, the direction of the stretching changes inside the satellite –> dissipation. Our aim is to stydy the system of coupled equations and prove asymptotic stability of the synchronous resonance. We want to model the orbital, rotational and internal degrees of freedom.

Internal friction –> circular orbit + 1:1 resonance is a (local) attractor.

Spherical case was done earlier by D. Bambusi.

J. Colliander: Normal Forms and the Upside-Down $I$-method

(joint work with Soonsik Kwon and Tadahiro Oh)

Rémi Carles: Interaction of coherent states for Hartree equations

This talk will involve semiclassical analysis and the equation might not be Hamiltonian.

Schrodinger equation in semiclassical regime ($\epsilon \ll 1$): $$ i \epsilon \partial_t \psi^\epsilon + \frac{\epsilon^2}{2} \Delta \psi^\epsilon = V(x) \psi^\epsilon. $$ We study semiclassical wave packets (coherent states), $$ \psi^\epsilon (0, x) = \frac{1}{\epsilon^{d/4}} a ( \frac{x-x_0}{\sqrt{\epsilon}}) e^{i p_0\cdot (x-q_0)/\epsilon}, ~ a \in S(R^d). $$ The potential $V$ is smooth, real-valued and at most quadratic: $ V \in C^\infty, \partial_x^\alpha V \in L^\infty, ~\forall |\alpha| \geq 2. $$ (No sign assumptions) The associated Hamiltonian flow is globally well-posed but might involve exponential growth.

Classical action: $ S(t) = \int_0^t (\frac{1}{2} |p(s)|^2 - V(q(s))) ds. $

Equation for $\psi^\epsilon$ is equivalent, via algebraic manipulations to $$ i \partial_t u^\epsilon + \frac{1}{2 } \Delta u^\epsilon = {\mathcal{V}}^\epsilon u^\epsilon. $$ where $ {\mathcal{V}}^\epsilon =$…ack slide switch.

Ehrenfest time. Validity of the approximation with $V$ is not a polynomial. The difference between the appoximation and the original solution solves an inhomogenous Schrodinger equation. This is studied via energy estimates.

He carries ont a derivation of the ansatz from “scratch” by comparing things at different levels of $\epsilon^j$. The analysis “explains” why this is a reasonable choice.

Hartree equation. Same equation as before with additional term $(K * |\psi^\epsilon|^2) \psi^\epsilon$. Here we need $K \in W^{\infty, \infty}$. The ansatz is adjusted by adding in a new prefactor $\epsilon^\alpha$. The importance of the nonlinearity emerges differently depending upon the value of $\alpha$.

Two initial coherent states. When $\alpha = 0$, the Hamiltonian flow is affected by nonlinear effects. Modified $\epsilon$-dependent actions.

Hmmm…..I should recast interction Morawetz in the semiclassical setting and see if an interesting estimate emerges in the semiclassical limt.

Main Result: The exact solution can be approximated by two coherent states but there is a required phase drift between the coherent states. There is a corollary about the Wigner measures. The Wigner measure does not see the nonlinear effect except when $\alpha =0$.

Tiphaine Jézéquel: Homoclinic orbits with many loops near a $O^2 i\omega$ resonant fixed point for hamiltonian systems

(joint work w. Patrick Bernard and Eric Lombardi)

$$ \frac{du}{dt} = L_\epsilon u + Q_\epsilon (u). $$

$u \in R^4$
$t \in R$
$u=0$ is a fixed point, i.e. $Q_\epsilon (0) = 0$.

This implies that the dynamics in a neighborhood of zero. The resonance configuration is captured by the $O^2 i\omega$ resonance. 4 eigenvalues on imaginary axis, 2 above real axis, 2 below.

Physical Context

Motivated by study of water waves. 3d gravity-capillary fluid modelled by the Euler equation and we look for 2d traveling wave solutions. The “spatial dynamics method” produces an infinite dimensional equation. Using the center manifold theorem, this problem is reduced to a 4-d invariant manifold. We look for particular soltutions in the manifold. This is the collapse to 4d.

Initial aim: existence of solitary waves. In the R4 setting, this corresponds to a homoclinic connection to 0 in the center manifold. This turned out to be hard. So, we transferred to a different study. We study the existence of generalized solitary waves. This corresponds to a homoclinic connecton to a periodic solution.

Lombardi 2000.

Beautiful pictures. Excellent exposition of the phase space portraits in R4. Pictures are getting even better.

OK, the strategy was nicely described. The last part of the talk begins to show how the nice pictorial overview of the proof strategy is actually implemented. The details look formidable involving KAM, lots of ODE manipulations. She quotes ideas from Moser 1958, Russman 1964.

(I learned later from Tiphaine that she created her figures using Adobe Illustrator.)

Thursday 2011-09-08

Sandrine Grellier: Integrable effective dynamics for a nonlinear wave equation

(joint work with Patrick Gérard; arXiv preprint)

Consider the half wave equation:

$$i \partial_t u - |D| u = |u|^2 u$$

Here $|D|$ is what you expect. This is a tyo model for NLS on degenerate geometries leading to a lack of dispersion. This equation admist the same kind of conservation laws as NLS:

$H(u)$
$p(u)$
$Q(u)$

Compared to the $NLS_3^+$, this is a nondispersive equation..

$\Pi_+$ is projection onto Fourier modes $k \geq 0$. $\Pi_-$ is the projection onto modes $k<0$.

The equation is $L^2$ critical but the first iteration of the Duhamel formula is not bounded in $H^s$ for $s< \frac{1}{2}.$ Despite this, we have a Cauchy theory in $H^{1/2}$. There exists a unique solution in $C(R; H^{1/2}(T))$. Persistence of regularity also holds for $s>1/2$. The proof uses some Brezis-Gallouet type logarithmic inequalities. It provides rather bad large time estimates:

$$ | u(t) |{H^s} \leq e^{e^{Cs t}}. $$

We compare this to the cubic Szegö equation: $$ i \partial_t u_+ + \partial_x u_+ = \Pi_+ (|u|^2 u). $$ Decoupling result: Assume that $\Pi_+ u_0 = u_0$….ack slide changed….

Theorem: Let $s>1$. Assume that $\Pi_+ u_0 = u_0 = O(\epsilon)$ in $H^s$. Then, $u =v + O(\epsilon^{3-\alpha})$ on a time scale of size $\epsilon^{-2} \log \epsilon^{-1}$ where $v$ solves the half-wave equation. Furthermore, the solution remains small in $L^\infty$.

Corollary (Weak Turbulence): Let $s>1$. There exists a sequence of data $u_0^n$ and a sequence of times $t_n$ such that the data converges to zer in $H^r$ for all $r$ while the $H^s$ size of the solutions at time $t_n$ exceeds a power of $\log \frac{1}{| u_0^n |_{H^s}}$.

Contrast this with the 1d cubic NLS. Zakharov-Shabat 1972: no such norm inflation. For 2d cubic, CKSTT 2010 construct small $H^s$ data which grows large.

The proof comes from the “weak turbulent property” of the cubic Szegö equation. If the approximation result in the Theorem held on a longer time interval, we could prove a stronger weak turbulence result for the half-wave problem, more analogous to the corresponding result for Szegö where divergence to infinity has been established.

Quick sketch of the sequential norm inflation property for cubic Szegö. This follows from a rather explicit analysis of solutions of the form constant + pure exponential.

Sketch of proof: Analysis similar to what I spoke about yesterday. Nonlinear term is explicitly represented in terms of Fourier coefficients.. The $L^4$ expression is separated into the positive and negative frequency components in $L^4$ plus another term related to the $L^2$ norm and another term.

Removal of trivial resonances with a change of phase, as in Bourgain. Birkhoff normal form transformation. Resonance set is identified and has some algebraic structure so that the resonant quartets can be identified. There is no problem with small divisors here. This leads to a new system after these transformations. Our task is to show that this transformed system is approximated well by the Szegö equation. Smallness in $H^s$ can be shown via bootstrap on a time interval of size $\epsilon^{-2} \log \epsilon^{-1}$.

Lax Pair and a priori bounds for Szegö: Hankel operator….Lax 1968, Gérard-Grellier 2010. Peller 1982 shows that the trace of the Hankel operator is equivalent to the $B^1_{1,1}$ norm. This space is an algebra which contains all the $H^s$ spaces and is a subspace of $L^\infty$. The solution stays bounded in $B^1_{1,1}$.

There are many things to understand. We would really like to understand NLS on the Heisenberg group.

Nice discussion at the end of the talk explaining how the half-wave problem is sort of in between the Szegö equation and the NLS on the Heisenberg group.

Dario Bambusi suggested that a normal form iteration method a la Bourgain might allow for an improved approximation result.

Oana Pocovnicu: The Szegö equation seen as the resonant dynamics of a nonlinear wave equation

Similar to the last talk. My study will be done on the real line, rather than the torus. Why do that? Normal form methods work nicely on the torus. In the setting of the real line, we can still have small divisors. Cutoffs like done in Jim’s talk would create other issues and the approximating result would involve an equation other than the Szegö equation. Instead of using the Normal forms approach, we use the renomalization group (RG) method.

SE: $ i\partial_t u = \Pi_+ (|u|^2 u).$

In the case of the real line, we have one solution with initial data of an explicit form, then we can prove that the Sobolev norms behave like $$ | u(t) |_{H^s} \thicksim t^{s-1}. $$

Theorem: Let $v$ solve the half wave equation with initial data $\epsilon W_0$ where $W_0 \in H^s_+ (R).$ Let $u$ denote the evolution from the same data under the Szegö equation. Assume that $| u(t) |_s \leq C \epsilon (\log \frac{1}{\epsilon^\delta})^\alpha$ for $0 \leq \alpha \leq \frac{1}{2}$ then we have a good approximation….ack slid changed.

Then we have a corollary which transports the weak turbulence property of Szegö over to the half wave equation.

Remark: In order to show arbitrarily large growth of the solution, we need a better approximation result for a time of size $\epsilon^{-2 - \beta}$ where $\beta >0$. The point here is that the Szego equation is the resonant subsystem sitting iside the half-wave equation, analogous to the way the Toy Model sits inside cubic NLS on $T^2$.

RG method: Chen-Goldenfeld-Oono 1994, De Ville, Harkin, Holzer, Josic, Kaper; Ziane Temam, Moise, Abou Salem.

How does this method work?

We make a change of variable to remove the $\epsilon$ prefactor in front of the data and to move into the interaction representation. This introduces an equivalent equation with some exponentials and $\epsilon^2$ in front of the nonlinearity. We make a naive perturbation expansion in powers of $\epsilon$ and a Taylor expansion of the nonlinearity in terms of the unknown $w$. We plug and chug to identify powers of $\epsilon$. There are no ad hoc assumptions….we can just try….so it has some flexibility over the Birkhoff normal form.

The manipulations allow us to identify resonance as a vanishing of the phase function inside the Duhamel integral. It gows in time as a secular term and will cause our approximation to break down. We consider then the renomalzation group equation. We define a new approximating object which includes the explicit secular term.

Many resonances in this half-wave equation. The resonant set of the half wave equation ${ \phi (\xi, \xi_1, \xi_2, \xi_3) = 0}$ has non-zero measure for fixed $\xi$. Nice remarks connecting the resonant set to the discussion from Bernicot’s talk.

She has also obtained a second order approximating equation to the half-wave equation. This equation is Szegö plus some 5-linear terms. The approximation degree is tighter ($\thicksim \epsilon^5$) but on the same time interval $\epsilon^{-2} \log {\frac{1}{\epsilon}}.$ This quintic extension of Szegö is not yet understood.

Zaher Hani: Long time strong instabililty and unbounded orbits

Consider cubic NLS on $T^2$. This problem is LWP in $H^s$ for $s>0$ and GWP for $s>2/3$. We are interested in dynamical aspects of global flow. How do the orbits behave in $H^s$? Are all the orbits bounded or doe they grow in time? Such questions are crucial in understanding the frequency dynamics of the energy.

Upper bounds on the $H^s$ norm. WE have ocnservation of mass and energy which give a prior bounds on $L^2$ and $H^1$. There are no unbounded orbits in $H^s$ for $s =0, 1$. The 1d analog possesses infinitely many conservation laws that control integer Sobolev norms. Polynomial-in-time bounds have been established by Bourgain, Staffilani, CKSTT< DSPST, Sohinger, CKO. We don’t expect poly bounds to be sharp.

Lower bounds Does there exist a global solution fo cubic NLS that satisfies $\sup_{t \in R} | u(t) |_{H^s} = + \infty$?

Conjecture (Unbounded orbits conjectore): For $s>0, s \neq 1$ there exist global solutions to cubic NLS on $T^2$ that satisfy $\sup_{t \in R} | u(t) |_{H^s} = + \infty$.

Think of the solution supported on three frequency scales: low, medium, high.

High frequences need to become larger. Medium frequencies have to decrease to balance the increase at high frequencies. Conservation of mass requires that the low frequencies become larger to compensate for the net loss of mass at medium and high frequencies.

Theorem (CKSTT 2008): $\exists$ solution of cubic NLS which is initially small in $H^S$ but at some later time the solution exceeds an arbitrarily large size.

Caution: This does not imply the existence of an unbounded orbit.

Another related result is due to Carles-Faou.

Long-time strong instability: $X$ banach space. $S(t)$ is a continuous dynamical system on $X$. CKSTT result shows long time strong instability near the zero solution. This notion generalizes the CKSTT conclusion around a point other than the zero solution.

Lemma (H 11): Suppose $D \subset X$ is dense. If $S(t)$…ack slide changed…

Lemma suggests that to prove existence of unbounded orbits, it is enough to prove that LTS instability holds near a dense subset of $H^s$.

While proving that generic orbits are unbounded seems ambitions, we can formulate a localized version of the lemma. We don’t strive to prove the genericity of unbounded orbits. The localized lemma recasts the lemma above onto a closed subset $F \subset X$.

The proof is a straightforward application of the Baire category theorem. This is the program. It remains open whether this approach applies to NLS. Instead, we will obtain results on some other systems inspired by NLS. We make a first nontrivial step towards the implementation of this program for the cubic nonlinearity.

Theorem (LTSI near single-frequency data H 11): NLS exhibits LTSI near $A e^{inx}$ in $H^s$, at least for $s \in (0,1)$.

Consier NLS with a trilinear Hamiltonian. In the limit $R \rightarrow \infty$ the nonlinearity $\mathcal{N}_R \rightarrow |u|^2 u $ and we have the LTSI property for this sytem.

NLS in Fourier space. Recasting NLS in Fourier space following CKSTT. Parallelogram of four frequencies is required to excite activity at a frequency $n$. Resonance corresponds to the requirement that the parallelogram be a rectange. The restriction of the 4-wave interactions to the rectangles is called RFNLS. If the initial data are supported on a subset $\Lambda \subset Z^2$ and for any three vertices $n_1, n_2, n_3 \in \Lambda$ on a rectangle, we will say that $\Lambda$ satisfies the closure property if we are certain that the fourth vertex is also in the rectangle.

A rectangle is a first example of a set which satisfies the closure property. Consider the rectangle (0,0), (N,0), (0,N), (N,N). We can calculate explicitly the associated ODE system. Suppose that at time zero, we have the mass cocnetrated mostly at (N,0) and (0,N) while there is a little bit at (0,0), (N,N). At a later time, the mass moves across to the other diagonal. Thus, the Sobolev norm increases by a factor $2^{s-1/2}$. The CKSTT construction is a concatenation of this idea.

Step 1: Build a set $\Lambda = \Lambda_1 \cup \dots \cup \Lambda_P$ satisfying structural and geometric properties and the norm explosion property.

Step 2. Construct a solution to RFNLS.

Step 3. Approximation result. RFNLS approximates FNLS.

Step 3 is the easiest one in the CKSTT paper. Recall that the passage from FNLS to RFNLS involved throwing away the nonresonant terms. An integration by parts argument allows CKSTT to show that these terms contribute very little to the FNLS dynamics. These observations are the key steps to prove the approximation Step 3.

When the ground solution is changed from zero to a pure single frequency data, we have to show that the complete solution $u(t)$ is approximated well by the ground solution evolution plus the (adapted) CKSTT solution $v(t)$. By galilean invariance, we can assume that $n=0$ and $u_0 = A$ so that $u_g (t,x) = A e^{i |A|^2 t}$ and we would like to limit the interactions between the zero frequency and those in the resonant set $\Lambda$. Nonresonant interactions with the zero mode create problems. He calls this effect a second order resonance. This analysis identifies why the theorem is restricted at this stage to $s \in (0,1)$.

We define the nonlinearity $\mathcal{N}R$ by throwing away all interactions for which $|\omega_4| > R$. $\mathcal{N}_0$ is the resonant nonlinearity we saw before.

$R$-closure, a natural generalization of the closure property…..wow I like this! There is more flexibility in the CKSTT construction than I realized.

Berti’s Question: How fast is the diffusion? Answer: You have to track it through CKSTT. You will find this is a four tower exponential. Therefore, the rate of growth suggested by the CKSTT example is about $\log \log \log \log t$ which is pretty slow….but does diverge.

Nice discussion afterwards speculating on applications of the Baire Category lemma to the periodic Szegö equation.

Erwan Faou: 2d Cubic NLS: Energy cascades vs. Sobolev stability of plane waves

(reporting on two joint works, one with Rémi Carles another with C. Lubich and Gauckler)

This work was inspired by some numerical simulations.

Cubic NLS on $T^2$. We won’t care if it is focusing or defocusing.

Rewrite the Hamiltonian wrt Fourier coefficients.

When we add a convolution potential (diagonal in Fourier), we can prove stability results for small intial data. Bourgain, Kuksin, Craig-Wayne, Poschel, Eliasson-Kuksin, Bambusi-Grébert, Faou-Grébert. Typical results imply preservation of the actions for a polynomial (or slightly longer) time in terms of the size of $\epsilon$, the data size.

Without potential in $d=1$, there are stability results. Complete integrability.

Without potential in $d=2$. Many small quasiperiodic solutions. CKSTT instability result (which was mentioned in all the talks today!)

Semi-discrete system:

Space approximation.. We use a Fourier pseudo-spectral collocation method. We look for a trigonometric polynomial satisfying NLS at the grid of $K^2$ points. There is an aliasing problem.

Splitting schemes. We use a symplectic integrator.

The numerical is very close to the dynamics of the modified energy. Two instability mechanisms: small divisor issue and the aliasing problem. To avoid the small denominators we use a Courant-Friedrich-Lwey condition $\tau K^2 < C$. The alsiasing is avoided by making sure that the frequencies remain localized. (See Dario’s talk tomorrow on the stabiilty of solitons.)

Numerical tests on NLS. Generic prservation of the actions over extremely long times for small initial data. Typical picture of $\log |\xi_j (t)|^2. The graph consists of horizontal lines. When we start with 5-mode data, we see some more interesting dynamics.

Energy Cascade (with Rémi Carles)

arXiv: Carles-Faou

Theorem: The solution $u$ in Fourier satisfies $u_j (t) = e^{-i t |j|^2} v_j (\epsilon t) + O(\epsilon)$ for $t \leq \frac{T}{\epsilon}$ where $v_j (\tau)$ solves the resonant system RFNLS.

Proof: integration by parts, contained in other works. The work is done in the Wiener algebra.

Quadruplets…rectangles. In dimension 1, there are no rectangles. therefore, the resonant Hamiltonian only depends upon the actions.

We have preservation of the actions in 1d over a time $\epsilon^{-1}$. If you try to reproduce this numerically, it is quite difficult due to the aliasing issue. Prime numbers help to avoid the aliasing issue…interesting.

Simulating energy Cascades. Consider data supported on 5 modes so that it forms a cross. He’s revisiting the construction I displayed in Napoli! Very cool. Dynamics of the extremal modes can be tracked.

Theorem (Carles-Faou 2010): Let $u_0 (x,y) = 1 + 2 \cos x + 2 \co y. Then….ack slide changed.

After n iterations, the mode that is turned on is such that $|j|^2 = 2^n$. For that mode, we have a lower bound on the size of that Fourier coefficient. I ran a long time simulation the other night. The same initial data (the cross) and he tracked the $H^4$ norm. You start at order 0.1 and after time 2000 you reach order 1.

More numerics needed (with R. Belaouar, CMAP) we are trying to do some very long simulations. This is a different mechanism from CKSTT.

He showed a movie which showed a slowly growing island of activity near the origin. Very cool 5 frequency model starting on a “cross”. The example reminded me of the cartoon version of the CKSTT construction I exposed in Napoli.

Plane Wave Stability (with Lubich and Gauckler)

preprint

When the $L^2$ norm lies inside some typical set. quantified preservation of the super actions….too fast for me to type. Orbital stability of the plane wave.

Plane waves stability:

Phase invariance
$|u_0|^2 is controlled by the $L^2$ norm of the $u_j, j \neq 0$.
Change of variables $(u_0, u_j) \longmapsto (a, \theta, v_j)$ …slide change.

Gauge invariance squeezes out the dependence on $\theta$. Preservation of the $L^2$ norm. The equations close under the change of variables. Even though the change of variables is not symplectic, there is some magic. The system for the $v_j$ turns out to be Hamiltonian. The normal form toolbox is applied and the genericity condition (in measure) is exploited….lots to understand here.

Discussion following the talk among Bambusi, Hani, Faou and me: Why doesn’t this contradict the theorem of Hani? Answer. The plane wave stability time here is limited. Hani’s effect takes place much much later. The FGL result is analogous to Nekoroshev and Hani’s shows diffusion after the stability time. There was a suggestion that there might be a KAM theorem lurking here which would allow the FGL stability type result to persist to infinite time provided that the condition on $\rho$ is implemented more cleverly. Very interesting….

Friday 2011-09-09

Dario Bambusi: Solitons in a numerical algorithm for NLS

(joint work with Erwan Faou and Benoit Grébert)

The point: when you compute you calculate the dynamics of a numerical approximate model of the problem. Solitons can sometimes be destroyed by the algorithm.

Consider focusing NLS on R. There are ground state, particular solutions. We know that these solutions are orbitally stable.

What happens if you try to put on the computer the dynamics of the NLS. We consider a large window and make a space discretization. We have reduced the problem into a finite dimensional system of ODEs. Then, you use a splitting method. You compose a flow associated with the vector field associated with the nonlinearity (called $P$) followed by the linear flow (called $A$).

The Euler method is not a symplectic method. We use an exact method for the nonlinearity and Euler for the linear. The soliton does not persist but is eventually destroyed. Under a CFL condition ($\tau/\mu^2$ is not too large), things improve. If this is too large there is a resonance and bad things can happen. He shows an example with CFL of size 19 which shows the growth of high frequencies. These grow higher and higher and after 3000 steps the soliton is completely lost. On the contrary, if you take a small CFL of size 1.9, then the soliton persists for a long number of iterations.

The question we want to address: Can we explain what is going on analytically here?

Space Discretization: Let $\psi (x)$ be substituted by $\psi_j = \psi (\mu j), ~ \mu \ll 1$ discretization. Substitute the Laplacian by the standard discrete version of it. You then find the usual discrete NLS. To put it on a computer, you cut off the number of points. So, instead of sampling at all points $j \in Z$, you wok on a large interval $[-K, K]$ with Dirichlet boundary conditions.

Time Discretization and preliminaries: Let $\phi^t_X$ denote the flow map of the vector field $X$. We use a splitting method. Substitute the flow of DNLS by $\phi^\tau_{A+P}$. WE have the soliton manifold $S$ and we have the discrete version of the $H^1$ space: $ | \psi |{E\mu}^2 = \mu ( \frac{1}{\mu^2} \sum_j |\psi_j - \psi_{j-1}^2 + \sum_j |\psi_j|^2). $

Theorem:

Assume that

$\mu \ll 1$ and $\tau \ll 1$
let $r \geq 4$ be an integer such that $\frac{r \tau}{\mu^2} < \pi$.

initial datum: $\psi$

If $d(\psi, S) \leq C (\mu + \frac{\tau}{\mu^{1/2}} + \frac{1}{\mu^2} e^{-C\mu K})$ then for $|n| \leq C \tau^{2-r}$ one has $d(()\phi^\tau_Q \circ \phi^\tau_P)^n, S) $ …ack slide changed.

Idea of Proof:

Conserved quantities. We know that $\phi_c$ realizes the minimum of the energy subject to a mass constraint. The minimum is unique assuming even in $x$.
space discretization (Bambusi-Penati): equal characterization with nearby functionals. space cutoff: idem.
time discretization (splitting): there exists a modified energy which is quasiconserved for the algorithm. (Benettin-Girogilli, Faou-grébert).

finite elements; discrete solitons;

Hamiltonian interpolation. Problem: does there exist $Z$ Hamiltonian such that $\phi^\tau_A \circ \phi^\tau_P - \phi^1_{\tau Z} = 0$? That is, can we show that the composition of two Hamiltonian flows is another Hamiltonian flow. (This seems obvious to me but I think the issue has to do with making this claim valid in the discrete setting.)

Galina Perelman: Contracting sphere blow up solutions for the 3D cubic NLS

(joint work with J. Holmer and S. Roudenko)

We consider the cubic focusing NLS in 3D.

conservation of mass, momentum and energy.

virial identity.

$\dot{H}^{1/2}$-critical. Scaling $\psi (t,x) \longmapsto \lambda \psi(\lambda^2 t, \lambda x)$

General facts:

global existence and scattering for small $H^{1/2}$ data.
Virial identity implies existence of blowup solutions
scaling lower bound on the blowup rate.
Merle-Raphaël has a more sophisticated blowup rate. The critical norm explodes faster than a power of the $\log (T-t)$.

Blowup scenarios:

*Self-similar blowup: *

Numerical experiments strongly suggest the existence of self-similar blowup solutions. It is expected that this is true however, the profiles are not in the critical space. Therefore, the asymptotic is valid only locally and we need some cutoff that will grow and will account for the explosion of the critical norm.

Merle-Raphaël-Szeftel: Rigrous justification of the self-similar blwup regime for slightly $L^2$ supercritical NLS.

* Circle blowup solutions (Holmer-Roudenko, Zwiers):*

Consider cylindrical coordinates on $R^3$. $Q$ denotes the ground state of the 2D cubic NLS…..dynamic is stable wrt to $H^1$ initial perturbation preserving the cylindrical symmetry. The idea of this construction is inspired by Raphaël who constructed solutions to the quintic NLS on $R^2$. In cylindrical coordinates, the problem resembles the 2D cubic NLS in $(z,r)$ apart from another term from the Laplacian. The extra term should not contribute to the dynamics provided that the explosion takes place away from the origin. This idea is built on the ring blowup solution of Raphaël. Should also mention Raphaël-Szeftel.

* Contracting sphere blowup solutions:*

Fibich-Gavish-Wang: numerical results an some heuristic arguments suggest the existence of radial finite time blowup solutions which explode on a contracting sphere. Assuming these exist, the behavior of the thickness scaling parameter and the contracting radius parameter can be calculated using the conservation laws.

Theorem: There exists a radial solution $\psi$ to cubic NLS on $R^3$ with $\psi \in C((0,t_0]; H^1)$ which explodes as a contracting sphere with radius $q(t)$ and scaling parameter $\lambda(t)$. with $q \thicksim t^{1/3}$ and $\lambda \thicksim t^{-2/3}$. The blowup rate for the $H^1$ norm is $t^{-2/3}$.

Remark: The choice of 3D cubic NLS is for its simplicity. One might expect that the same result holds true for other $L^2$ supercritical and $H^1$ subcritical problems with adjusted behavior for $\lambda$ and $q$. In our calculation, we use the fact that the nonlinearity is $C^\infty$.

Outline of Proof

Step 1. Construction an arbitrarily good approximate solution (up to any order) Step 2. Construct exact solution with small remainder.

….slides moving fast

New parameter is $E = \lambda^{-1}q^{-2}$.

We expect that $q \thicksim t^{1/3}$ and $E \thicksim 1$ so that $\lambda \thicksim t^{-2/3}$.

Build a formal solution….solvability conditions are trivial for even parameter $j$ but for odd $j$ these are nontrivial and allow to determine the coefficients $q_l$.

Build an approximate solution. We cutoff the iteration process used to define the formal solution at some stage. The error of the approximate solution is quantified to be small, like $t^{2N=2}$, after $N$ steps in the formal iteration.

Construction of an exact solution. Proposition: There exists a solution of the cubic NLS with is $t^N$-close to the approximate solution $\psi^{(N)}$. Also $\psi \in C([t_1, t_0], H^1 \cap ^{-1} L^2)$.

Main theorem follows from the proposition. Snapshots and a profile extraction, LWP….

How to prove the proposition? Bootstrap arguments based on energy type estimate.

Almost conservation of a quantity $G$. Coercivity of $G$. Control of missing directions: Conservation laws control two of them. Two others are controlled using the equation.

Q: Can you build concentric rings that collapse? Or are the parameters rigidly linked?

Maybe. There is some flexibility in the construction…. Matryoshka Doll Blowup?

Evolution Labels Needed on Medicines

Thu, 25 Aug 2011 05:20:49 +0000

Self-qualified intellectuals, like Texas Governor Rick Perry, have a right to know! I was born in El Paso: I have a right to know! Parents of American kids have a right to know! Does the Theory of Evolution have anything to do with our medicines? Which ones? Flu shots? Cancer treatments?

Medicines which are designed based on the Theory of Evolution should have a visible disclosure label. Here are two suggestions for images that might be appropriate for the evolution disclosure label: (Image no longer available: Monkey Sillhouette)

Some medicines which are based on this scientific theory are described on the Evolution 101 website at the University of California at Berkeley.

Individuals not wishing to use medicines based on the Theory of Evolution should, naturally, have the right information available to select a different course of treatment. Believers in evolution will take comfort knowing their medicine is based on science (… a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe). We all have a right to know: Medicines should have evolution disclosure labels.

Wisdom from Vannevar Bush on Science Research Policy

Sat, 30 Jul 2011 05:19:00 +0000

Timeless and timely extracts from Science: The Endless Frontier by Vannevar Bush:

Scientific Progress is Essential

Advances in science when put to practical use mean more jobs, higher wages, shorter hours, more abundant crops, more leisure for recreation, for study, for learning how to live without the deadening drudgery which has been the burden of the common man for ages past. Advances in science will also bring higher standards of living, will lead to the prevention or cure of diseases, will promote conservation of our limited national resources, and will assure means of defense against aggression. But to achieve these objectives - to secure a high level of employment, to maintain a position of world leadership - the flow of new scientific knowledge must be both continuous and substantial.
Moreover, since health, well-being, and security are proper concerns of Government, scientific progress is, and must be, of vital interest to Government. Without scientific progress the national health would deteriorate; without scientific progress we could not hope for improvement in our standard of living or for an increased number of jobs for our citizens; and without scientific progress we could not have maintained our liberties against tyranny.

The Importance of Basic Research

Basic research is performed without thought of practical ends. It results in general knowledge and an understanding of nature and its laws. This general knowledge provides the means of answering a large number of important practical problems, though it may not give a complete specific answer to any one of them. The function of applied research is to provide such complete answers. The scientist doing basic research may not be at all interested in the practical applications of his work, yet the further progress of industrial development would eventually stagnate if basic scientific research were long neglected.
One of the peculiarities of basic science is the variety of paths which lead to productive advance. Many of the most important discoveries have come as a result of experiments undertaken with very different purposes in mind. Statistically it is certain that important and highly useful discoveries will result from some fraction of the undertakings in basic science; but the results of any one particular investigation cannot be predicted with accuracy. Basic research leads to new knowledge. It provides scientific capital. It creates the fund from which the practical applications of knowledge must be drawn. New products and new processes do not appear full-grown. They are founded on new principles and new conceptions, which in turn are painstakingly developed by research in the purest realms of science.

Today, it is truer than ever that basic research is the pacemaker of technological progress. In the nineteenth century, Yankee mechanical ingenuity, building largely upon the basic discoveries of European scientists, could greatly advance the technical arts. Now the situation is different. A nation which depends upon others for its new basic scientific knowledge will be slow in its industrial progress and weak in its competitive position in world trade, regardless of its mechanical skill.

Industrial Research

The simplest and most effective way in which the Government can strengthen industrial research is to support basic research and to develop scientific talent. The benefits of basic research do not reach all industries equally or at the same speed. Some small enterprises never receive any of the benefits. It has been suggested that the benefits might be better utilized if “research clinics” for such enterprises were to be established. Businessmen would thus be able to make more use of research than they now do. This proposal is certainly worthy of further study.

Remove the Barriers

Higher education in this country is largely for those who have the means. If those who have the means coincided entirely with those persons who have the talent we should not be squandering a part of our higher education on those undeserving of it, nor neglecting great talent among those who fail to attend college for economic reasons. There are talented individuals in every segment of the population, but with few exceptions those without the means of buying higher education go without it. Here is a tremendous waste of the greatest resource of a nation - the intelligence of its citizens.
If ability, and not the circumstance of family fortune, is made to determine who shall receive higher education in science, then we shall be assured of constantly improving quality at every level of scientific activity.

We must renew our scientific talent

The responsibility for the creation of new scientific knowledge - and for most of its application - rests on that small body of men and women who understand the fundamental laws of nature and are skilled in the techniques of scientific research. We shall have rapid or slow advance on any scientific frontier depending on the number of highly qualified and trained scientists exploring it.

Troublesome Trends at NSERC

Sat, 30 Jul 2011 05:17:03 +0000

I was troubled to learn recently that:

1. NSERC awarded far fewer postdocs and grad student fellowships in 2011 vs. 2010.

The official statistics reveal that NSERC awarded less than half the number of PDFs in 2011 than were awarded in 2010. Master’s awards are down. Doctoral awards are down. NSERC communicated an official explanation in reply to N. Ghousshoub’s post on this news, but the numbers still trouble me.

2. NSERC staff appears to have reallocated funds across disciplines, without consultation.

In the past, changes to NSERC’s investment strategy took place in consultation with the academic scientific research community through reallocation exercises. NSERC President Suzanne Fortier’s reply to the public letter by the math/stats community reports that the reallocation exercise was "determined to be very demanding with limited return" and has been discontinued. It is reassuring that the Ministry of Industry and NSERC have requested input from an expert panel on science performance and research funding. It is unclear what role the report and recommendations from the expert panel will have on NSERC policies. What is clear, in light of the data assembled by N. Ghoussoub, is that NSERC staff has substantially changed their investment portfolio across disciplines over the past five years.

Fields Institute Establishes Fields Medal Symposium

Sat, 16 Jul 2011 05:15:36 +0000

I was very happy to receive an email yesterday announcing that the Fields Institute has established a symposium to honor the Fields Medalists. Please find the text of the announcement I received posted below:

FIELDS MEDAL SYMPOSIUM

The Fields Institute is delighted to announce the establishment of the Fields Medal Symposium. The Symposium will take place annually in Toronto at the Institute, celebrating the achievements of one of the recently announced Fields Medalists. The Fields Medal Symposium will be a three-day event featuring an address or a series of lectures for a general scientific audience by the Medalist, as well as lectures and panel discussion by other invited participants on themes related to the Medalist’s work. The Symposium is intended also as an inspiration to young people, and will include public activity with the participation of high-school or undergraduate students. The Symposium will be promoted in the Canadian and international press, and will be broadcast live throughout the world via the Fields Institute’s interactive videoconferencing facilities.

The Fields Medal Symposium is endorsed by the International Mathematical Union. It will be inaugurated in October 2012, on the occasion of the twentieth anniversary of the Fields Institute. The first Medalist honoured in this way will be Ngo Bao Chau (University of Chicago, Fields Medal 2010). The program for the first Symposium will be organized by a Committee consisting of Jim Arthur (University of Toronto, Chair), Bill Casselman (University of British Columbia), Edward Frenkel (Berkeley) and Gerard Laumon (Orsay).

The Fields Medal is the world’s most prestigious prize in mathematics. It is awarded by the International Mathematical Union every four years, to two to four mathematicians (recently four). The Fields Medal, first awarded in 1936, and the Fields Institute are both named after John Charles Fields (1863–1932), who was born in Hamilton, Ontario, and was a faculty member at the University of Toronto. Fields took a strong interest in the global world of Mathematics, and endowed the Medal in order to create an award comparable to a Nobel Prize. (There is no Nobel Prize for Mathematics.) The awards, however, have an interesting difference. The Nobel Prize is usually awarded to mature scientists to crown their careers. The Fields Medal, on the other hand, is awarded to researchers at most forty years old. It is intended not only to crown pioneering achievements but also to encourage further visionary work.

The Fields Medalist honoured on the occasion of the Symposium will receive an honorarium of $25,000. The Fields Institute has been successful in raising private sponsorship support of the Fields Medal Symposium for aneinitial eight-year period. This funding will cover the honorarium, as well as promotion of the event and the Medalist’s expenses. The Institute will provide the venue for the Symposium and cover the expenses of other invited Symposium participants from its scientific budget. The initial eight-year sponsorship will enable the Symposium to become established and attract continued funding to build an endowment.

The annual Fields Medal Symposium will be one of the highest profile events in the global mathematics community. The Fields Institute is extremely grateful to the sponsors whose generous support has made the initiative possible. Following are the current private sponsors of the Symposium. The Institute is actively pursuing corporate sponsorships. The Institute welcomes further support of the Fields Medal Symposium, at any level, as well as any other inquiries about the program. More information about the Fields Institute can be found at http://www.fields.utoronto.ca .

Sponsors of the Fields Medal Symposium

Silver Level $100,000–199,000

James Stewart, Prof. Emeritus, McMaster University, text book author, donor of the Fields Institute Library

Bronze Level $25,000--99,000

Edward Bierstone, Fields Institute and the University of Toronto
George Elliott, University of Toronto and the Fields Institute
John R. Gardner
Philip Siller, BroadRiver Asset Management, L.P.

Canada Should Leverage its Connection to the Fields Medal

Thu, 09 Jun 2011 05:12:17 +0000

John Charles Fields was born in Hamilton Ontario in 1863, graduated from the University of Toronto in 1884, and got his PhD from Johns Hopkins University in 1887. Fields helped establish the National Research Council, the precursor to NSERC. Canadians should be proud that John Charles Fields diplomatically unified the International Mathematical Union during the time between WW1 and WW2 to create the International Medal for Outstanding Discoveries in Mathematics, known colloquially as the the Fields Medal. This award is regarded as the highest honor a mathematician can receive. It is often described as the Nobel Prize of Mathematics although I don’t like that moniker.

The Nobel Prize is 10M Swedish Kronor, about \$1.5M.
The Shaw Prize is worth \$1M. (Yesterday, the 2011 Shaw Prizes were announced. Congratulations to Mathematical Sciences winners Demetrios Christodoulou and Richard Hamilton.)
The Abel Prize is worth six million Norwegian Kroner, about \$1M.
The Chern Medal is worth \$0.25M and includes the opportunity to direct up to \$0.25M to certain charities.
The Fields Medal is worth \$0.015M. Yup, \$15,000. But, it is regarded as the most prestigious mathematical award.

The Fields Medal is awarded to mathematicians under age 40 for outstanding discoveries. As a result, the Fields Medal often highlights emerging mathematical fields with recent breakthroughs made by a vibrant young researcher. The awarding of the Medal often energizes the research activity in these fields.

The Abel Prize was founded in 2003 by Norway with an initial investment of $23M. How should one assess the value of the Nobel Prize system for the country of Sweden? How should Norway value the Abel Prize? Based on the prize amounts, Canada appears to value research differently than her northern neighbors.

A Public Proposal

A one-time-only Canadian investment of ~\$5M (assuming a ~5% return on investment) would generate enough interest to raise the value for 4 Fields Medals each valued at $250K given every 4 years. In the past, the IMU’s prize committee has sometimes awarded fewer than 4 medals so \$5M might generate a fund that sustains value against inflation. Through philanthropy or government investment (and subject to peer review), Canada should raise level of the Fields Medal.

(Why $250K and not $1M? The Fields Medal is awarded when a researcher is in full creative bloom, and before the age of 40 years old. The 40 year rule fundamentally distinguishes the Fields Medal from the Nobel Prize. My view is that the award should be substantial but not life-altering, assuming a typical academic salary and lifestyle. )

Canada should leverage its connection to the Fields Medal starting at the next International Congress of Mathematicians in Seoul in August 2014:

The Prime Minister of Canada should award the medals at the next ICM to be held in Seoul, Korea in August 2014.
The raised award amounts and the participation by Canada’s government will generate interest in mathematics, the Fields Medal and in Canada, as a country that values the advancement of knowledge.
Canada’s mathematical research institutes (PIMS, BIRS, Fields, Perimeter, CRM) should jointly organize a year of concentration targeting the rapidly developing areas highlighted by the Medals during calendar year 2015. The four month interlude between the announcement of the Medals at the ICM and the beginning of 2015 can be used to plan the programs. Summer schools should be designed to bring advanced undergraduate and graduate students into the stream of ideas and discoveries highlighted by the medals. In principle, funds for these activities are already committed by NSERC through its long term commitment to the institutes.
Canada should aim higher and build new mathematics education practices intertwining research universities and the public schools using interactive technologies towards long-term goals like:
- Canada should set a national strategy to win the International Mathematical Olympiad. Milestones: consistent top ten finishes within five years; top five finishes thereafter.
- Canada should rejuvenate K-12 enriched mathematics education toward growing (not importing) a Fields Medalist in the next 12 years.
- Canada should produce a female Fields Medalist.

The International Medal for Outstanding Discoveries in Mathematics, colloquially named after its Canadian founder John Charles Fields as the Fields Medal, is the world’s top honor for mathematical advancement. Canada should make the value of this award commensurate with its prestige and make long term plans to leverage that investment.

Unfortunately, the window for public submissions to the six member expert panel reviewing Canada’s federal research and development policy has closed. Nevertheless, I hope that this proposal is considered by that review panel and by the Science, Technology and Innovation Council. The recent budget announcements by the Harper and McGuinty governments include stunning investments in research on the brain ($100M), the genome ($65M), optics ($45M), the Perimeter Institute (Federal $50M, Ontario $50M) and the SR&ED program (~$4B, yes, Billion) giving tax breaks to business claiming R&D expenses. Research and development investments by governments, like the one proposed here, and those recently announced should be made strategically and through a transparent and robust peer review system.

A New Dawn for Math and Stats in Canada

Sat, 21 May 2011 05:53:36 +0000

The recently released 2010 International Review of Mathematical Sciences for the UK has a timely quote for the Canadian statistical and mathematical communities to consider (see page 10):

"A longstanding practice has been to divide the mathematical sciences into categories that are, by implication, close to disjoint. Two of the most common distinctions are drawn between ‘pure’ and ‘applied’ mathematics, and between ‘mathematics’ and ‘statistics’. These and other categories can be useful to convey real differences in style, culture and methodology, but, in the panel’s view, they have produced an increasingly negative effect when the mathematical sciences are considered in the overall context of science and engineering, by stressing divisions rather than unifying principles. Furthermore, such distinctions can create unnecessary barriers and tensions within the mathematical sciences community by absorbing energy that might be expended more productively."

Yesterday, a majority of the Evaluation Group (1508) which adjudicated the 2011 NSERC Discovery Grants competition posted a public letter to NSERC President Suzanne Fortier. This letter finally provides some official insights into the 2011 competition that I publicly requested on April 11. The letter also validates the concerns expressed by the public statement by the Math NSERC Liaison Committee from April 26 which has (so far) been signed by:

Yesterday's letter clarifies the responsibilities of the Evaluation Group, its Executive Committee, and NSERC staff in evaluating the proposals and assigning dollar amounts. We now know that the Executive Committee was allowed to give different dollar amounts to proposals with equal merit. Stats proposals were given more money than math proposals despite being evaluated as having equal merit. We also know that proposals evaluated in 2010 were given substantially larger money amounts than equal merit proposals in 2011. These outcomes are unfair. The 2011 aftermath provides an opportunity for NSERC to evaluate and improve the new evaluation system used for adjudicating the Discovery Grants competition.

The statisticians on the Evaluation Group did not sign yesterday’s public letter to the NSERC President.

The unequal treatment of statistics and mathematics during the 2011 competition has provoked a rift between these two communities of Canadian researchers. These two communities were forced to merge by NSERC, against the wishes of the statistics community, in 2009. Although the statisticians on the Evaluation Group did not sign yesterday’s statement, three notable signatories to the public statement by the Math NSERC Liaison Committee are prominent statisticians:

President of the Statistical Society of Canada
Chair of the Research Committee of the Statistical Society of Canada
Chair of the Long Range Plan for the merger of math and stats.

Now is the time for mathematicians and statisticians to collaborate to improve the system for distributing federal investments in research. Let's work together to avoid being divided and conquered.

NSERC Peer Review System is Broken for Mathematics

Tue, 12 Apr 2011 05:49:49 +0000

Anomalous results of the 2011 NSERC Discovery Grants competition in mathematics have provoked a loss of confidence in the NSERC peer review system. To avoid a substantial loss of Canada’s scientific talent, which has been enhanced through the Canada Research Chairs program and other spectacular hiring over the past ten years, scientific policymakers need to quickly fix the broken peer review system. In the absence of an effective peer review process setting the strategy for research investment, Canada will miss out on the rewards made over the past decade’s recruitment of scientific talent.

What is happening in other sciences? Anecdotal reports from the following sources suggest the anomalies are not restricted to the math department at Toronto:

Toronto: MATH, CHM, EEB, PHY, STA, Engineering
UBC: CS, MATH
Queens: MATH

What happened to other disciplines? I’d like to know but NSERC won’t reveal the 2011 data until after the federal election. I’d like to hear from other scientific disciplines about their confidence in the recently transformed NSERC peer review system.

In 2007, NSERC commissioned a review by an international committee culminating in this report. (Please find my annotated version here and a marked up version of the NSERC Management response to the 2007 International Review Committee Report.) This report made recommendations leading to fundamental changes in the peer review process for all disciplines starting in 2009. The implementation of these changes (involving the so-called conference model and binning system) and other forces have provoked a loss of confidence in the peer review process at NSERC among mathematicians at Toronto, and across Canada.

Toronto Math Results are Anomalous

The results (names omitted) of the 2011 NSERC Discovery Grants Competition for the Department of Mathematics at the University of Toronto are anomalous:

Professor A. \$29k/y to \$18k/y
Professor B. 40 to 15
Professor C. 42 to 30 to 42 to 18
Professor D. 26 to 18
Professor E. 40 to 40
Professor F. 38 to 47
Professor G. 0 to 0
Professor H. 15 to 13

(The numbers represent annual research grant amount in dollars for the past 5 years and the new number for the next 5 years. For a description about how mathematician’s use these funds, go here.)

About Professor C.

Consider the case of CMS award winning Professor C. In 2010, this researcher’s grant was cut from 42 down to 30. After an appeal, the grant was reinstated for one year back to 42. In this year’s competition, one year after the appeal, NSERC drops it to 18, a 57% cut. Meanwhile, Professor C’s frequent collaborator (each had more than 50% overlap of their research with the other during 2006-2011), Professor K., received 45 staying at 100% of the previous level in this competition. Will the real opinion of NSERC on Professor C’s research please stand up? Professor C’s story is quite similar to Don Fraser’s personal account. (Within the conference model, I understand that a different group of only 5 experts may have reviewed the proposals of Professors C and K. This remark can account for the inconsistency but reveals aspects of larger issues that need to be fixed.)

About Professor B.

Imagine running a successful research operation (success, former students get awards, 13 major pubs in 2006-2011) for the past five years like Professor B using 45K/y. Students are in the pipeline; postdoc candidates have been scouted; Professor B has new ideas. NSERC rewards this person with a drop from 45 down to 18, a 60% cut. This researcher is confused with the outcome: “What did I change? What should I have done differently?”

About Professor G.

This person is a (perhaps the) world leading expert on a substantial research area. It seems this person, despite spectacular research success, is unworthy of a Discovery Grant because they don’t produce enough students. It is as though Canada has a sports car and they don’t put tires on it.

Secondary Effects Scenarios

Mathematicians, of international calibre, with a steady research production stream and surrounded by young researchers have had their grants slashed by nearly 60% during the 2011 NSERC Discovery Grants Competition.

Now, imagine you are an assistant professor in Canada. You might have nice support right now, like a Sloan or an ERA. You are building a research group, spending money on HQP, scouting talent. But your funding has a finite time horizon and the Discovery Grants look unstable, unpredictable. So, would you leave Canada if you could? Fix NSERC or the young talent will leave Canada.
Now, imagine you are a recently recruited Canada Research Chair: if you just concluded that your junior faculty member might be wise to leave Canada, how do you see your department in 10 years? Would you leave Canada if you could? Fix NSERC or the CRCs will leave Canada.

The system is broken and needs to be fixed.

There are (at least) two main problems:

Math in Canada is treated unfairly compared to other disciplines
The Peer Review system is broken

Math in Canada is treated unfairly

The main problem with mathematics funding in Canada is the amount invested is too low. I’ve written about this before. Consider the data from 2009 of NSERC Discovery Grants (2010 is similar, 2011 is not available) over the disciplines:

The average math and stats grant is $20K/y while the average over all disciplines is $41K/y. Why is it that the average scientist in Canada can expect more than double the amount a Canadian mathematician can expect? Keep in mind that Discovery Grants are primarily used to fund research personnel not expensive labs.

David Wehlau’s data (posted and discussed here) reveals the trend: over the past twenty years, mathematics investment as a percentage of the total amount in Discovery Grants funding has declined from nearly 4% down to 2%. Math has received less and less funding compared to other disciplines. This subtle reallocation needs to be abruptly reversed.

The unfairness then multiplies. Consider the following snippet from the 2007 international review report: Other disciplines are benefitting more from other industrially targeted NSERC programs and other sources compared to pure mathematicians. NSERC views Discovery Grants as grant-in-aid: a precursor grant leading to other sources of funds. The international review committee reports that is not the case for mathematics AND mathematicians receive on average less than half the funds received by other scientists. This is an implicit funding reallocation away from mathematics toward other disciplines and is unfair.

Broken Peer Review System

The outcome of the 2011 competition, and consistent reports (like Don Fraser’s) from the past two years, have provoked a loss of confidence in the peer review system at NSERC. To rebuild trust and avoid the departure of talented scientists, the mathematics community of Canada needs to know what happened in 2011. We need to understand why the peer review system produced the 2011 funding allocations.

There has been considerable chatter in the mathematics community about the 2011 competition. However, we need people with official roles to speak officially at this time. In addition to the forthcoming data from NSERC, I hope that Section 1508, the Mathematics and Statistics Evaluation Committee will explain the 2011 evaluation process and actively participate in discussions leading to an improved system that regains the confidence of mathematicians and statisticians working in Canada. What happened? How can we fix it?

Mathematics and the Library at the University of Toronto

Wed, 06 Apr 2011 05:47:23 +0000

Overview

UTL has built, and is in the process of expanding, a suite of web-based tools that have the potential to revolutionize the research driven by scholars at the University of Toronto. Some of these tools are designed to open the library’s archival system to new content streams generated within the University’s daily research activities. Other tools provide new methods to collect, share and discuss library resources enabling new ways to collaborate and develop research ideas. The technologies underpinning these advances are already in place or soon to be deployed. The next main challenge is the principal obstruction: we need to stimulate the researchers to use them. My view is that they won’t unless they see a payoff for their research goals.

New tools and their potential

A quick summary of web-based resources provided by UTL organized along four key categories:

Archival
Scholarship
Collaboration
Broadcast

Archival

The library’s tspace is an online repository for storing content generated by the University of Toronto. This resource is a bit like arXiv but localized to UofT and it is open to a wider variety of content types (movies, scans of hand-written notes, etc.) than arXiv. Once uploaded to tspace, the content is permanently archived, given an immutable web location address and is strategically exposed to web search engines. All PhD theses produced here end up on tspace. The library wants scholars to upload preprint and postprint versions of published papers to tspace. Notes should be uploaded to tspace enabling researchers to later make references to the ideas emerging from seminars, etc. The remarkable feature of tspace is that it enables UofT researchers to directly influence the Library’s collection. However, the success of the tspace repository is limited without the active participation by UofT researchers who are plenty busy with other tasks.

Scholarship

The library has piloted and will soon deploy BibApp, a web gateway organizing library content around the scholars at UofT who produced it. (Here is my page inside the test server.) BibApp’s database of citations is automatically filled (with UTL sometimes doing massive amounts of data cleaning) using content streams from resource amalgamators like Scholar’s Portal. However, BibApp also leverages the expertise of the scholar’s who appear inside BibApp by allowing them to edit and improve BibApp’s content about their scholarship. The application shows scholars information about the copyright restrictions of their published works:

Those references which appear green can be enriched with postprint and preprint versions of the published work. BibApp will be intertwined with tspace so that uploads into tspace can take place within the BibApp pages.

Collaboration

Peter Clinton reports to me that the library will soon provide storage space enabling faculty to build and share citation collections through zotero. Zotero provides rapid collection of reference data on published works, preprints on arXiv, web pages, etc. and stores the collected data in a database that is accessible through the web. Each citation is stored inside a container into which other data can be developed. This technology allows for precise exchanges of ideas to take place among the references shared by a group of scholars. This can be extremely useful for a research group led by a UofT faculty member or for a broader group of researchers with similar interests.

Broadcast

The library has invested in broadcast video streaming technology and built MyMedia. MyMedia is UofT’s version of youtube. Videos generated by UofT faculty can be uploaded in their native format. The MyMedia server converts and makes available the video content in various formats so that the content can be watched on iPhone, Android, through a web browser, etc. The library has also made recording and screen capture technologies available to UofT faculty so it is easy to generate video content for instruction and research purposes.

But, will people use this technology?

The technology is in place. However, the experience of libraries across the world over the past ten years building these types of tools spawns a common observation: to make it really work requires changing the behavior of people.

UTL and Toronto Math share some resources. These resources will be deployed in pilot programs to encourage and make it easier for math faculty to take advantage of the Library’s technologies. The shared resources should be spent strategically to create short term examples that mathematicians can recognize as valuable within their research plans. We need examples that show how these tools enhance the research potential of faculty-led research groups to inspire adoption of the technology into the daily work flow of researchers at UofT. Updates to follow….

The Lucky Few of Waterloo: Does the Perimeter Institute deserve $50M Times Two

Thu, 31 Mar 2011 05:44:16 +0000

There is chatter (here is more) suggesting that the $50M from the Conservative federal government (over 5 years) and the additional $50M (also over 5 years) from the Ontario Liberal Government to the Perimeter Institute is based more on politics than on scientific merit. These funding announcements emerge just a few weeks after the news that Neil Turok, Director of the Perimeter Institute, joined the Science, Technology and Innovation Council which advises the government on science policy. The chatter resonates with other statements that scientists from Western Canada get more than their fair share, that scientists from Toronto are discriminated against in funding decisions, and that Quebec scientists get funding just to allay separatist agitations. The key difference is that scientists outside of Perimeter must compete for funds through the Tri-council granting process which, at least in principle (but not in practice for mathematics), provides accountability and selects for scientific success.

The peer review system leverages the expertise of leading scientists to assess proposals for research investment by the government. When it works well, the process is trusted by the community of scientists to be based on scientific merit. A trusted peer review process is the ecosystem in which scientific creativity and excellence flourish. When the selection of scientific investments is perceived to be based upon factors other than scientific merit, the entire system is destabilized. The goals of research funding will not be met if the distribution system provokes poisonous comments against fellow scientists. The politicization of research funding and the ensuing unscientific conversations inside the community of scientists distract us from our agenda to advance the basic understanding of everything.

Scientific research investments selected by scientists through effective peer review are strategically superior and less risky than research investments made by politicians.

Consider the numbers:

The annual amount for 2009-10 Discovery Grants funding for biology, chemistry, physics, mathematics, computer science, all types of engineering,....was 64 Million dollars distributed over around 2000 researchers. The average individual grant amount was $33K.

In 2009-10, the Perimeter Institute had 12 Faculty, 12 Associate Faculty, 47 postdoctoral fellows, and 25 PhD students. The planned 100 million dollar investment over five years yields 20 million dollars per year. Let's imagine PI has 50 principal investigators. This averages to an individual grant amount of $ 400K ~ 12 X $ 33K. (Keep in mind that Perimeter Institute faculty also have Discovery Grants, that we are making underestimates, and that PI has other substantial recent investments.)

It is not obvious that the average Canadian scientist merits 12 times less research funding than the average scientist at the Perimeter Institute.

Canadians should insist on an accountable distribution system of government investments in research. We should demand an effective peer review process in the distribution of all research and development investments made by the government, even those funds distributed outside the Tri-council umbrella like the allocations for Perimeter and the $4.7 Billion in SR&ED tax breaks. The Perimeter Institute might merit these investments but the scientific innovation system in Canada is threatened by earmarked research investments chosen without peer review.

Vannevar Bush

Wed, 30 Mar 2011 05:41:18 +0000

Vannevar Bush was the inventor of government investment in research innovation. He was founder of the US National Science Foundation, founder of Raytheon, main organizer of the Manhattan Project which influenced Berkeley leading to Silicon Valley, etc. Here is a timely extract from his letter to President Roosevelt entitled Science: the endless frontier:

Five Fundamentals
There are certain basic principles which must underlie the program of Government support for scientific research and education if such support is to be effective and if it is to avoid impairing the very things we seek to foster. These principles are as follows:

Whatever the extent of support may be, there must be stability of funds over a period of years so that long-range programs may be undertaken.

The agency to administer such funds should be composed of citizens selected only on the basis of their interest in and capacity to promote the work of the agency. They should be persons of broad interest in and understanding of the peculiarities of scientific research and education.

The agency should promote research through contracts or grants to organizations outside the Federal Government. It should not operate any laboratories of its own.

Support of basic research in the public and private colleges, universities, and research institutes must leave the internal control of policy, personnel, and the method and scope of the research to the institutions themselves. This is of the utmost importance.

While assuring complete independence and freedom for the nature, scope, and methodology of research carried on in the institutions receiving public funds, and while retaining discretion in the allocation of funds among such institutions, the Foundation proposed herein must be responsible to the President and the Congress. Only through such responsibility can we maintain the proper relationship between science and other aspects of a democratic system. The usual controls of audits, reports, budgeting, and the like, should, of course, apply to the administrative and fiscal operations of the Foundation, subject, however, to such adjustments in procedure as are necessary to meet the special requirements of research.

Basic research is a long-term process - it ceases to be basic if immediate results are expected on short-term support. Methods should therefore be found which will permit the agency to make commitments of funds from current appropriations for programs of five years duration or longer. Continuity and stability of the program and its support may be expected (a) from the growing realization by the Congress of the benefits to the public from scientific research, and (b) from the conviction which will grow among those who conduct research under the auspices of the agency that good quality work will be followed by continuing support.

Rotman Dean to Government: Give the basic research funding to business schools not scientists

Fri, 18 Mar 2011 05:38:26 +0000

Dean Roger Martin’s remarks in the Globe and Mail yesterday threaten Canada’s intellectual infrastructure and therefore merit the attention of all Canadians, especially policymakers planning the upcoming federal budget and researchers in Canada’s universities. Amazingly, he asserts:

“What makes a country prosperous is not investment in science and technology. It is businesses producing high paying jobs by having unique products and processes that a customer needs.”

Who does he think creates those products and processes? Business majors? His observation will come as a shock to the Chinese, who are working hard to build a competitive modern economy by investing heavily their research resources in the basic sciences and engineering in their universities, not in business schools. They know these investments will create the industries of the future, just as they have for the west in the past.

What is amazing about Martin’s claim is that virtually all of the “high paying jobs” in industries with “unique products and processes” – those leading edge industries like communications technology, pharmaceuticals, bio-technology, etc. that Martin so casually dismisses, have products that are the result of investment in university-based research in science and technology.
In the United States, Silicon Valley, which houses many of the companies he says are now led by CEOs with business degrees, was created by scientists and engineers from Stanford and the University of California.
Indeed, it would be difficult to identify a single industry today that was not based on a scientific or technological innovation of the past.
Business does not create business; creative innovation creates business.

After claiming that Canadian business schools are broke ($200M Campaign, New building), Martin cites statistics comparing the percentage of students in a discipline versus the percentage of tri-council research funding given to that discipline. Martin argues that the percentage of students in a discipline should equal (or at least influence) the percentage of tri-council funding given to that discipline. This is a terrible idea. Martin appeals here to the value of fairness (“In business….it’s a 10 to 1 cut”). If taken seriously, Martin’s reallocation plan restricts policymakers from defining the government’s research investment portfolio; it prevents them from making targeted investment decisions. The plan potentially forces money to be poured into disciplines which happen to be popular with undergraduates at the time (whether frivolous or not). Science policy should not be determined by a popularity contest among undergraduate major choices. I prefer the old ways to think: scientifically, strategically.

Roger Martin’s viewpoint will hurt Canada in the short and long term if taken seriously now. Basic scientific research has produced (among other things):

electricity (why you aren't in the dark, can take an elevator in a high rise, and have clean laundry)
vaccines (partly why you are not already dead)
transistors (performing the basic functions on which computers are built, enabling the internet)
encryption (why you can safely make financial transactions online and make other secure communications)
combustion engines (how your car generates power for motion)
financial derivatives (so that insurance companies and banks can prepare for risks)
search engines (how we manage information overload on the internet)

What percentage of the workforce relies on these developments? In contrast, what basic elements of the modern economy can be attributed to MBAs? (Perhaps MBAs can claim to have invented mortgage backed securities and credit default swaps leading to to the financial meltdown of 2008- 2009?)

Martin highlights a list of technology companies (Hewlett-Packard, IBM, Microsoft, Apple, Cisco and Intel) and reports that many of their CEOs have MBAs. Look up the history of these companies and you will find:

Bill Hewlett and David Packard were electrical engineers.
Microsoft
- Bill Gates was a math major/computer scientist at Harvard before founding Microsoft.
- Paul Allen also studied computer science.
- Steve Ballmer majored in mathematics and economics.
Apple
- Steve Wozniak is a computer scientist and an electrical engineer.
Cisco
- Len Bosack started as a computer scientist.
- Sandra Lerner has a Bachelor’s degree in political science, a master’s degree in econometrics and a master’s degree in statistics and computer science.
Intel
- Gordon Moore PhD in Chemistry and minor in Physics.
- Robert Noyce Ph.D. in physics
Same story for google (Sergey Brin is a computer scientist/mathematician, Larry Page is a computer scientist).
Same for facebook.
Same for Research in Motion, ….

MBA CEOs may find efficient ways to bring widgets to the market and make a profit in the process. But the widgets are invented by scientists and engineers through basic research targeting science and engineering, not business education or business research. Business majors and MBAs are important in the operations of businesses that create new products, so business education is important. But to suggest that it is primarily responsible for new products and innovations creating business is just plain wrong. Misconceptions about the impact and role of basic research funding by business leaders may be one of the largest obstructions currently contributing to Canada’s innovation gap.

The challenges we face in Canada, and as human beings on earth, require new ideas. The long research lines of science (pioneered by Galileo, Newton, Gauss, Euler, Darwin, Riemann, Einstein, Curie, Schrödinger,…,Banting, Polanyi,…) have consistently produced innovation and prosperity, and therefore merit vigorous and consistent government funding.

Mathematics Discovery Grants are Insufficient and Broken

Thu, 10 Mar 2011 06:35:15 +0000

Beginning this academic year, I have served the Department of Mathematics as the Associate Chair [Research]. The principal responsibility of this position is to administer the hiring process, for tenure stream and postdoctoral appointments. I also recently served on the Connaught Physical Science Review Panel which reviews applications for Connaught funds and adjudicates the McLean Award. Serving on the Connaught Panel has given me a perspective on the research climate in fields outside of mathematics. Metaphorically, these responsibilities have required me to look up from my research papers and out the window at the infrastructure supporting researchers at UofT. I am troubled by what I have seen:

Here is a table generated by NSERC showing the Discovery Grants (DG) data across disciplines.

Discovery Grants are Insufficient

On average, Canadian mathematicians receive about $20K/y to sustain their research program. This post by Izabella Laba explains how mathematicians typically use this money at UBC. As part of the planning exercise, I suggest we consider how much money a mathematician needs to run a research program. Simple considerations show that $20k is insufficient.

At Toronto, faculty are encouraged to contribute one quarter of their NSERC DG to support graduate students. This amounts to $6K from a typical $20K NSERC DG.
Faculty pay some incidental expenses related to computing infrastructure costs, printing, etc. Let's suppose these total about $1K.
Collaborative mathematical research requires travel to bring the coworkers together. A typical research visit requires an airfare purchase and a hotel room. Each research visit costs between \$1K and \$2K.
The development of graduate students and postdocs into researchers often requires conference participation. The associated travel costs are frequently paid using the advisor's Discovery Grant. Local expenses for sponsored graduate students are often paid by the conference. Each field trip by a graduate student or postdoc costs about $500, sometimes more if the conference is outside of North America.

What does minimal research activity look like? What does it cost? Minimal research activity might involve, say, three to four research visits over the calendar year, and one graduate student research field trip. These might add up to $4K.

So, a minimally active researcher with a typical research grant of $20K spends $6K supporting a graduate student, $1K on incidentals, and $4K on research visits leaving $9K.

At Toronto, postdoctoral positions require between \$32K and \$40K from faculty research grants. This money is supplemented with a teaching stipend to complete the salary package for the postdoc.

The minimally active mathematician with average funding can barely afford one third of a postdoc. Therefore, it is necessary to combine funds with like-minded colleagues to generate a postdoc position. The need to build funding alliances prevents typical faculty members from choosing the postdoctoral candidate with the most synergy with their research program. Instead, the typical researcher needs to look for a candidate that two of their colleagues will also like enough to spend \$10K or more to have in the department. Instead of recruiting postdocs with potentially explosive overlapping research interests, we make deals just to get someone in the department and hope for some resonance after training the Postdoc on our research topic. Of course, all three faculty members want the person they hire to contribute to their research program. So the Postdoc is encouraged to learn background materials in three (hopefully related) areas which are not strongly linked to their thesis area. Except in rare cases with unexpected synergy, this arrangement is a failure factory.

Mid-Career Funding Gap

Early research awards (ERA, Sloan, etc.) fund research activities by new faculty. These opportunities are restricted by a time horizon typically around 10 years after the PhD. Young faculty at Toronto with these grants can often solely fund the research component of a Postdoc funding package.

Similarly, eminent senior members of our department with large grants can solely fund postdocs.

Midcareer mathematicians typically must make a deal with their colleagues to assemble the funds to to hire a Postdoc.

Canada’s Discovery Grants funding policies do not adequately support research activity by mathematicians and especially hurts researchers in the middle of their career. Young researchers who win early research awards in Canada are capable of starting a research program. The present funding structure does not allow these emerging mathematicians to fully develop through the mid-Career phase into world class research leaders. Canada has been very effective at recruiting talented young mathematicians during the past decade. When young research stars in Canada start to realize the mid-Career funding gap prevents them from carrying out their research plans, they will leave and go elsewhere.

HQP training is not necessary for spectacular research success.

Why does NSERC require an HQP (Highly Qualified Personnel) component in research plans? I expect the answer has to do with the general goal that the government policies should encourage the training of a highly skilled population. This is a good goal. However, the HQP requirements for Discovery Grants fail to envision the secondary effects of world leading research on the training of future scientists. Consider, for example, the case of Jean Bourgain. Bourgain had one Ph.D student so his HQP production would be viewed as insufficient to merit a Discovery Grant by NSERC. However, Bourgain's advances have created new fields of mathematics where a generation of mathematicians (e.g. Izabella Laba, Wilhelm Schlag, Gigliola Staffilani, Terry Tao ...) has blossomed. The NSERC definition of HQP fails to envision the effects of ground breaking research in the development of future scientists.

Henri Poincaré had five graduate students. John von Neumann only had three students. I wonder if they’d qualify for a Discovery Grant given the highly qualified personnel requirements in the funding formulae? My colleague Victor Ivrii made an historical advance by proving Weyl’s conjecture about the eigenvalues of the Laplacian on a domain. Due to the HQP requirement, Victors’s DG is zero. Similarly, my colleague Michael Goldstein advances the frontier with big results which appear in Annals, Annals (again), Acta, etc. and his efforts are rewarded with a $14K/y NSERC DG. Goldstein’s research excellence has been recognized outside of Ottawa.

In my opinion, the fact that Canada fails to invest in the scholarly activity of its world-class researchers is a disgrace. The Discovery Grants system needs to be fixed.

Research Program Profiles should be Defined

The mathematical community, perhaps through the long range planning exercise, should formulate typical budgets required for mathematician’s operating at different levels. How much money is required to effectively run the research enterprise for a mathematician? Of course, the answer depends upon career stage, level of engagement with students and postdocs, etc. The mathematical community should define funding requirements for a collection of research program profiles (parametrized by quantity and quality, and HQP level ranging through zero, some, to lots of junior collaborator participation in the research). If NSERC wants to put us in bins, mathematicians should define those bins, not NSERC staff, and we should insist on adequate funding to allow Canada's mathematical research community to emerge as world leading. Presently, there are too many government generated obstructions preventing our ascension.

Edinburgh Meeting Notes 3

Fri, 21 Jan 2011 06:33:06 +0000

Galina Perelman: 2 soliton collision in NLS

$$ i \partial\_t \psi = - \psi_{xx} + F(|\psi|^2) \psi, ~ x \in R $$ where $F(\xi) = -2 \xi + O (\xi^2), ~ \xi \rightarrow 0.$

This family of equations has solitary wave solutions $$ e^{i \theta(x,t) \phi (x - b(t), E)} $$ where $\theta(x,t) = \omega t + \gamma + v \frac{x}{2}, ~b(t) =vt + c$ (all reall parameters). The profile $\phi$ is the associated ground state, which is $C^2$, decays exponentially, is even, …

If I set $\epsilon^2 = E$ and write $\phi(y, \epsilon^2) = \epsilon \hat{\phi}(\epsilon, \epsilon).$ We have then that $\hat{\phi}(z, \epsilon) = \phi_0 (z) + O(\epsilon^2)$ where $\phi_0$ is the standard soliton for cubic NLS. A calculation shows that $$| \phi(\cdot, \epsilon^2) |_{H^1} = O(\epsilon^{1/2}).$$ Let’s collect the parameters $\sigma = (\beta, E, b, v) \in R^4.$

The question I’d like to address:

Question: As $t \rightarrow -\infty$, suppose that $\psi(t) = w(\cdot, \sigma_0 (t)) + w(\cdot, \sigma_1 (t)) + o_{H^1} (1)$. Because of the galilean invariance we can arrange so that $\sigma_0$ does not move and we assume that $v_1 > 0$. So, we can arrange this data to have completely decoupled solitons as $t \rightarrow - \infty$. The question is then to understand the soliton collision and also what happens afterwards.

Perturbative regime: $$\epsilon^2 = E_1 \ll 1, E_0 \thicksim 1, v_1 \thicksim 1.$$

Collision Scenario:

$w(\cdot, \sigma_0 (t))$ is ‘preserved’.
$w(\cdot, \sigma_1 (t))$ splits into two outgoing waves of the cubic NLS. The splitting is controlled by the linearized operator associated to the large soliton $w_{\sigma_0}$.

Collision: $|t| \lesssim \epsilon^{-1-\delta}, ~ \delta > 0$. pre-interaction: $t leq - \epsilon^{-1-\delta}$ post-interaction: $t leq - \epsilon^{-1-\delta}$

She draws a picutre:

Long wide soliton to the left of a big soliton at the origin before the collision. After the collision the small soliton splits into two waves, one moving left and one moving right. The big soliton at the origin is drawn not centered at the origin.

$s = s(\frac{v_1}{2}), r = r (\frac{v_1}{2})$ where $s(k), r(k)$ are the translation and reflection coefficients of the linearized operator corresponding to $w(\cdot, \sigma_0 (t))$. Here we have $|s|^2(k) + |r|^2 (k) =1$. The only trace of nonlinearity appears in the phase.

This phenomena has been observed before by Holmer-Mazuola-Zworski and earlier by physicists. H-M-Z conisdered the cubic NLS with an external delta potential. For small incoming solitons, they have observed the small soliton splitting caused by the Dirac function potential.

Hypotheses:

(H0): $F \in C^\infty, F(\xi) = - 2 \xi + O(\xi^2), \xi \rightarrow 0.$ $F(\xi ) \geq - C\xi^q, C>0, q<2, \xi \geq 1$. (GWP in $H^1$) $\exists !$ ground state.

Linearization around $w(x, \sigma(t)) = e^{i\theta} \phi(x - b(t), E)$. We substitute $\psi = w + f$ and expand to obtain the following equation for $f$:

$$ i {\bf{f}}_t = L(E) {\bf{f}}. $$

Here ${\bf{f}}$ is a (column) vector $(f, \overline{f})$. $$ L(E)= (-\partial_y^2 + E) \sigma_3 + V(E). $$ Here $\sigma_3$ is the Pauli matrix and $V$ is a certain matrix involving $V_1 = F(\phi^2) + F’ (\phi^2) \phi^2$ and $V_2 = F’ (\phi^2) \phi^2$.

She draws a spectral plane. Essential spectrum along real line in region $|x| > E$ and some eigenvalues drawn as x’s inside the gap and one above and below the real line on the imaginary axis. 0 is an eigenvalue. We have two explicit eigenfunctions $\xi_0$ and $\xi_1$.

$M(E)$ is the generalzied null space of $L(E)$. We have the following equivalence:

$$\sigma(L(E)) \subset R, {\mbox{dim}} M(E) = 4 \iff \frac{d}{dE} | \phi(E) |_2^2 > 0.$$

These conditions imply the orbital stability of $\Phi$.

$Lf = \lambda f, ~\lambda \geq E, \lambda = E + k^2, ~ k \in R$. If $k^2 + I \notin \sigma_p (L(E))$ then $\exists ~! ~ f(x,k) = s(k) e^{i k x} (1, 0)^t + O(e^{-\gamma x})$ as $ x\rightarrow + \infty, ~ \gamma > 0$ and $f(x,k) = e^{ikx} (1,0)^t + r(k) e^{-ikx}(1,0)^t + O(e^{\gamma x}), x \rightarrow - \infty$.

$w(x,\sigma, t), ~ j=0,1$ normalized as before.

(H1): $$\frac{d}{dE} | \phi(E) |2^2 |{E=E_0} > 0$$

(H2): $\epsilon^2 = E_1$ sufficiently small

(H3): $M(E + \frac{v_1^2}{4}) \notin \sigma_p (L(E_0))$ (Nobody knows how to prove no embedded eignevalues.)

Proposition: $\exists ~! ~ \psi \in C(R, H^1)$ such that ….

Theorem: For $\epsilon^{-1-\delta} \leq t \leq \delta \epsilon^{-2} | \ln \epsilon |$

$$ \psi (t) = w (\cdot, \sigma(t)) + \psi_+ (t) + \psi_{-} (t) + h(t)$$

$\sigma(t) = (\beta(t), E_0, b(t), v_0), ~V_0 = \epsilon \kappa$ where $\kappa$ is an explicit constant and $$ |\beta(t) - \beta_0 (t)|, |b(t) - v_0 t| \leq C \epsilon^2 t. $$
$\Psi_{\pm} (x,t) = ….ack too fast to type… $$\Psi_{\pm}$$

is expressed as an explicit phase times a function $S^{\pm}$ which solves cubic NLS emerging from data built using thre reflection, transmission coefficients and $\phi_0 (y)$. 3. error estimates in terms of $\epsilon.$

Edriss Titi: Loss of smoothness in 3d Euler Equations

(joint work with Claude Bardos)

Overview:

Background
- Euler
- Classical
- Nonuniqueness: De Lellis - Sh…
Shear flow
- DiPerna Majda example: weak limit of Euler solutions whose limit is not a solution
- Illposedness of Euler in C^{0,\alpha}
Vortex sheets induced by 3d shear flows
- Examples
- Differences between 2d and 3d Kelvin-Helmholtz problems
- Comments on numerics

Euler equations

Euler equations on the 3-torus. $\omega$ is the vorticity. Recast using Biot-Savart.

Vorticity stretching term distinguishes 2d and 3d.

Classical Wellposedness:

global existence and uniquenes for initial data $\omega_0 \in L^\infty$. This result is due to Yudovich (1963). Some extension….
For data in $C^{1,\alpha}$, Euler equations are short time well-posed and the solution conserves energy. [Lictenstein (1925)]
The same result holds the context of Sobolev spaces $H^s, ~ s > \frac{5}{2}$. (Basically same result in more modern spaces)

Question: Does there exist a regular solution (say in $C^{1,\alpha}$) of the 3d Euler equation that becomes singular in a finite time (blows up problem)? This is in osome sense as difficult as the millenium problem. There are different opinions….

“I spoke with Necas about this…near end of his life…on Wendesday’s he thinks it blows up and on Thursdays he thinks no…so he has bad dreams about it…”

DeLellis-Szekelyhidi: There exists a set of initial data $u_0 \in L^2 (\Omega)$ (not explicitly constructed, Baire argument) for which the Cauchy problem has, for the same inital data, an infinite family of weak solutiosn of the 3d Euler equations: a residual set in the space $C(R; L^2_{weak} (\Omega))$.

These are also in $L^\infty$ so they have finite energy. (Built on Shnirelman and others….). This is a breakthrough…but it is not so physical. Maybe a selection mechanism….for NS we don’t have such a result. Leray solutions are not known to be unique. Any result like this for NS would be extremely important….connect it with turbulence. The lack of uniqueness, according to Leray, relates to turbulence.

Shear flows:

$$u(x,t) = (u_1 (x_2), 0, u_3 (x_1 - t u_1 (x_2))).$$

For $u_1, u_3 \in C^1$, the above shear flow is a classical solution of the Euler equations with pressure $p=0$. Yudovich used these to show the existence of solutions with exponentially growing high regularity norms.

This example due to DiPerna-Majda (1987).

Theorem (DiPerna-Lions): Norm explosion in $W^{1,p}$ for Euler, for any $p \geq 1$.

Idea of the proof: $ \partial_{x_2} u_3 (x_1 - t u_1 (x_2))=…$

Theorem: The shear flow is a weak solution of the Euler equations in the sense of distribtuions in $R^3$, provided $u_1, u_3 \in L^2_{loc} (R^3)$. On the periodic box, we can do same thing and in this case we have finite energy.

Why do I stress the finite energy? This relates to the Onsager conjecture.

Theorem [Ill-posedness of the Euler equations in $C^{0,\alpha}$]:

The shear flow with $C^{1,\alpha}$ components $u_1, u_3$. However, for $u_1, u_3 \in C^{0,\alpha}$ then the above shear flow is always in $C^{0, \alpha^2}$ which is a much larger space. We instantly lose the $C^{0, \alpha}$. There exists a shear flow which starts in $C^{0, \alpha}$ which, at any positive time, is not in $C^{0, \beta}$ for any $\beta > \alpha^2$.

This family of solutions is compactly supported in space and time.

Other spaces and optimal spaces:

There are many layers of spaces between these H"older spaes. He writes a tower of inclusions between $C^{1,\alpha } \subset C^{0, \alpha}$. In fact, there is well-posedness [Pak and Park] vs. failure of wp in $B^1_{\infty, infty}$ (Zygmund class) and failure in certain Triebel-Lizorkin spaces.

Weak limit of oscillating initial data:

DiPerna-Majda example…

Shear flow with vorticity interface. Vortex sheet flows are irrotational off an interface. To build such solutions he takes $u_1, u_3$ as (parametrized) Heaviside functions.

…wow…this talk is coming pretty fast, slides are changing…I stop typing and start to just try to keep up.

Numerical investigation of blowup for the 3d Euler

John Gibbon gave a talk a few years ago on the history of these investigations. Tom Hou and Bob Kerr are competing and disagreeing in this direction….is there a singularity…maybe not?

Question: Does the soluton of the following PDE blow up? $$ \partial_t u - \nu \Delta u = |\nabla u |^4? $$

What would you try numerically to determine if it blows up or not? You can even collapse it to the corresponding 1d problem?

Postlude Discussion: Yudovich explored the DiPerna-Lions shear flow examples to see that norms measuring high regularity can grow exponentially in time. Chemin has studied the vortex patch and shown some measures of regularity of the boundary of the patch grow doubly exponentially fast. It was not explicitly clear to me yet how to relate Chemin’s rough patch boundary example to the growth of norms measuring regularity of the solution. Also, Chemin’s examples emerge from non-smooth initial data. I remain interested in the question: Does there exist nice data for 2D Euler which evolves with high regularity norms growing doubly exponentially?

Benoit Grébert: Hamiltonian Interpolation for Approximation of PDEs.

(joint work [Grébert-Faou] with Erwan Faou)

Aim:

Take a PDE with solution u. Consider a numerical approximation $u^n$ built with a symplectic integrator which approximates $u(nh)$. We build a hamiltonian $H_h$ such that $$u^n = \Phi_{Hh}^{nh}(u_0) + very ~small.$$

I am concerned with the long time behavior of the numerical trajectory.

My concern right now is not in estimating the quality of the approximation. Instead, I want to understand the numerical flow.

Outline:

Finite dimensional Context (ODE)
PDE Context
Ideas of the proof (time permitting)

Finite Dimensional Context

We go back to Moser’s theorem. A discrete symplectic map close to the identity can be approximated by a Hamiltonian flow. Consider an analytic symplectic map $$ R^{2n} \ni (p,q) \longmapsto \Psi(p,q) \in R^{2n} $$ with $\Psi = Id + O(\epsilon)$. Then $\exists~ H_\epsilon$ such that $$\Psi = \Phi_{H\epsilon}^\epsilon + O(e^{-\frac{1}{c\epsilon}}).$$ ([Moser 1968], [Benettin-Giorgilli 1994])

Numerical Context: Suppose I have a Hamiltonian ODE system $$ (\dot{p}, \dot{q}) = X_H (p,q) $$ and an associated numerical discrete-time-step symplectic integrator $$ (p_n, q_n)= \Psi_h^n (p_0, q_0). $$ We then have that $\Psi_{h} = \Phi_{Hh} + O(e^{-1/ch}).$ We obtain that $H_h (p_n, q_n) = H_h (p_0, q_0) + n e^{-1/ch}$. So, we are observing that the modified energy is essentially conserved for exponentially long times.

Backward Error Analysis

[Hairer-Lubich 1997]
[Reich 1999]

PDE Context

$$ H = H_0 + P$$ Here we imagine $H_0$ is the linear part and P is the nonlinear part. As an example, consider the cubic NLS on $T^d$. We can treat other equations as well. Let’s recall the Hamiltonian formalism in the Fourier variables:

Expand $u$ to get $$ u = \sum \xi_j e^{ijx}, ~ {\overline{u}}= \sum \eta_j e^{-ijx}. $$ We can then write, for each $j \in Z^d$, $$ {\dot{\xi}} = -i \frac{\partial H}{\partial \eta} $$ $$ {\dot{\eta}} = i \frac{\partial H}{\partial \xi}. $$ For the cubic NLS case, we obtain $$ H = \sum |j|^2 \xi_j \eta_j + \sum^* \xi_{k_1} \xi_{k_2} \eta_{l_1} \eta_{l_2} $$ where $\sum^*$ is the sum over all the parameters subject to the constraint $k_1 + k_2 = l_1 + l_2$.

The problem we face here is that the linear part is unbounded, and we have infinitely many dimensions as first obstructions in passing from the ODE to the PDE context.

Splitting Method: $$\Phi_{P+H0 } \thicksim \Phi_p^h \circ \Phi_{H0}^h =^? \Phi_{Hh}^h.$$

First naive idea: Use the Baker-Campbell-Haussdorf formula. We can then expand as a Lie series… to write $$\Phi_p^h \circ \Phi_{H0}^h = e^{h\mathcal{L}p}e^{h\mathcal{L}H0} = e^{h\mathcal{Hh}} $$ with $H_h = H_0 + P + \frac{h}{2}{ P, H_0 } + \dots. $$

To proceed, we will need conditions $small = h^N C(N,, | num sol |_H^N)$ NOT FAIR! So we need to work harder.

First Idea: Replace $hH_0$ by $A_0$ by cutting off to low frequencies. We can splt and impose the CFL condition. Midpoint + split. He considers different cutoffs.

We then consider $\Phi_p^h \circ \Phi_{A_0}^1$.

Second Idea: Use the Wiener Algebra. Space of functions with Fourier coefficients in $l^1$.

Theorem (Grébert-Faou): For the approximation scheme $\Phi_p^h \circ \Phi_{A_0}^1$ there exists a (polynomial) modified energy $H_h$ such that

$$ | \Phi_p^h \circ \Phi_{A_0}^1 (\xi, \eta) - \Phi_{Hh}^h(\xi, \eta) |_{l^1} \leq h^{N+1} (cN)^N $$

uniformly for $|(\xi, \eta)|_{l^1} \leq M.$

So, assuming that the numerical trajectory is bounded in $l^1$ (as opposed to the stronger claim that it is bounded in $H^k$ for $k$ large) then $$ H_h (u^n) = H_h (u_0) + Cn h^{N+1}. $$

Of course, I have to explain: what is $N$? This is related to a regularization condition. We know that $N = \frac{r-2}{r_0 - 2}$ where $r_0$ is the degree of $P$ (so 4 for cubic NLS). The parameter $r$ is determined by the condition: $ \forall ~ j = 1, \dots, r$ and for any $j$-tuple of integers $(k_1, \dots, k_j) \in Z^d$, we have $$|\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj}| \leq 2 \pi.$$

CFL: $|\lambda_k | \leq C$.

He describes some examples where $N = 3, 4$ and $N=7$.

For cubic NLS, we end up obtaining $$Hh = \frac{1}{h} A_0 + Z_1 + h Z_2 + \dots $$ where $$ Z_1 = \sum^* \frac{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})}}{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})} - 1}. $$ You can now see how the zero divisor issue emerges and is resolved.

Edinburgh Meeting Notes 2

Thu, 20 Jan 2011 06:30:32 +0000

Edinburgh Meeting Notes 2

Sergei Kuksin (École Polytechnique): Nonlinear Schrödinger Equation

We consider Hamiltonian PDE. This is of course very interesting. In physics, there is a class of pdes which is also of interest:

Hamiltonian PDE = small damping + small forcing

Why is it so important?

This class contains a very important equation: Navier-Stokes. $$ \dot{u} + (u\cdot \nabla) u + \nabla p = \epsilon \Delta u + force; ~ \nabla \cdot u = 0. $$ We are interested in cases $d = 2,3.$ For $d=3$, this problem seems impossible. So, let’s collapse to the 2d case.
Nonlinear Schrödinger equation with some damping and forcing $$ \dot{u} + i \Delta u - i |u|^2 u = \epsilon \Delta u + force. $$ Similarly, we might want to study the PKdV equation $$ \dot{u} + u_{xxx} + u u_x = \epsilon u_{xx} + force. $$

We are interested in the small viscosity $\epsilon \ll 1$ and $t \rightarrow \infty$ extremes. At least we want to study $t \gtrsim \epsilon^{-1}$.

Two papers on my web page:

We introduce the slow time $\tau = \epsilon t$.

Perturbations of linear Hamiltonian PDEs

$$ \frac{\partial u}{\partial \tau} + i \epsilon^{-1} (- \Delta u + V(x)u ) = \Delta u - \gamma_R |u|^{2p}u - i \gamma_I |u|^{2q} u + (random force). $$ Both of the parameters $\gamma > 0$ and satisfy $\gamma_R^2 + \gamma_I^2 =1$. The parameters $p,q$ are natural numbers, possibly 0. WE weill look at the case $d=1$ on $x \in [0,\pi]$ with Dirichlet boundary conditions.

Some more information about the random force, $$ (random force) = \frac{d}{d\tau} \sum_{j=1}^\infty b_j \beta_j (\tau) e_j (x) $$ Here the $\beta_j$ are complex valued standard, independent random variables.

We will work in the Sobolev space $H^2$.

Theorem 1: If $u_0 \in H^1$ then $\exists ~! ~ u^\epsilon (\tau, x)$ such that $$ E ( |u|1^2 + \int0^\tau | u(s)|_2^2 ds) < \infty. $$

Let $u_0^\omega \in H^1$ be a random.

Let $\mathcal{P}(u_0^\omega) = \mu$ denote the measure in H^1
Calculate $u^\omega (\tau)$.

Definition: A measure $\mu$ is called a stationary measure if $\forall ~ \tau$ we have $\mathcal{P} (\mu_\tau ) = \mu.$

Bogolyubov-Krylov: A stationary measure almost always exists.

Theorem 2 (Hairer, Odasso, AS): If $b_j \neq 0 ~ \forall ~j$ then $\exists ~ !$ stationary measure $\mu^\epsilon.$ For any solution $u(\tau)$, we have $$ \mbox{dist} (\mathcal{P}(u(\tau)), \mu^\epsilon) \rightarrow 0 ~\mbox{as}~ \tau \rightarrow 0. $$

The measure $\mu_\epsilon$ depends upon the force but not on the data.

Fourier Tranform

For the operator $A = - \Delta + V(x)$ consider the eignefunctions $\phi_1, \phi_2, \dots$ with associated eigenvalues $\lambda_1, \lambda_2, \dots$. Assume that

$\lambda_1 >0$
$\lambda \cdot s \neq 0 ~ \forall s \in {\mathbb{Z}}^\infty, ~ 0 <|s| < \infty.$

For any $u \in H^1$, we can expand $u$ w.r.t. the basis and denote the associated coefficients by $v_1, v_2, \dots$. The Fourier transform is the map $u \longmapsto v$ and the inverse goes the other way.

We can pass from $v_j$ to polar coordinates $I_j, \phi_j$. He recasts the dynamics w.r.t the polar coordinate variables and started speaking about averaging lemmas.

Effective Equations

These objects are somehow analogs of the kinetic equations in the theory of weak turbulence….some notation….I want to understand this better….an average of the nonlinear potential energy term. This is a semilinear heat equation with a nonlocal heat equation. The term proportional to $\gamma_I$ does not influence the effective equation. This equation really takes complete control when $\epsilon$ is very small.

The advance obtained here uses randomness in the forcing. “I expect that the effective equation is relevant even without the randomness but I don’t know how to prove it.”

J. Colliander: Numerical Simulations of Radial Supercritical Defocusing Waves

My slides

F. Bouchet (ENS-Lyon): Invariant measures

Collaborators:

A. Venaille
E. Simonnet
H. Morita
M. Corvellec

Physical phenomena. I am interested in self-organization in turbulent flows. Examples: stripes and spots on Jupiter. Ocean currents. Height differences in ocean surface. Stable jets.

I will mainly speak about the 2d Navier-Stokes equation with random forcing. This is not such a good model for these phenomena. There are others that are quite similar that might be better to describe the phenomena listed above like the quasigeostrophic and shallow water layer models.

Equilibrium will be related to 2D Euler. For 2D, we have the vorticity-stream formulation. Steady solutions to the Euler equation satisfying $\omega = f(\psi)$ or, equivalently, ${\bf{u}} \cdot \nabla \omega = 0,$ play a crucial role in describing the dynamics. Degeneracy: what is the selection mechanism leading to $f$? The main advance is that $f$ can be predicted using equilibrium statistical mechanics ideas.

Outline:

Invariant measures of the 2D Euler equation
- Equilibrium stat mech
- applications of equilibrium stat mec
- invariant measures of the 2d euler equation
Irreversible relaxation of the 2D Euler equations
- irreversibility in fluid mechanics
- …..slide switched….ack
2D stochastic Navier-Stokes equation: non-equilibrium phase transitions

Statisitical mechanics for 2d and geopphysical flows.

Statistical equilibrium. very old idea. famous contributions

Onsager 1949
Joyce-Montgomery 1970
Caglioti Marhioro plvirenti lions 1990
Robert-Sommeria 1991
Miller 1991
Eyink-Spohn 1994

Robert-Sommeria-Miller (RSM) theory:

The most probable vorticity field. We want to measure the number of microscopic fields $\omega$ which correspond to a probabiility $\rho$. The number of such configuarations is quantified by the Boltzmann-Gibbs Entropy. This is the mixing entropy. Microcanonical RSM variational problem. Critical points are startionary flows of the QG model.

Microcanonical measures for Hamiltonian systems:

Hamilton’s equations
Liouville Theorem
Define the microcanonical measures which are the natural invariant measures taking into account the constraints in the dynamics.

Detailed Liouvilles thEorem for 2D Euler:

Lee 1952, Kraichnan JFM 1975, Robert 2000

We want to take into account the casimirs and the constraints. He describes a limiting process based on galerkin approximations. Mean field behavior? Large deviations? Sanov theorem?

……lots of discussion…..ideas vs. proofs…..nontribvial…what’s going on? Audience is confusing me…speaker seems clear.

Young measures….entropy…

The claim is that the theory he and his collaborators hav developed explains the emergence and stabiltiy of coherent structures like the great spot on Jupiter. Similar statements about ocean structures.

Are microcanonical measures invariant measures for the 2D Euler dynamics? Is the setof invariant Young measures for the 2D Euler dynamics larger than the set of microcanonical measures?

Two conjectures:

Weak perturbations of the 2D Euler equations close to steady states converge to invariant Young measures.
The 2D Euler equations converge to invariant Young measures.

Wave breaking is an irreversible mechanism in fluids that does not require viscosity.

Sebastian Reich: Data Assimilation

Data Assimilation

Nature Physical Laws

Measurements Model

 Data
Optimal prediction

He drew arrows between these frameworks of understanding and highlights the assembly of processing at the data assimilation level.

Sequential Data Assimilation in a nutshell.

Model + Observations $\longmapsto$ Prediction

Ingredients of Data Assimilation:

Mathematical and numerical model. solutions and their undertainties caused by approximation errors as well as state and parameter undertainties.
Data/observations with measurements as well as approximation (forward operators) errors –> Inverse problems
Numerical approximations to the data assimilation problem within a statistical (Bayesian) framework, assessment of the induced predictions and their uncertainties.

Mathematical problem statement

Consider an evolution problem for which the initial state is treated as a random variable with some given probability density function. For simplicity assume finite-d phase space. The uncertainty in the initial conditions will generally lead to unpredictability over long time intervals. Weather prediction is a nice example.

To counterbalance this increase in uncertainty, we collect observations at discrete times subject to some random measurement errors. We wish to find a trajectory that makes optimal use of the available information in terms of initial data, observations and model dynamics. The task of data assimilation is to combine the model, the measurements and then we want to make the optimal prediction.

Theoretical solution

i) Model dynamics

Lift the dynamics to the level of the Liouville equation on the probability distribution function.

ii) Data assimilation

Assimilate data using Bayes’ theorem $$\pi (x|y) \thicksim \pi(y|x) X \rho_{pr} (x). $$

Here $\pi(x|y)$ is the know conditional PDF (likelihood) for observing $y$ given a state $x$. Given an actual measurement, we can correct and proceed.

Under Bayes’ theorem, we always reuce uncertainty.

Ensemble Prediction…

ack….slides are changing fast.

Particle filter. We give better weight to points that are closer to the observed data. If we repeat this a few times, there will be very few particles contributing to the final answer.

Assimilation as a continuous deformation of probability: McKean-Vlasov

We can think of Bayes theorem as an optimal transportation problem.

Crisan-Xiong 2010 did something similar in the context of continuous time filter problem.

Otto 2001 for an application in gradient flow dynamics.

We started with an ODE, spoon fed the measurement data to update the dynamics, and encounter a more complicated dynamical description of the system. We encounter a McKean-Vlasov system, a modified Liouville equation, which is closed by an elliptic PDE.

Numerical filter implementations will now rely on appropriate approximations to the lliptic PDE. We use the ensemble of solutions to define an appropriate statistical model and then solve via numerics or by quadrature.

Obvious choices for the numerical version of $\rho$ include a Gaussian PDF parameterized by the ensemble mean and covariance matri (ensemble Kalmna filter) or Gaussian mxture modes.

N. Faou: 2d Submarines

2D Euler equation on 2-torus….I was a bit tired and did not type notes during this talk.

Edinburgh Meeting Notes 1

Wed, 19 Jan 2011 06:24:35 +0000

These are notes from a meeting entitled Advanced Numerical Studies in Nonlinear PDEs in Edinburgh, Scotland.

Walter Craig (McMaster): Water Wave Interactions

I’m an analyst but I’m going to talk about numerics and experiments as well as analysis. We will discuss the problem of water waves and then I’ll talk about two specific settings in which the theory has led to good and quite elegant numerics and the numerics have started to answer some questions.

(joint work with P. Guyenne and C. Sulem)

Outline

Free surface water waves
Hamiltonian PDEs
Periodic Traveling wave patterns
Solitary wave Interactions
The KdV scaling limit

Free surface water waves

Euler’s equations of hydrodynamics, incompressible and irrotational flow. This is therefore given as a potential flow. The irrotational assumption is really an oceanographers assumption. Of course, there is vorticity but we follow the models of oceanographers.

The fluid domain is $-h < y < \eta (x,t)$. So, the domain is changing. Free surface boundary conditions hold on $y = \eta (x,t)$.

Zakharov’s Hamiltonian

The energy functional

$$ H = K+P $$ $$ K = \int_x \int_{-h}^{\eta(x)} \frac{1}{2} |\nabla \phi|^2 dy dx. $$ $$ P = \int_x \frac{g}{2} \eta^2 dx. $$

This could also include surface tension effects.

Zakharov’s choice of variables, $$ z = ( \eta(x), \xi(x) = \phi(x, \eta(x))), $$ for which we consider $\phi = \phi[\eta, \xi] (x,y)$.
Express the energy in terms of $\xi$ and $\eta$. This involves the Dirichlet-Neumann operator $G(\eta)$.

Dirichlet-Neumann operator

Laplace’s equation on the fluid domain: $\Delta \phi = 0$ subject to bottom Neumann boundary condition. Free surface boundary data $\phi (x, \eta(x)) = \xi(x)$, for which the D-N operator is given by $$ \xi(x) \longmapsto \phi(x,y) \longmapsto N \cdot \nabla \phi (1+ |\nabla_x \eta|^2)^{1/2} := G(\eta) \xi(x). $$
In these coordinates, we can rewrite the boundary conditions in a new (and nicer) form. This reexpresses the water wave problem as a Hamiltonian system in Darboux coordinates.

Hamiltonian PDEs

KdV is a Hamiltonian PDE with a different symplectic structure.
Other Hamiltonian PDEs
- shallow water equations
- Boussinesq
- KP
- NLS
- Dysthe equation

Many of these problems arise in scaling limits of the water wave problem.

Lemma (Properties of D-N operator):

$G(\eta) \geq 0$ and $G(\eta) 1 = 0$.
$G(\eta)^* = G(\eta)$ Hermitian Symmetric
$G(\eta): H^1_\xi \rightarrow L^2_\xi$ is analytic in $\eta$ for $\eta \in C^1$. There is an operator valued power series expansion of $G(\eta)$ (using a theorem of Christ-Journé 1987).
Some explicit calculations of the Taylor expansion (I couldn’t keep up….)
Conservation Laws
- Mass: $M = \int \eta dx$ (He shows the calculation using properties of $G$.)
- Momentum: Similar calculation
- Energy: Easy since the commutator of $H$ with itself vanishes.
Taylor expansion of the Hamiltonian
Linearized equations; comparison with the harmonic oscillator.

Periodic Traveling wave patterns

Can I find traveling wave solutions? $$ \eta(x,t) = \eta ( x-tc); \xi(x,t) = \xi(x - tc) $$
Spatially periodic, $\Gamma \subset {\mathbb{R}^{d-1}}$. $$ \eta(x + \gamma, \cdot) = \eta(x, \cdot), \xi(x+\gamma, \cdot) = \xi(x, \cdot), ~ \forall \gamma \in \Gamma. $$

On such domains, we can use the Fourier tranform.

Rk: Notice this is a mathematician imposing a period rather than the physics making that selection. More can be said in this direction, but let’s proceed this way.

Rk: These (time independent) traveling wave patterns can be imagined to emerge in transient interactions in seas. The nonlinear actions create large amplitudes and this might be related to the phenomena of freak waves.

Equations for traveling waves.

Periodic traveling wave patterns are critical points of the Hamltonian on the variety $I = const$, with Lagrange multiplier $c \in {\mathbb{R}^{d-1}}.$

This leads to a bifurcation problem.

brief history (dimension $d=2$):

Levi-Civita 1925; existence of traveling waves
Struik 1926; traveling waves case
Zeidler 1971
Beale 1979
Jones-Toland 1985

brief history (dimension $d=3$):

Reeder Shinbrot 1981
Sun 1986
Craig-Nicholls 2000
Iooss-Plotnikov-Toland 2000 (small divisor problem)

He shows a picture from the wave tank at Penn State. He then shows some numerics which are trying to model those observations and they look beautiful.

Kuksin Question: Stability of these patterns?

Craig Answer: This is a very good question. I don’t know results like that. This is related to Benjamin-Feir. McLean showed instability for $d = 3$. Some further discussion….We need the Bloch theory of stability for these wave patterns. This appears to be difficult analytically so might need some numerical studies at first. There are instability zones….

Solitary wave Interactions

Solitary waves in 2-dimensions (Friedreichs-Hyers 1954, Amick-Fraenkel-toland 1980s)

Head-on collisions of solitons.

The numerics reveal some inelasticity in the collision. We’d like to understand those. If we make the amplitude of the solitons bigger, the dispersive ripples are more visible.

The KdV scaling limit

Titi’s Question: Can we reduce to the surface equations including rotation?

Craig’s Answer: Yes and No. You can make a rotation depending purely on y and impose that. Then it is reducible. But this is rather artificial. There is stuff that happens in the middle which is not a surface effect. Therefore, this problem requires a more complete analysis of the Euler equation and will not collapse to a system on the surface.

Sergey Nazarenko (Warwick): Assumptions, Techniques, Cahllenges in Wave Turbulence

This is not so much about new result. Instead, this is an attempt by a physicist trying to explain wave turbulence ideas being explored by physicists to mathematicians. My view is that there is a lot of interesting work to be done. Lots of open problems….

What is wave turublence?

He shows a picture of a relatively calm seashore from Nice. He emphasizes there is a wide range fo scales in these problems. WT is a statistical system of nonlinear waves.

Examples:

Water waves
Waves in rotating and stratified fluids (internal and inertial waves, Rossby waves)
Plasma waves
Waves in Bose-Einstein condensates
Kelvin waves on quantized vortex filnments
MHD turbulence in interstellar turbulence and solar wind
Nonlinear optics
Solids: phonons, spin waves. Kinetics of phonons in weakly anharmonic crystals is a first example of study in tis direction (1920s). I didn’t catch the name….

He shows a picutre of a wave take of Lukaschuk.

Waves in fusion plasmas. Shows a picture of a Tokamak. Drift wave turbulence causes anomalous heat and particle loss - major problem for fusion. The devices have grown larger and larger basically to carry out the confinement for a longer period of time.

MHD turbulence in astrophysics. He shows some data from the Ulysses/Swoops (los alamos) solar wind studies.

Bose Einstein Condensates Nazarenko-Onorato 2006:

Inverse cascade - condensation
Condensate strongly affects WT

Quantum Turbulence (see Lvov et. al. 2007) (Superfluid turbulence)

Kelvin waves on quantized vortex filaments
Interaction with hydro eddies (vortex bundles) is important
Kelvin Wave Turbulence

Optical Turbulence

Bortolozzo et. al 2008
This project studies nonlinear corrections (coming from the optical physics) which are included beyond the 1d NLS model.

Kuksin Question: Which corrections? Can you write them down?

Nazareknko: Something like a DNLS correction…not so clear.

Ingredients in the approach

He writes $NLS_3^\pm (T^d)$ and comments that this is a physically reasonable model but we are really interested in the study in infinite space with finite energy density.

He reexpresses the NLS equation in Fourier language.

Set of wave modes: amplitudes and phases.

N-mode joint probability density function. Some notation….probability…sectors in the wave modes setting.

Random Phase (RP) and Random Phase Amplitude (RPA) systems

RP:

All phases are independent random variables such that uniformly distributed on $S^1$.

RPA:

All amplitudes and all phases are independent random variables.
All phases are uniformly distributed on $S^1$.

Note: RPA does not mean Gaussian. Nevertheless, we have obtained successful closures without assuming the Gaussian statistics.

Frog Jumps!

expanding in small nonlinearity
Assuming RP at $t=0$.
Taking limit of a large box followed by the limit of small nonlinearity.

(The order of these steps is important.)

Evolution of joint PDF? We can derive the evolution equation under these assumptions. The derivation is rather systematic, in fact it is perhaps rigorous.

Mathematical Challenges:

WT is formally derived for $t=0$.
Does it work at the long time of nonlinear evolution?
Does RPA survive over this time?
Adding forcing and dissipation: will WT describe the steady state?

Hmmmm….This RPA condition at $t=0$ reminds me a bit of the assumption of product wave function in the QMB theory. The dynamics in the Hartree derivation might drive the multiparticle wave function away from the product case. Here we have a dynamic that might drive us away from the RPA condition.

Evolution of 1-mode PDF.

Kinetic equation (Hasselmann 1962).

Kolmogorov-Zakharov state.

Explained a steady state spectrum corresponding to energy cascade.
Exact solution of the asymptotic closure.

Numerics and Analysis of KE.

What is the role of KZ solutions with respect to the thermodynamic Rayleigh-Jeans state?
Similar issues for the classical Boltzmann equation.

Zakharov was awarded the 2003 Dirac Medal for “putting the theory of wave turbulence on a firm mathematical ground”! What is it that we want to do?

Gregor Tanner (Nottingham): A wave chaos approach towards describing the vibro-acoustic response of engineering structures

(Joint work with D. Chappel, Stefano Gianai, Hanya Ben Hamdin, Dmitrii Maksimov)

This talk is more directed toward engineering applications. inuTech is an industrial collaborator.

Overview:

Introduction - the need for numerical short wavelenght methods in vibroacoustics
From wave equations to the Liouville equation
Solving the Liouville equation - a boundary integral approach (Dynamical Energy Analysis - DEA)
Tackling the Midfrequency problem - hybrid methods
Numerical results

Aim: predicting wave intensity distributions for the vibro-acoustical response of mechanical structures. Think of a car. Companies like Bombardier and Airbus use these methods. It is a difficult problem. You want these structures to be quite and with no noise in the interior.

Where is the problem?

Techniques:

Low frequencies - wavelength around the size of the object
Finite Element method
Boundary elemtn method
plane wave methods

High Frequencies:

Ray tracing
Statistical energy analysis
…

Midfrequency problems:

Structures with large variations in the local wavelength. (Large variations in the stiffness of components, ie body frame and side panels.)
Hybrid methods. Try to connect exact numerical methods with the statistical methods.

Short wavelength approximations - from wave chaos to statistical methods

Wave chaos -short wavelength asymptotics
- Keller
- Gutzwiller
- Berry
- Bogolmolny
- Smilansky
Nonlinear dynamics - thermodynamic formalism
- Ruelle
- Arnold
- Sinai
- Eckmann
- Cvitanovic - chaosbook.org
Wave transport - statistical methods in vibro-acoustics
- Lyon - SEA (1967 paper)
- Langley - WIA
- Heron
- Weaver - diffusion equation
- Le Bot - radiative transformation

Linear wave equation. WKB ansatz. Hamiltoninan equations for the amplitude and phase. Characteristics of JH; nonlinear ODE; Liouville equation (linear).

Linear wave –> WKB –> HJ equation –> Liouville Equation

Think of polygonal billiards, not necessarily convex.

We want to understand the influence of a source (transfmitting at frequency $\omega$) at one location on the wave amplitude at another point. He writes this as a green’s function $G(r, r_0, \omega)$.

Small wavelength limit, so low frequency waves.

Write things as sums over all paths.

Perron-Frobenius operator…

(Image no longer available: Tandem Satellite images of coastline of Madagascar)

I started wondering about connections between these ideas and quantum ergodicity…

Postlude:

I had a nice conversation with Gregor after the break. I learned from him about microlasers. The idea is to build a circular region out of a lasing material. We energize the material somehow with hopes to excite the whispering gallery mode. The laser light propagates near the boundary but can be arranged to exit the medium by raising the curvature at a specific location. These appear to be rather hard to control to create a unidirectional beam. Since the losses take place all along the boundary, there is very little power in the output beam. Some web searching revealed an advance made by the Capasso group at Harvard. . (Image no longer available: Microlaser Cavity (Artistic Rendition))

David Dritschel (St. Andrews): CLAM, The Combined Lagrangian Advection Method

(Many many collaborators)

(Image no longer available: Courbet’s "The Wave")

I’ll be speaking a bit about a numerical method. I’ll focus mostly on the results we’ve obtained to understand the large scale atmospheres, like Jupiter and perhaps also the ocean.

The numerical method (CLAM) emerges from a Lagrangian method from the 50s for studying fluid dynamics. Zabusky then built from these developments to develop new methods in plasmas. We’ve been extending these ideas to treat certain geophysical fluid flows.

The atmosphere and the oceans are extremely complex, turbulent flows. Accurate computer simulation is immensely difficult to achieve. However, much of this difficulty is inherent in the computational methods employed:

None take direct advantage of the natural inherent Lagrangian advection of dynamical, chemical and biological tracers. (Exploit Lagrangian Descriptions.)
None seek to separate slow vortical (eddying) and fast wave-like motions and use appropriate, optimal numerical methods for each. (Slow Rossby waves interacting with fast waves inertial-gravity waves.)

We can build the mathematical theory of the separation into the numerical methods and this will lead to better predictions.

Contour Advection (CASL) Dritschel & Ambaum 1997

geostropic and hydrostatic balances are basic features for describing atmospheric wave dynamics.

This talk reminds me somehow of Bourgain’s high/low method for proving low regularity GWP.

The idea is to use the advection of the vorticity to resolve some (especially relevant) sub-grid scales.

Dugald Duncan: IDE equation

Overview:

full IDE equation and how it looks like, where it arises
Linear part of the IDEbehaviour and approximation
the full problem - behanviour and approximation
examples

$$ u_t = \sigma \int_\Omega J(x-y) [u(y,t) - u(x,t)]dy + f(u) dx ~ \forall x \in \Omega, t>0. $$ Typically, $f(u) = u - u^3$. This should be contrasted with the Allen-Cahn equation

$$ u_t = \sigma \Delta u + f(u) dx ~ \forall x \in \Omega, t>0. $$

There are no spatial derivatives. Therefore, there are now boundary conditions. Instead, this is some kind of integral dynamical equation. It is similar to the Cahn-Allen equation.

This equation is also related to sandpiles, neurons, phase transitions.

Other variations recently: Rossi, Perez-Llanos, Andreu, Mazon, Toledo et. al. They study a nonlocal version of the $p$-laplacian.

Linear IDE:

Ignore the nonlinear reaction term for now and take $\sigma \geq 0$ and $\Omega \subset {\mathbb{R}}$: $$ u_t = Lu.$$
L is a linear operator - partly a convolution: $$ Lu = \int_\Omega J(x-y) [u(y,t) - u(x,t)]dy = J * u $$ …ack slide changed….

Discontinuities don’t move. The solution collapses to the average value. Ther eis acomparison principle.

Snapshots of linear behavior.

He does a Fourier analysis of the behavior of plane waves. Instead of having an $\omega^2$, we have $$\hat{J} (\omega) - \hat{J} (0) \thicksim \frac{\omega^2}{2} \frac{d^2}{d \omega^2} {\widehat{J}} (\omega).$$

Peter Bates and Paul Fife did some of the earliest analysis on this equation.

Confronting Bias in Hiring Committees

Tue, 11 Jan 2011 00:55:54 +0000

Jennifer Berdahl of the Rotman School of Management recently shared some slides with me that I thought should be more widely circulated among mathematicians. This presentation was created by MIT Professors Barbara Liskov and JoAnne Yates and is based on materials compiled by Abby Stewart for the University of Michigan’s STRIDE (Strategies and Tactics for Recruiting to Improve Diversity and Excellence) Committee. There are a lot of links to interesting materials on the STRIDE page such as guidelines about how to write a recommendation letter, and discussion on the advancement of women in science careers. Here is the link to the slides… FACULTY DIVERSITY TRAINING SLIDES.pdf

Notes on Nonlinear Dispersive Wave Equations Workshop in Oberwolfach

Mon, 04 Oct 2010 22:39:37 +0000

This page contains notes by J. Colliander taken at the workshop:

Nonlinear Waves and Dispersive Equations
Organizers:
Date: September 12th - September 18th, 2010

I apologize for any mistakes! If any of the speakers would like me to post (or link to) their slides, please send me the file. –Jim Colliander

Jérémie Szeftel: Alas, I missed this talk….thanks Air Canada. ↩

arXiv ↩

slides ↩

Sebastian Herr: Small data theory for energy critical periodic NLS ↩

(joint work with Tataru and Tzvetkov)

Energy critical NLS focusing or defocusing on a manifold M. Specific examples with Laplace Beltrami operator. Mostly intersted in manifolds with periodic geodesics. For example $\mathbb{T}^3$ or tori crossed with $\mathbb{R}^d$.

Target is LWP.

Warm-up remarks ↩

Warm up: $M = {\mathbb{R}^d}$. Strichartz, dual Strichartz, dispersive decay $\implies$ (Cazenave-Weissler) LWP.
Non-Euclidean cases: Asymptotically Euclidean and nontrapping metrics have been studied.
Failure of sharp Strichartz estimates on torus and on sphere.
Trapping creates geometric obstructions to dispersion.
Trapping can create instabilities and failure of Strichartz estimates.
Known estimates: Strichartz with a loss of derivatives.

$NLS^{\pm}_5 (T^3)$ ↩

Available estimates have some loss. The loss obeys the scaling but it is insufficient to control the quintic nonlinearity. We end up needing an $L^4$ estimate, which is unavailable.
Our strategy is to use multilinear, scale invariant versions of Strichartz estimates to better share the derivatives.
Use almost orthogonality wrt spacetime to reduce estimates to smaller scales.
Replacements/refinements of the $X^{s,1/2}$? We use the critical function spaces $U^p, V^p$.
We will need refinements of these spaces which are sensitive to finer than dyadic frequency localizations.

New Strichartz Estimates ↩

We have the Strichartz estimates on functions supported on cubes in Fourier space
For all rectangular sets of arbitrary orientation and center, we get a better bound!
This boils down to a classical estimate (Landau 24) for counting the number of lattice points on 6d ellipsoid.

Perturbative Analysis ↩

$U^p$: Definition involving all partitions of the line using $U^p$-atoms.
These are Banach spaces which embed into $L^\infty$.
$V^p$: We need another type of space. These are functions of finite $L^p$ variation over the partitions of the line.
$U^p \rightarrow V^p_{rc} \rightarrow L^\infty$ (Embeddings)
$\| u \|{U^p{\Delta} H^s} = \| e^{-it \Delta} u \| {U^p (R; H^s})$. (Similarly wrt $V^p$, as in Ginibre’s Asterisque.)
We choose then $p=2$ and call the resulting spaces $X^s$ and $Y^s$.
Properties: $U^2_{\Delta} H^s \rightarrow X^s \rightarrow Y^s \rightarrow V^2_{\Delta} H^s$ (Embeddings)
We define restrictions to smaller time intervals….
$X^s$ and $Y^{-s}$ have a nice duality relationship.

Trilinear Strichartz ↩

Refinement which generalizes Bourgain’s $p=6$ Strichartz estimate.

Sketch of proof ↩

Decompose the largest frequency $N_1$ annulus in cubes of the second largest frequency $N_2$.
We can replace $Y^0$ by $V^2_{\Delta} L^2$.
We deduce control on the quintic nonlinearity using the trilinear estimate. Some gain is obtained by playing with the exponent $p$ in the $U^p$ spaces, which he attributed to elementary properties of these atomic spaces.
This gain and some other slack in the other trilinear estimate allows one to sum up over the dyadic scales.

I am confused at this point? Do we have some derivative slack or are thing really tight? Since we are considering an $H^1$ critical problem, there can be no slack….I discussed this with Sebastian a bit after the talk. I was confused; there is no derivative slack.

Next, there is a new localization (the rectangle decomposition). The cubes are decomposed into almost disjoint strips of a certain width. The almost orthogonality is gained from the temporal frequency! (This reminded me of the ideas from Koch-Tzvetkov and later developed by Ionescu-Kenig)

Contraction estimate ↩

It is not necessary to use the rectangles to get this estimate. For the qunitic case, we can avoid the rectangles. For the cubic NLS, by duality you have a 4-linear estimate and by Cauchy-Schwarz you are reduced to bilinear estimates. For the cubic case, it is necessary to use the rectangle decomposition.

Remarks ↩

With similar ideas, they can treat the cubic case on $R^2 \times T^2$ or $R^3 \times T$.
This involves bilinear refinements instead of cubic refinements.
small data GWP for energy critical NLS on certain manifolds where arguments of the Euclidean setting fail.
Large data is a very interesting problem.
This is the first critical result for NLS on a compact manifold.
Quintic NLS on the 3-sphere? Strichartz estimates fail but possible to control second Picard iteration.
Cubic NLS on $T^4$.

Questions ↩

Flat waveguides?
$L^2$ critical case?

Benjamin Schlein: Effective evolution equations from many body quantum dynamics ↩

Resources: Schlein’s talk at ICMP 2009, Schlein’s Zurich Lectures

Introduction ↩

Consider $N$ particles moving in 3d. These particles can be described in quantum mechanics by a wave function $\Psi_N \in L^2 (R^{3N})$. The probability density $| \Psi_N (x_1, x_2, …, x_N)|^2$ represents the probability of finding particle 1 at location $x_1$ and so forth. Bosons are symmetric wrt particle interchange. Fermions are antisymmetric. We will restrict in this talk to Bosonic symmetry: For all permutations $\pi$, $$ \Psi_n (x_{\pi_1}, \dots, x_{\pi_N}) = \Psi (x_1, \dots, x_n)$$

The dynamics of the wave equation is governed by the Schrödinger equation $$ i \partial_t \Psi_N = H_N \Psi_N $$ Typically, $$ H = \sum_{j=1}^N (-\Delta{x_j} + V_{ext} (x_j)) + \lambda \sum^N V(x_i - x_j). $$ We have well-defined local dynamics. The problem is that we have way too many particles in typical physical systems. We want to find effective descriptions of the dynamics. In certain regimes, we can approximate this complicated but linear evolution using effective equations

Mean Field Regime

The particles interact with many other particles. The strength of each of these many interactions is small so that the effect of all of them is of order 1: $N \gg 1, \lambda \ll 1$. We will assume that $N \lambda \sim 1$. The dynamcis generated by the mean field Hamiltonian: $$ H^{mf} = \sum (-\Delta{x_j} + V_{ext} (x_j)) + \frac{\kappa}{N} \sum^N V(x_i - x_j). $$ We study the dynamics emerging from a product wave function: $$ \Psi_N (x_1, \dots, x_N) = \prod_{j=1}^N \phi (x_j) $$ Because of the interactions, we can’t expect that the product wave function remains of product form. But, in the mean field case, we might expect that $\Psi_N (t) \sim \phi(t)^{N}$. If we assume this, we obtain a self-consistent Hartree equation. Here is the heuristic step: $$ \frac{\kappa}{N} \sum^N V(x_i - x_j) \sim \frac{\kappa}{N} \sum^{i \neq j} \int V(x_i - y) |\phi(y)|^2 dy \sim \kappa (V * |\phi(t)|^2) (xj). $$

Reduced Densities

$$ \gamma_N (t) = |\psi_N (t)\rangle \langle \psi_N (t)|$$
Partial traces
- When we take partial traces, we lose some information. It is integrated out. However, we are only interested in the data that can be extracted based on measurements of finitely many particles.

Theorem: (Under suitable assumptions on $V$). Let $\phi \in H^1 (R^3), \Psi$ a pure product wave function, $\Psi_N (t)$ the linear evolution of the many body system. The for all fixed $k \in {\mathbb{N}}, t \in R$, the reduced density matrices converge to the projectors build on the $\phi$ evolutions where $\phi$ solves the Hartree equation.

The more singular the potential, the more difficult it is to prove the theorem.
Spohn 1980: proved this for bounded $V$.
Erdös, Yau 2000: $V(x) = \pm \frac{1}{|x|}$.
Rodnianski, Schlein 2008: $V(x) = \pm \frac{1}{|x|}$, gives quantitative convergence with control by $\frac{C}{N}e^{kt}$.
- The RS work was based on an approach by K. Hepp.
- The approach is based on a representation of the problem on Fock space.
- Coherent states and quantum field theory ideas.
Knowles, Pickl 2009: Improved to more singular potentials.
Grillakis, Machedon, Margetis 2009 I, II: Second order corrections to the mean field dynamics, giving norm convergence.

Boson Stars ↩

$N$ particle Hamiltonian $$ H = \sum^N \sqrt{1 - \Delta {xj}} - G \sum \frac{1}{|xi - xj|}$$
$N \gg 1, G \ll 1, NG = \kappa$
$\forall N \exists \kappa(N)>0$ (kappa critical) such that:
- $\inf \frac{\langle \Psi , HN \Psi \rangle }{\| \Psi\|_2^2} = 0 ~if~ \kappa \leq \kappa(N)$
- $\inf \frac{\langle \Psi , HN \Psi }{\| \Psi\|^2} = - \infty ~if~ \kappa \geq \kappa(N)$
- Lieb, Yau proved that $\kappa(N) \rightarrow \kappa^H$ as $N \rightarrow \infty$
Look at the corresponding effective field equation.
- For $\kappa \leq \kappa^H$, we have global well-posedness.
- For $\kappa \geq \kappa^H$, there exists finite time blowup solutions (Fröhlich-Lenzmann 2006).

Theorem (Michelangeli-Schein 2010): Let $\phi \in H^2 (R^3)$ and form product wave function $\Psi_N$ and let $\Psi_N(t)$ evolves according to the regularized Hamiltonain (where the singularity is tamed by adding a small positive term to denominator which vanishes as $N \rightarrow \infty$). If we have $H^{1/2}$ control on the nonlinear level by constant $k$ over a time interval $[0,T]$ then we have convergence. Moreover, if the nonlinear problem explodes then the energy per particle in the linear problem also blows up. (Hypotheses were a bit strange to me….I asked about it after the talk and need to look at the paper.)

Dynamics of Bose-Einstein Condensates ↩

Drop the external potential. Effective dynamics in this case is described by the Gross-Pitaevskii equation: $NLS_3^+$ The derivation of effective dynamics in this setting has only been established for the defocusing case.

Adrian Constantin: Camassa-Holm ↩

Physical Background ↩

2d water waves over a flat bed. He draws a curve above a flat bottom at $y = - h_0$ and the free surface is given by the graph $y = \eta (x,t)$. He writes the Euler equations, mass conservation, imposes reasonable boundary conditions. These are generally accepted to be the right model. I will work with one other assumption: $u_y - v_x = 0$: irrotational. There are various scales you can plug into the problem and then you can non-dimensionalize. The problem can then be written in terms of just two parameters $\epsilon$ and $\delta^2$ where: $$ \epsilon = \frac{2}{h_0}, $$ $$ \delta = \frac{h_0}{\lambda} $$ Small amplitudes $\epsilon \ll 0$ and $\delta$ is the shallowness parameter so shallow water wave theory means that $\delta$ small. Shallow water small amplitude is when $\delta \ll 1$ and $\epsilon = O(\delta^2)$. If you study this, you get KdV and BBM equations. In this regime, these model equations enjoy global existence. The nondimensional form of the KdV equation is $$ \eta_t + \eta_x + \frac{3 \epsilon}{2} \eta \eta_x + \frac{\delta^2}{6} \eta_{xxx} = 0. $$ Here is the emerging BBM: $$ \eta_t + \eta_x + \frac{3 \epsilon}{2} \eta \eta_x + \delta^2 (\beta + \frac{1}{6})\eta_{xxx} - \beta \delta^2 \eta_{xxt} = 0, \beta \geq 0. $$ KdV is completely integrable and has solitons. BBM has some nice analytic features but only 5 conserved quantities. These derivations are $O(\delta^4)$.

Where does Camassa-Holm come into this business? Since all the waves that are physically reasonable, we have global existence. We would like to have a simple model that captures the phenomenon of wave breaking: $\eta$ is bounded, $|\eta_x|$ becomes unbounded in finite time. (This is described as a desirable extension in the book Linear and Nonlinear Waves, by Whitham.)

Emergence of Camassa-Holm Equation ↩

Moderate amplitude (shallow water): $\epsilon = O(\delta), \delta \ll 1.$

Johnson found a path like this to see Camassa-Holm emerge. (“Unfortunately, the original derivation of that equation was not correct.” “They assume that $\epsilon$ is small and later they assume that $1/\epsilon$ is small…”) You can derive an equation for the horizontal velocity at a particular depth. If you do this derivation at depth $\frac{1}{\sqrt{2}} |h_0|$. Another equation called Degasperi-Procesi emerges when you consider this at a different depth: $$ u_t - u_{txx} + 3k u_x + 4 u u_x = 3 u_x u_{xx} + u u_{xxx} $$ Both of these equations are integrable! (I did not know about the D-P equation before…) Both of these equations have solution which break down, in the fashion of wave breaking described above.

For CH, we have some conservation laws which implies the solution stays in $L^\infty$. Fokas and Fuchsteinner found the CH equation in a list of 12 equations that are the only completely integrable equations. It is difficult to use the integrable systems machinery to study the wave breaking. For DP, if you start with data in a nice enough space (say $H^{3/2}$) and you can then prove the solution stays in $L^\infty$.

Tzvetkov Q: Can the solution be extended after the wave breaks? A: You might be able to extend the solution like shocks. But the relevance of the wave breaking event in CH, it is not clear whether the CH is a good approximation of the Euler equations. Therefore, even if the PDE theory for CH can be extended, this does not mean you have a relevant extension modeling the water wave problem.

Geometric viewpoint as a geodeisc on the diffeomorphism group ↩

There is this famous paper of Arnold that shows that Euler may be viewed as a geodesic flow on the diffeomorphism group. CH and KdV can be similarly interpreted as a geodesic flow on the Bott-Virasoro algebra. This geometry thing is very nice, very appealing. However, this geometric point of view does not give a useful consequence from the viewpoint of analysis.

The best result for CH is that when the solution does not change sign it stays global. This is built from Nöther’s theorem, which provides a different view on the CH equation

Write ${\mathcal{D}} = [ \phi: C^\infty ~orientation~ preserving~ diffeos ]$. This is a Lie group and the tangent space at the identity is $C^\infty (S)$. We can then move this tangent plane around using Lie algebra properties by right-translating. The geodesic equation looks like $\phi_t = u(t, \phi)$ where $u \in \mathcal{D}$.

If I do this for $L^2$, I get $u_t + 3 u u_x = 0$. However, the Riemannian exponential map $exp_R$ is not a local chart.
If I do this for $H^1$ ( I believe this is referencing the Riemannian structure imposed on $C^\infty$ diffeos) we get CH.
Consider the Bott-Virasoro Algebra $Vir = C^\infty \times {\mathbb{R}}$ and you do some Bott cycle thing which looks like a diffeo flow with a twist, you get KdV. (This is a result of Olsheyenko(?) and Khesin.)
DP equation also has some interpretation this way but it is more complicated.

Integrable Sturcture ↩

CH Lax Pair. This is an isospectral problem. For CH, we have $\psi_{xx} = \frac{1}{4} \psi - \lambda m \psi, m = u - u_{xx} + k$. This is a weighted spectral problem. If $u$ solves CH, then the eigenvalues of this equation are time independent.
DP Lax Pair. $\psi_{xxx} - \psi_{x} - m z^3 \psi = 0, z \in {\mathbb{C}}, m = u - u_x +k$. When $m$ is strictly positive, we can perform certain Liouville substitutions which allow us to recast this as a regular Sturm-Liouville problem.

This talk did not properly survey the literature. Instead, the talk was intended to highlight a physically relevant derivation of the CH and to show that this equation is mathematically and physically interesting.

Tzvetkov asks: Is there a Miura transformation? Seems to be no….although AC appeared to me to answer a different question.

Merle asks: Can you track the blowup using the integrable machinery here? We need estimates on the eigenvalues hold if $m>0$ and you can give examples which show that sign-changing $m$ breaks down the needed estimates on the eigenvalues.

Ponce asks: What is the best LWP theory for CH? Answer: Kato’s theory needs $H^{3/2}$. You have existence, uniqueness and continuous dependence in $H^1$, but the continous dependence is weaker.

Ponce asks: Is the peakon stable? A: Yes, this is a result of Molinet and El Dika.

Claudio Muñoz: Dynamics of gKdV solitons under perturbations by potentials in front of nonlinear term ↩

My computer ran out of battery….

arXiv: Muñoz on KdV

arXiv: Muñoz on NLS

Mihalis Dafermos: Superradiance, trapping and decay for waves on Kerr spactimes in the general subextremal case $|a| < M$. ↩

(joint work with Igor Rodnianski)

Kerr family $(0 \leq |a| \leq M)$ of metrics (in Boyer-Lindquist coordinates) $$ g_{M,a} = - \frac{\Delta}{\rho^2}(dt - a \sin^2\theta d\phi)^2 + \frac{\rho^2}{\Delta}dr^2 + \rho^2 d\theta^2 + \frac{\sin^2 \theta}{\rho^2}(a dt - (r^2 + a^2) d\phi)^2. $$ Here $\rho^2 = r^2 + a^2 \cos^2 \theta, \Delta = r^2 - 2 M r + a^2 = (r - r_{-})(r-r_{+}), r_{+} \geq r_{-}.$ This is a vacuum solution $(R_{\mu \nu} =0)$ and has Killing fields $\partial_t, \partial_{\phi}$. The domain of outer communications is $r > r_{+}$. The case $a=0$ is Schwarzschild 1916. The Kerr case is $a \neq 0$ and was discovered in 1963.

Penrose diagram for Kerr $(0 < |a| N M)$. What is a black hole? A spacetime has a black hole if the past of this infinity is not the entire spacetime. Both Kerr and Schwarzschild are expected to be unstable and the structure of the singularity should be something in between.

Penrose Diagrams (images taken from Dafermos-Rodnianski)

Penrose diagram of Schwarzschild spacetime

Penrose diagram of Kerr spacetime

Boundedness and decay for $\square_g \psi =0$ on Schwarzschild and Kerr ↩

These are natural questions from several points of view. One important application of these ideas is to address the stability properties of these solutions of the Einstein equations.

Current state of the art for the quantitative study of $\square_g \psi = 0$ ↩

Boundedness in general class of $C^1$ stationay axisymmetric spacetimes [DR].
“Integrated local energy decay” for exactly Kerr:
1. Slowly Rotating Case $|a| \ll M$ [DR], Tataru-Tohaneanu, Andersson-Blue
2. $|a| < M$, [DR] this talk!.
Pointwise-in-time decay from 1. and 2. (energy based method [DR] based on resolvent method of Tataru)

We typically think of proving decay as a two step process. We first prove that some spacetime integral of energy to the future an arbitrary hyperboloidal space-like hypersurface controlled by the energy on the hypersurface. This type of result has been shown in the slowly rotating case. Once we have items 1. and 2. (boundedness and integrated local energy decay) we can use the vector field method to get pointwise-in-time decay. These methods appear to be robust and might be applicable to nonlinear problems.

Review of the main features of Kerr Spacetimes ↩

Red-shift (associated to the event horizon)
Superradiance
Trapping (trapped null geodesics)

Red-shift

Two observes move in spacetime. You think of observer A emitting constant frequency signals and you imagine these being received by observer B so the frequency is shifted to the red. First discussed in 1939 by Oppenheimer-Snyder. Extremal case $a = M$: The red-shift factor at the horizon vanishes. The positivity of the surface gravity is a geometrical object underlying the red shift.

Superradiance

In Schwarzshild, the killing v.f. $\partial_t$ is timelike n the exterior becoming null on the horizon. Thus there is a conserved (by Nöther) non-negative definite energy by the time-like condition. The only subtlety is that the energy degenerates at the horizon.

In stationary perturbations of Schwarzschild, $\partial_t$ in general becomes spacelike near the horizon. this happens already for Kerr with $0 \neq |a| \ll M$. The corresponding energy is conserved but does not have a sign. For particle motion, this leads to the so-called Penrose Process. For waves, this leads to the phenomenon of superradiance (Zel’dovich).

In particular, using the conservation law associated to $\partial_t$ one cannot prove a priori boundedness, even away from the horizon. The energy radiated to null infinity might be bigger than the initial energy and this is called Superradiance.

For Schwarzschild, the only trouble is near the horizon because we have a useful energy control for radiated energy to null infinity. For Kerr, we don’t have that because of the superradiance phenomenon and this creates new difficulties. We need to prove boundedness and decay everywhere, not just near the horizon.

Trapping

On Schwarzschild, the photon sphere $r = 3M$ has the property that it contains null geodesics. These null geodesics thus neither escape to null infinity nor to the horizon. In Kerr, the behaviour persists, but it is more complicated! It is not obviously located in physical space but can be thought of more easily inside phase space. One can concentrate energy for arbitrarily large times near trapped null geodesics. One has to capture this to prove dispersive results. In particular, pointwise-in-time decay estimates for energy must lose derivatives (Ralston).

Proof of integrated local energy decay. ↩

We will only discuss the first energy. Higher order estimates require commutation with the redshift vector field, the Hawking v.f. and $\partial_t$.

The method of proof will exploit energy currents.

In the large $a$ case, the construction of these currents will need to frequency localized for two reasons.

To distinguish between non-superradiant and superradiant frequencies.
To degenerate at the correct value of r.

A convenient way of doing both at the same time is frequency localizing via Carter’s celebrated separation of the wave equation. Kerr geometry only has two killing fields. This is not enough to separate the equation. However, there is some extra symmetry there that helps you. In view of Ricci flatness, this separability is equivalent to separability of Hamilton-Jacobi equations and the existence of a Killing tensor. These three objects are devices to extract this hidden structure.

Separation.

Big display….can’t keep up with that. We are studying $\square_g \psi = F$ (where $F$ arises from cutoffs) and we take $\widehat{\Psi}$ and rewrite it using some structure of an oblate spheroidal metric in the $\theta, \phi$ variables. The content of what Carter noticed is that when you do this, you can show that there is a hidden ODE lurking in this decomposition…..

More big display…working pretty hard here, lots of indices….new coordinate $r^{*}$ so that things look more like the Regge-Wheeler coordinates in Schwarzschild. With this decomposition, we can identify the superradiant frequencies. The superradiant modes should be thought of as the modes which send infinite negative energy through the horizon.

Completely separated energy current identies (analogues of $\nabla^\mu (T_{\mu \nu}(\psi) (y \partial_{r^{*}})^\nu)$, etc.)

Lots of notation with symbols I don’t know how to make….

General Idea

From the above currents, produce integral identities with positive definite underlined bulk terms and (upon summation) we get the integrated decay except in regions which can’t be handled this way, basically because these frequency ranges are associated with trapping.

Kerr for small $|a|$. The constructions for all the other frequency ranges can be easily perturbed to yield positive definite bulks. The boundar terms, however, are not a priori controlled, this is the problem of superradiance. Since $|a|$ is small, this can be remedied by adding on a small amount of the redshift identity. Basically, for the small $|a|$ case, we have very little superradiance and can control it with the red shift. For large $|a|$, we need another idea.

The key observation seems to be that superradiant frequencies are not trapped. You can accomodate the superradiance using this idea by using the red shift.

Remark 1: In the small rotation case, the relationship between superradiance and red shift was the key idea. Remark 2: There are no trapped null geodesics which are orthogonal to $\partial_t$. This is a phenomenon identified by Alexakis-Ionescu-Klainerman in their works on uniqueness properties.

Some other important results

Positive and negative cosmological case
Ohter equations, like Dirac, Maxwell instead of wave equation. (Blue, Hafner, Finster et. al)

Open Problems ↩

Extremal case $a=M$ (recent results of S. Aretakis)
Higher dimensions (Schlue, Laul-Metcalfe)
Other measures of decay, Strichartz, …
Robust additional decay
Maxwell equations on Kerr (Blue) (Earlier work by Blue on Schwarzschild)
Equations of gravitational perturbation
Nonlinear stability of Kerr?

Stephen Gustafson: Dynamics on near-harmonic Schrödinger and Landau-Lifschitz maps ↩

(w. Nakanishi, Tsai) The paper that precedes the new stuff here is posted.

Landau-Lifschitz (30s), magnetizations $u(t,x) \in R^3$ with a constraint $|u(t,x)| = constant. The Landau-Lifschitz equation is:

$$ u_t = a_2 u \times \Delta u - a_1 u \times (u \times \Delta u), a_1 \geq 0. $$

Broader context:

$u(\cdot, t): R^2 \rightarrow S^2.$
energy $E(u) = \frac{1}{2} \int_{R^2} |\nabla u|2 dx$
heat flow $u_t = \Delta u + |\nabla u|^2 u = - E’(u)$
Schrödinger Map: $u_t = u \times \Delta u = J E’(u)$, $J$ is a complex structure.
Landau-Lifschitz is a combination of these equations
Also related to wave maps

Regurlarity vs. Singularity: energy critical problems ↩

Energy is scale invariant in $R^2$. $E = \int_{R^2} |\nabla u |^2 \geq 4 \pi |degree (u)|$ iff Harmonic map. So, what can you say?

Heat flow:

$E< 4 \pi \implies$ global smooth solutions Struwe 1985.
$E > 4 \pi \implies$ singularities may form, follows from Chang-Ding-Ye 92 via subsolution construction.

Wave map:

$E< 4 \pi \implies$ global smooth solutions Sterbenz-Tataru 09
$E > 4 \pi \implies$ singularities may form, follows from Chang-Ding-Ye 92 via subsolution construction. Rodnianski-Sterbenz 06, Kreiger-Schalg-Tataru 08, Raphael-Rodnianski 09

Schrödinger Map:

$E$ small $\implies$ global Bejenaru-Ionescu-Kenig-Tataru 08
Open: larger energy, singularity, for LL also.

Equivariant Maps ↩

Simples setting: near harmonic, equivariant maps.

$m \in Z^+$ is the degree
$(r, \theta) $ are polar coordinates
$R = {\hat{k}}\times$ (rotation about ${\hat{k}}$)

2-parameter family of harmonic maps (at minimal $E = 4 \pi m$). The energy is constrained. We work in a small energy shell above the $4 \pi m$ level. This is a restrictive class but known blowups are in this class. For heat flow case, we only have blowups with $m=1$ but for wave maps we have examples with $m \geq 1$.

Theorem Gustafson-Nakanishi-Tsai 09: For $m \geq 3$, solutions are global and converge to a (nearby) harmonic map (asymptotic stability):

$$ { {| u(t) - H^{\mu} |{L^\infty}} } + { { { {a1}} E (u(t) - H^\mu) \rightarrow 0 ~(t \rightarrow \infty). $$

Remarks:

includes the pure Schrödinger map case $a_1 =0$
also for $m=2$ heat flow ($a_2 = 0$) in a symmetry sub-class:
- solutions are global and converge to a harmonic map family
- the parameters can drift, eg to give infinite-time blowup: $s(t) \rightarrow 0$. In particular asymptotic stability fails. (Here $s$ is the length scale of the harmonic map.)

He draws a picture. A $\delta$ neighborhood of the harmonic maps in the energy space, viewed as an infinite graph over $s>0$. For $m \geq 3$, solutions in the $\delta$ neigborhood move dynamically back down to the harmonic maps. For $m=2$, you can drift all over the place in the case of heat flow.

New results: global solutions for degree 2 (LL) with $a_1 > 0$. ↩

Setting, as above with dissipations.

Theorem GNT: For $m=2$, solutions are global and converge to the harmonic map family, but not to one particular harmonic map.

The harmonic map family parameter $\mu(t)$ does not have to converge in general. But it will converge if the initial perturbation has a slightly faster spatial decay: $$ u_{x_1} - u \times u_{x_2} \in |x| L^1 \implies \mu(t) \rightarrow \mu_\infty. $$
For $m=1$ and only for the heat-flow, finite time blowup can occur but not for more localized perturbations.

(We expect this holds for (LL) but we have no proof.)

Remark: New results of Bejenearu-Tataru for $m=1$ Schrödinger map of degree 1. Harmonic mapps are unstable, stable with more localization. You should think of the $m=1$ Schrödinger map case as the most delicate.

Standard “modulation theory” approach ↩

Take your solution and split it into the harmonic map piece plus a remainder. Rewrite things for this remainder term. You look at the linear part of this equation driving the remainder dynamics. Because of invariances of the equation, the linearized operator has zero modes. What would you do to kill the kernel? Choose the parameter at time $t$ so that the perturbation is orthogonal to the kernel of the linearized operator. This leads to an ODE on the parameter dynamics. It remains to get dispersive/diffusive estimates to prove that the parameter converges as time goes to infinity. Do we have these estimates?

Dispersive/diffusive estimates

The remainder $z$ is controlled by a derived quantity $q$, so the analysis is a bit indirect and we now fight to control this related quantity. The $L^2$ norm of $q$ measures the energy gap above $4 \pi m$. The (recast) remainder $q$ satisfies a reasonable nonlinear Schrödinger-(heat) equation. This equation has a potential which depends upon $m$. For $m>1$, we have a lower bound estimate on the potential $V$. In this case, we can get “Strichartz” estimates on $q$.

For $m \geq 4$, this standard approach works G-Kang-Tsai 08, [Guan-G-Tsai 08].
For $m \leq 3$, the orthogonality condition is incompatible with the desired $L^2_t$-decay estimates and the standard approach fails.
For $m \leq 2$,, the orthogonality condition makes no sense.
For $m=1$, we don’t even have an $L^2$-eigenfuction but rather a resonance.

A remedy for $m \leq 3 $ and its cost. ↩

Change the orthogoanlity condition. Instead of demanding that the remainder be orthogonal to the kernel of the linearized operator, you require that the remainder be orthogonal to a localized function (unrelated to the kernel). Of course, there is a penalty for this change. The parameter dynamics ODE transforms then to involve another term and we have different parameter dynamics. This extra term is analyzed in some way.
Solution 1: “Normal form” for $m=3$. By integrating by parts a few times, we get good control on a modified quantity $[\mu (t) - (\psi^s /s | q)]$. We need control on the correction term. For $m \geq 3$, the correction basically does nothing so things work as before.
Parameter drift for $m=2$ heat flow. In this case, the “normal form” correction need not be bounded. For the heat-flow case, it is possible to simplify up to converging errors and the nonintegrability of the correction term can be exploited to drive the blowup, blowdown and oscillation properties of the scale parameter $s(t)$.
Solution 2: Take $a_1 >0$ and exploit the dissipation “We need to somehow stop pretending that the Schrödinger and heat equations are the same…” In the dissipative case, we can extract some dissipation on the correction term. (Probably, this decay is not available in the Schrödinger case.) There are some factorization tricks where the operator is recast, some Duhamel tricks…and an iteration on dyadic time intervals where the time dynamics of the parameter $\mu(t)$ are updated. What emerges is an upper bound by $\log t$ on the parameter $\mu(t)$.

Conclusions ↩

Near harmonic dynamics for (LL) for degree $m \geq 3$.
For $m =2$, more complex behavior.
For $m=1$, do finite time singularities form? This is only known for the heat-flow.

Ioan Bejenaru: Near soliton evolution in 2d Schrödinger Maps ↩

(joint work with Tataru)

Much of this will be a deja-vu since it overlaps with Gustafson’s talk.

Schrödinger map, Heisenberg model in ferromagnietism or the conservative part of the Landau-Lifschitz equation. Energy Conservation, scale invariance $s_c = n/2$ which is the threshold for the well-posedness theory. The $n=2$ case is energy critical.

Main Question: Global regularity of smooth solutons?

Sulem-Sulem 86 established existence of local solutions for $s> [n/2] + 2$, and this was improved to $s> [n/2] +1 $ by McGahagan. An “easier” problem is the global regularity for “small” initial data. Chang-Shatah-Uhlenbeck 00, ….Bejenaru-Ionescu-Kenig-Tataru 08 established GWP of the SM for small data in the critical Sobolev space in $n=2$.

Small data now resolved. What happens for large data?

Large Data Theory ↩

The dynamics depend upon the target manifold. For the sphere target, the problem is called “focusing” and for the hyperbolic target, the problem is called “defocusing”. This terminology makes good sense for wave maps but is not as explicitly understood in the case of SM.

A key feature in these problems is played by the existence of solitons: $ u \times \Delta u = 0$, which are known as harmonic maps. there are no nontrivial finite energy harmonic maps for the hyperbolic target. There are nontrivial harmonic maps with finite energy for the $S^2$ target.

A SM which fails to be regular at one time, bubbles like a HM.

Main Conjecture: In the hyperbolic case, the problem is globally wellposed independent of the size of the data. In the spherical case, solutions emerging from data below $4 \pi$ will be globally wellposed while the problem with higher energy may develop singularities.

The above conjectre is known for the harmonic map flow Eells-Sampson 64, Struwe 85, Chang-Ding-Ye 92.

Singularity formation for the WM problem. Recent works RS08, KST08, RR10.

There is some progress on more general targets.

Equivariant Harmonic Maps on $S^2$. ↩

These are maps from the plan into the sphere. Think that the origin is mapped to the south pole. The point at infinity is mapped to the north pole. Think of the image of the positive x axis as a curve connecting the south and north pole. When you move in the domain around wrt theta one time, the curve connecting n and s pole moves around the sphere some number of times. Once you have these maps, you can fatten them up into a two parameter family of maps.

Basic setup for stability/instability ↩

Define the two parameter family of $m$-equivariant harmonic maps. If you have slightly more energy, then you float around the harmonic map family. But do you stay nearby a particular harmonic map or can you float far away? If you move far, the higher derivatives do not stay under control. We want to describe the trajectory of these maps.

Modulation Theory ↩

Linearize near a soliton, study the zero eigenvalue, and these solutions do not disperse. You want to get rid of this eigenvalue. There is room to do that because we have some choice about which soliton you linearize around. This approach has been developed by Gustafson-Kang-Tsai 06 and Gustafson-Nakanishi-Tsai 09. This has been pushed further recently but involves higher degree hypotheses. We have decided to concentrate on the $m=1$ case.

Theorem (Bejenaru-Tataru):

Let $m=1$ and $\gamma \ll 1$. The for each 1-equivariant initial data $u_0$ satisfying $ \| u0 - Q(0,1)^1 \| X \leq \gamma$, there exists a unique global solution $u$ so that $u - Q(0,1)^1 \in C(R,X)$ and $ \| u_0 - Q(0,1)^1\|_{C(R,X)} \lesssim \gamma$.

There exists a solution $u$ with the additional property that $ \| u(0) - Q(\alpha, \lambda)^1 \|$ …..ack……slide switched I could not keep up. It is an instability result with a large upper bound. It was not clear to me if the assertion was that this big drift really occurs but certainly this is suggested.

Ionescu-Gustafson-Bejenaru conversation: localizations of the perturbations can restore the stability for the heat flow case…

Frank Merle: Isolatedness of characteristic points for blow-up solutions of semilinar wave equation ↩

Background References (Incomplete)

(joint work with Hatem Zaag)

I want to give a talk about a series of works I have done with H. Zaag on the semilinear wave equation.

Semilinear Wave Equation, Blowup Surface ↩

$$ u_{tt} = \Delta u + |u|^{p-1}u$$ Here $p>1$. Let’s collapse to dimension 1. We have initial data $(u0, u1) \in H^1 \times L^2$.

Summary of the results:

Local Existence: We have local existence until a blowup time $[0,T)$.
Existence of blowup via ODE method. There is a more refined condition due to Levine: If a (not the same as mine) energy is negative then $T< \infty$.
The blowup phenomenon can be spatially localized. Therefore, as in the book of Alinhac, you can produce a blowup surface. The solution is well defined on all backwards cones behind the blowup surface.
Question: We want to understand the blowup surface. We don’t know anything about it besides that it is 1-Lipschitz.
A point is called non-characteristic on the blowup surface if the surface has smaller than slope 1 at that point so it does not touch the boundary of the light cone. Let us denote the set of characteristic points on the curve by $S$. The other points on the curve are non-characteristic and the set of such points is called $R$. Let us denote the blowup curve by $x \rightarrow T(x)$ so it is given by a graph $(x, T(x))$.

Caffarelli-Friedman 85: For $u_0 \geq 0, u_1 \geq 0$ and use monotonicity of the wave flow in 1 dimension to prove that $\partial_t u \geq (1 + \delta )|\partial_x u|$ and you can prove then that no characteristic points don’t exist.

This result is a bit misleading. We tried to prove the nonexistence of characteristic points and could not do it. So, we turned our attention to proving the existence of characteristic points.

Summary of Results ↩

Existence of characteristic points. There exist initial data $(u_0, u_1)$ which has $S$ nonempty.
$S$ is isolated, $R$ is open.
$T(\cdot)$ is $C^1$ on $R$.
The only way that a characteristic point can arise is like a “hat”. $T’$ from the right and from the left are well defined. (Alinhac has examples for quasilinear equations which can blowup at all points along a line segment of slope 1.) At points $S$ we have $T’$ of slope 1 on the right and slope -1 on the left.
At points along $R$, the solution is of one sign and points in $S$ are points where the solution changes sign.

Characteristic points are cusps along the graph of $T(\cdot)$.

A Lyapunov functional (Antonini-Merle).

He shows that the solution extends outside the light cone behind noncharacteristic points. This gives you the $T’$ well-defined (with same value from left and right) at a noncharacteristic point.

The talk was hard for me to type up and explain well….Frank emphasized that the proofs are quite intricate and not presentable in a linear fashion.

Ben Dodson: Defocusing $L^2$-Critical NLS ↩

References

Mass-Critical NLS ↩

$$ i u_t + \Delta u = \mu |u|^{4/d}u, u(0,x)= u_0 (x), x \in R^d$$

We concentrate on the defocusing case where $\mu =1$. This equation conserves the quantities

$M(u(t)) = \int |u(t,x)|^2 dx$
$E(u(t)) = \frac{1}{2} \int |\nabla u(t,x)|^2 dx + \frac{\mu d}{2(d+2)} \int|u(t,x)|^{\frac{2(d+2)}{d}} dx$

Strichartz Pairs $(p,q): \frac{2}{p} = d( \frac{1}{2} - \frac{1}{q} ), d \geq 3, p \geq 2.$

$$ A(m) = \sup [ { {| u |}{L^{2(d+2)/(d)}} } (R \times R^d): { {| {u0} |}_{L^2}} =M ] $$

Minimal Mass Blowup Solution Strategy ↩

Theorem (Tao-Visan-Zhang 08): If $u(t,x)$ is a minimal mass blowup solution on $I$ then $\exists ~x(t), \xi(t): I \rightarrow R^d, ~ N(t): I \rightarrow (0, \infty)$ and $$ u(t,x) = \frac{1}{(N(t))^{d/2}} e^{i x \cdot \xi(t)} Q_t ( \frac{x - x(t)}{N(t)}) $$ where $Q$ changes with time but ranges only in a precompact set. For any $\eta > 0, ~ \exists C(\eta) < \infty$ such that

$$ \int_{|x- x(t)| > \frac{C(\eta)}{N(t)}} |u(t,x)|^2 dx < \eta, $$

$$ \int_{|\xi- \xi (t)| > {C(\eta)}{N(t)}} |{\widehat{u}} (t,\xi)|^2 d\xi < \eta. $$

Theorem (Killip-Tao-Visan): To prove GWP it suffices to exlude three scenarios:

$N(t) \sim t^{-1/2}, ~t \in (0, \infty)$,
$N(t) =1 , ~ t \in (-\infty, \infty)$,
$N(t) \leq 1, {\liminf }_{ {t \rightarrow \pm \infty}} N(t) =0, ~ t \in (-\infty, \infty).$

Then he writes and doesn’t really explain……

$\int_{1}^{\infty} N(t)^3 dt < \infty$
$\int_{-\infty}^{\infty} N(t)^3 dt < \infty$

Collapse to $d=3$ for now.

Theorem (CKSTT 04): Interaction Morawetz Estimate

$$ | u(t) |^4_{ {L^4 (J \times R^3)}} \lesssim { {| u |^3_{L^\infty (L^2)}}} {| u |_{ {L^\infty(H^{1})}}} $$

He quotes some estimates from KVZ linking time integrated (over slabs) powers of $N(t)$ with Strichartz size on same slabs.

On LWP time intervals $J_k$ (defined by diagonal Strichartz norm of size $\epsilon$), we have $N(t_1) \sim N(t_2)$ on $J_k$.

Galilean Invariance Observations ↩

Using Duhamel formula, he claims that the galilean invariance $\xi(t)$ does not move too rapidly. This allows him to localize things near the frequency center and in this way tames the galilean invariance.

Planchon-Vega paper on interaction Morawetz describes why the interaction Morawetz estimate is galilean invariant. All these expressions involve galilean invariant right sides and left sides. He then explains that the Morawetz Action leading to the interaction estimate is galilean invariant.

This allows him to claim that $ i \partial_t (Iu) + \Delta (Iu) = |Iu|^{4/3} Iu + [|Iu|^{4/3} (Iu) - I(|u|^{4/3}u)]$ enjoys some nice control (if we could ignore the error term in square brackets). So, we turn our attention to the error term.

For $N \leq CK$, we have

$$| P_{ {|\xi - \xi(t)| > N }} u(t) |{ {L^2(L^6)}} \lesssim (\frac{K}{N})^{1/2} \rho(N) $$ where $\rho(N) \leq 1$ with $\lim{N \rightarrow \infty} \rho(N) = 0.$

$L^2_t$ interval decomposition induction argument ↩

Bust up $[-T, T]$ into small intervals on which we have good Strichartz control and….not clear what he is doing to me right here…. Sort the intervals into good and bad intervals where a bad interval is where $N(J_k) \geq \frac{\eta_1 (d) N}{2}$. He makes a crude estimate on the bad intervals and pays for them by adding up their contributions. On the good intervals, he changes the organization of the decomposition.

Either $$ \eta_1 (d) N \geq \sum_{J_k \subset G_j} N(J_k) \geq \frac{\eta_1 (d) N}{2} $$ or $G_j$ lies to the left of a bad interval or $G_j$ is ont he end of $[-T,T]$. This allows him to claim that the number of $G_j$ is bounded by $C(d) \frac{K}{N}$.

….not clear to me….but hopefully it will be after I work some more.

Decomposition of nonlinearity ↩

He expands the nonlinearity wrt the decomposition around the moving frequency center $\xi(t)$ and the moving spatial center $x(t)$. He dismisses some parts of the nonlinearity based on the induction hypothesis, and the smallness in $L^2$ on the frequency regime far from the moving galilean center. The bad term that remains needs further study.

Questions ↩

Colliander: What are the main new ideas beyond the works of Killip, Tao, Visan and Zhang? ###
- Galilean invariance taming trick.
- Barely slipping under the wire.
- Induction argument using $L^2_t$…
Colliander: And for lower dimensions?
- The critical spaces of Koch-Tataru $U^p, V^p$.
- Harder work on the decomposed nonlinearity due to the absence of the endpoint Strichartz estimate in $d=2$.
- $d=1$ is easier than $d=2$, which is a nightmare.

Postlude ↩

I had a nice conversation with Fabrice Planchon who reported having a longer discussion with Dodson in June. Fabrice suggested that the new elements are the galilean invariance trick, the induction argument exploiting the $L^2_t$ control on left and right side of the Duhamel estimate (only available in $d \geq 3$) and the role played by the time integrals of powers of $N(t)$. Technical difficulties in 3d emerge because the nonlinearity is not multilinear and the analysis there would be simpler if it were. In 2d, we have a nicer nonlinearity but the absence of the endpoint Strichartz estimate in 2d obstructs the $L^2_t$ induction argument. Dodson in fact uses the double endpoint! This conversation made me think that it might be a nice exercise to try to revisit the 2d argument under the false assumption that the forbidden endpoint Strichartz estimate and following the 3d strategy. Alternatively, there might be a streamlined (but incomplete) proof which exposes the strategy more cleanly if we assume somehow that the nonlinearity in the 3d case were multilinear.

Killip: Energy Supercritical Wave Equation in 3d ↩

Background References

Introduction ↩

$$ u_{tt} - \Delta u + u^7 = 0, ~ u: R \times R^3 \rightarrow R$$

$ E(u^\lambda) = \lambda^{-1/3} E(u)$ so that the energy does not control the small scale behavior. This is very alarming.

Definition: $$\cal{E} (t) = { {| u (t) |}^2_{ {\dot{H}^{7/6}}}}+ { {| u_t |}_{\dot{H}}^{1/6}}.$$

Theorem (Killip-Visan 2010): $\cal{E} (0 )< \infty $ then

$\cal{E}(t)$ diverges.
$u(t) - u^{\pm} (t) \rightarrow 0$ as $t \rightarrow \pm 0$, where $u^{\pm}$ is a solution of the linear wave equation.

Radial case was done by Kenig-Merle. Minimal blowup solutions have good spatial decay properties. This is really the main point of their work and ours. Two essential points in the KM work:

Radial Sobolev embedding: ${\dot{H}}^{7/6} \ni u \implies |u| \leq r^{-1/3}$.
$r u(r)= u_{out} (t-r) + u_{in} (t-r)$.
If the solution is small intially ${\cal{E}}(0) < \eta$ then scattering holds.

Scattering is equivalent to the finiteness of some spacetime Strichartz $L^{12} (R \times R^3)$.

Step 1: Minimal Criminal ↩

Keraani first proved the existence of minimal blowup solutions and there were used by Kenig-Merle. At each moment of time, this object has certain localization properties. It is frequency localized on a characteristic frequency scale $N(t)$ and is spatially localized near $x(t)$ at the Heisenberg dual scale $\frac{1}{N(t)}$. Here $N(t)$ defines a multiplier which captures 99% of the norm. Riesz’ interpretation of the Arzela-Ascoli theorem shows this object is precompact. This is why we call these objects almost periodic.

Ionescu: What does minimal mean?

Answer: Samllest ${\| {\cal{E}} (t)\|}{L^\infty{t}}.$

I can apply symmetries and subsequential limits to these minimal objects.

Step 2: Minimal Criminal satisfies one of three scenarios: ↩

$N(t) =1$ soliton-like
$N(t) \geq 1, N(t) \rightarrow \infty$ as $t \rightarrow \infty.$
Finite time blowup.

Step 3. No finite time blowup solutions. ↩

How could blowup occur? The norm lives on smaller and smaller sets. By finite speed of propagation, we can deduce that there is a point where concentration occurs. Suppose we have a minimal blowup solution. We then look at the backwards light cone. Outside the light cone, $u=0$ by minimality. We know that $u$ has finite 7/6 norm and it lives on a small set. But this means that the energy must go to zero and this means the solution is actually the zero solution so is not a finite time blowup solution.

Soliton and Cascade Solutions have finite energy.

Step 4. Solutions move more slowly than light speed. ↩

He draws a forward light cone. There is no energy at the apex. We know that the energy inside the ball defined by the light cone at time $T$ is bounded by $T^{1/3}$. Energy can come into the cone but nothing can go out due to light speed. This tells us that $$ \int_0^T \int_{|x| =t} |u|^8 dS dt \lesssim T^{1/3}. $$

This argument works well if $N(t)$ is not changing too fast. For varying $N(t)$, this can be shown to violate speed of propagation.
There are some other variations to get this nailed down.

Step 5. $L^p$ decay. ↩

We are worried that our super smooth function does not decay fast enough.

$\dot{H}^{7/6} \rightarrow L^9$ (embedding), but we can actually prove that it is in $L^6$. At any time, we can represent $u$ using a Duhamel formula: $$ u(0) = - \int_0^\infty \frac{\sin(t|\nabla|)}{|\nabla|} u^7 (t) dt. $$ You can use the energy-flux identity to turn this into the $L^6$ control. How? You split low into high and low frequencies. We are only afraid of the very low frequencies. $(u_l + u_h)^7$ so $u_l$ is small and some interpolations give you the control.

Why? There can be no other term at null infinity since we would be wasting stuff and this would not be minimal.

Step 6. A more quantitative $L^p$ estimate. ↩

$$ \int_{|x - x(t)| \geq R} |u|^8 dx \leq R^{-\gamma}. $$

Split the time interval $[0, \infty]$ into $[0, R/3]$. We can then set up a geometric bootstrap. Everything is fine on the short time interval. If we look far into the future, we get smallness in $L^\infty$ and can then interpolate against the $L^6$ control to get the target $L^8$ estimate.

Step 7. Climax $E(u) < \infty.$ ↩

We gain regularity.

Write the $H^1$ norm as an inner product: $\langle \nabla u(0), \nabla u(0) \rangle + \langle u_t, u_t \rangle$. Now play with Double Duhamel. We have a kernel decay like $|t - s|^{-1}$ which will not converge when integrated over $dt ds$. We would actually need $|t - s|^{-2 - \epsilon}.$ We introduce a Whitney ball. We decompose $R^3$ in Whitney balls w.r.t. the origin. The negative powers of $R$ gained above from the quantitative decay estimate allows us to sum over the Whitney balls.

This shows the energy is finite, after a lot of bookkeeping.

How do we use this to wrap things up and prove the theorem.

Step 8. Completion of Theorem ↩

No Soliton: $\frac{x}{|x|} \cdot p$ leads to the Morawetz identity which implies the estimate:

$$ \int \int \frac{|u(t,x)|^8}{|x|} dx dt \lesssim E(u).$$ This kills the soliton.

No Cascade:

Using the Whitney balls slack, we can in fact get tightness:

$$ \int \langle x - x(t) \rangle^\epsilon [|\nabla u|^2 + |u_t|^2] dx \leq \infty.$$

Questions/Comments: ↩

Nakanishi: Do you have the same result if you have bounded critical Sobolev norm hypothesis is only true in one direction of time?

Killip: If this nemesis existed, then I can time translate it to create a nemesis that I have just shown can not > exist. So, I believe this relaxed hypothesis can be made with the same conclusion.

Colliander: Peter Pang (an undergraduate at U. Toronto) has recently numerically simulated this problem in the radial case and observed that the critical Sobolev norm remains bounded and is not monotone in time.

Colliander: Can you relax the bounded critical norm hypothesis to one with very slow, say logarithmic, growth and maintain the scattering conclusion?

Killip: This makes my head spin. The minimal object approach, a la Kenig-Merle, is not amenable to this relaxation. It might be possible to approach this with the (more quantitative) gopher strategy of CKSTT.

Wilhelm Schlag: Global dynamics above the ground state energy ↩

(joint work with Kenji Nakanishi NLW, NLS)

Klein-Gordon and Schrödinger Equations ↩

$$u_{tt} - \Delta u + u = u^3, R^{1+3}$$

$$ i \partial_t \psi + \Delta \psi + |\psi|^2 \psi = 0, R^{1+3}$$

LWP in $H^1$. $T_* (| u(0)|_{\cal{H}}) >0 $ where $\cal{H} = H^1 + L^2$.

$E(u) = \int \frac{1}{2}(|\nabla u|^2 + |u_t|^2 + u^2) - \frac{1}{4}|u|^4 dx.$

If $E<0$ then you have finite time blowup.

Scattering set: $S_+ = [(u_0, u_1) \in \cal{H}: T_* = \infty, | u |_{ST} < \infty]$

$S_+$ is open, path connected, $S_+$ contains a small ball $B_\delta (0)$.

Questions and Answers ↩

Questions

$S_+ $ bounded in $ \cal{H} $.
$\partial S_+$: Is this smooth or very rough?
What is the dynamics of solutions on the boundary?
Does $\partial S_+$ separate regions of global existence versus finite time blouwp?

Answers

Recall $\exists ~ Q >0$ satisfying $-\Delta Q + Q = Q^3$.

Statements

Theorem (Nakanishi-Schlag 2010): (Radial Case for now)

$S_+$ is unbounded.
$\partial S_+ \cap [(u_0, u_1) \in \cal{H}: E(u_0, u_1) < E(Q, 0) + \epsilon^2]

Trichotomy: If you are slightly above Q, you either

Scatter to $Q$.
Scatter to 0.
blowup.

Computer Simulations ↩

(done with R. Donninger)

These were beautiful and provoke lots of ideas and wonder.

Structures in Phase Space ↩

$S_+ \cap surface$. Take $(Q+ Af, Bg), (Af, Bg)$, here with $f,g$ are radial fixed functions. Here $A,B$ are parameters and we draw a rectangle in (A,B) space and we color based on (numerical) GWP vs. Blowup.

$K(u) = \int | \nabla u|^2 + u^2 - u^4$

$ PS_{\pm} = [ (u_0, u_1) \in {\cal{H}}: E(u_0, u_1) < E (Q,0), K(u) \geq 0 (for +)]$ and $K(u) < 0$ for -.

PS denotes the Payne-Sattinger (1978) sets. What is up with these sets?

$-\Delta \phi + \phi = \phi^3, ~ J’ (\phi) = 0$ where $J(\phi) = \int \frac{1}{2} (|\nabla \phi|^2 + \phi^2) - \frac{1}{4} \phi^4 dx.$

$ K(\phi) = \langle J’(\phi) | \phi \rangle = 0.$

$\partial_{\lambda}|_{\lambda = 0} J(e^\lambda \phi) = K(\phi).$

Find the minimal height of the potential well. You do some mountain pass work.

$\inf [J(\phi): \phi \in H^1, \phi \neq 0, K(\phi) = 0)] = J(Q) = \inf [ J(\phi) - \frac{1}{4}K(\phi): \phi \in H^1, \phi \neq 0, K(\phi) \leq 0]$

Cor: $PS_{\pm}$ are invariant under the flow.

$PS_{+} \implies$ global existence.
$PS_{-} \implies$ finite time blowup.

$K(\phi) \geq 0 \implies K(\phi) \gtrsim \min (1, \| \phi \|_{H^1}^2 )$

Cor: $Q$ is unstable.

….as usual, Wilhelm is fast….deductions are rapid fire.

Ibrahim-Nasmoudi-Nakanishi proved that you not only have global existence in $PS_{+}$, but using the Bahouri-Gerard-Kenig-Merle compensated compactness machinery, you actually have scattering.

Final State Descriptions near $Q$ ↩

Theorem (Nakanishi-Schlag): ${\cal{Hrad}}^\epsilon = [ (u0, u1) \in {\cal{Hrad}} : {\cal{E}} (u0, u1) < J(Q) + \epsilon^2 ].$ Then, this set is a disjoint union of 9 nonempty sets. $\| (u, u_t) - (\pm Q, 0) \|_{\cal{H}} < C\epsilon.$

-: Scatter, Trapped by $\pm Q$, Finite time blowup
+: Scatter, Trapped by $\pm Q$, Finite time blowup

(Choptuik and Bizon have explored similar pictures in studying the GR setting.)

ack….too fast for me to type….grazing solutions…penetrating solutions…..exit mechanism….and now he is speeding up…..mind like a ferrari….beautiful phase space portraits

Jeremy Marzuola: Scattering and soliton stability in ${\dot{H}}^{-1/6}$ for quartic KdV ↩

(joint work with H. Koch)

The goal is to outline the ideas in this work.

The problem ↩

$$ \partial_t \psi + \partial_x (\partial_x^2 \psi + \psi^4) = 0 $$ with initial data $\psi_0$.

Quartic KdV is the first integer power gKdV that is not completely integrable. Also, we use multilinear estimates.

small data case: $| \psi0 |_{ {\dot{H}}^{-1/6}} \ll 1$.

$\psi0 = Q_{c} (x-x_0) + v_0, | v0 |_{ {\dot{H}}^{-1/6}}.$

Questions:

Scattering and GWP for small data (Yes)
Scattering and Asymptotic stability (Yes)
Existence of inverse wave operators (Almost)

Previous Results ↩

Pego-Weinstein 1994, Asymptotic stability with exponential weights.
Martel-Merle 2001-…, Asymptotic stability in energy space $H^1$ in a moving reference frame.
- Virial Identities
- Monotonicity properties
Côte 2006, Constructs multiple soliton solutions for gKdV.
Grünrock 2005, Multilinear estimates.
Tao 2006, Asymptotic stability in $H^1 \cap {\dot{H}}^{-1/6}$.

Function Spaces ↩

I don’t want to construct spaces in as much detail as done in the paper here. The convergence in the wave operators takes place in a Besov refinement of ${\dot{H}}^{-1/6}$.

$(U^p, V^p)$

These spaces are nicely presented in a paper by Hadac-Herr-Koch 2009. Tataru, Koch-Tataru.

Steps of Proof ↩

Improved linear estimates, there are many linear equations meriting detailed study.
- Airy $(\partial_t + \partial_x^3) \psi = 0.$
- The $u$ problem: $(\partial_t u + \partial_x ({\cal{L}} u))=0.$
- The $v$ problem: $(\partial_t v + ({\cal{L}} \partial_x v))=0.$
- Refined Kato smoothing estimates for Airy
- ${\cal{L}} = (-\partial_x^2) + c - p Q_c^{p-1})$ Refined (weighted) elliptic estimates for $\cal{L}$
- Virial identities (Martel-Merle) for the $v$ problem $\implies$ energy spaces for the linear evolution.

First Result:

$P_{Q’}^{\perp} \psi = \psi - \frac{\langle \psi, Q’ \rangle}{\langle Q’, Q’ \rangle} Q’$
${\tilde{P}}_{Q’}^{\perp} \psi = \psi - \frac{\langle \psi, Q \rangle}{\langle Q, {\tilde{Q}} \rangle} {\tilde{Q}}$ where ${\tilde{Q}} = x \cdot Q’ + \frac{2}{3} Q.$
$\cal{L} (\partial_x Q) = 0$
$\partial_x (\cal{L} Q’) = 0$
$\partial(\cal{L} \tilde{Q}) = Q’$
Variable coefficient operators (small modulations)
$U, V$ spaces/Littlewood-Paley.
Multilinear Estimtes
- Rely heavily upon the $L^6$ estimate: $\| u \|{L^6{t,x}} \leq \| |D|^{-1/6} u \|_{L^2}.$
- Bilinear Estimate…long expression hard to read….
- Example: $$\| \partial (v1 v2 v3 v4) \| ({ {\dot{Y}}^{-1/6}{\infty, T}}) \leq c \prod{j=1}^4 \| vj \| ({ {\dot{X}}^{-1/6}_{\infty, T}}).$$
Full nonlinear problem requires delicate modulation. If you do so, you can’t close the multilinear estimates. Instead, we only require orthogonality asymptotically, rather than at all times.
More multilinear estimates involving $Q, {\tilde{Q}}, Q’$.
GWP for small data/scattering in scaling spaces
Inverse wave operators.

Energy spaces ↩

Virial identity for the $v$-problem: $\eta (x) = -\frac{5}{3} \frac{Q’}{Q}$. Claim: $$ - \frac{d}{dt} \int \eta(x) v^2 dx + c \| [sech]^2 (\frac{3}{2} x) v \|_{H^1}^2 \leq 0.$$

So, we have some monotone decrease in this weighted space.

$$\partial_t \langle v, Q’ \rangle = \langle {\cal{L}} (\partial_x v), Q’ \rangle $$.

Kato Smoothing:

$\gamma_0 (x) = 1 + \int_{-\infty}^x (1 + |y|^2)^{-(1+\epsilon)/2} dy.$
$\gamma_\mu = \gamma_0 (\mu^{-1} (x - \mu^{-2} t))$

$ \frac{d}{dt} \int \gamma_\mu u^2 dx + \int (\gamma_\mu)’ (u_x^2 + \frac{1}{3 \mu^2} u^2) dx \leq 0.$

$ \partial_t \langle {\cal{L}}^{-1}v, v\rangle = 0.$

Lemma:

$$ E(v) = \int \gamma (x) (v_x^2 + v^2) dx + \lambda_E\int \eta(x) v^2 dx + \Lambda_E \langle {\cal{L}}^{-1} v , v\rangle

We define then our “natural” Energy spaces.

$X^s = L^\infty H^s \cap L^2 H^{s+1}_{\sqrt{\gamma’}}$
$Y^1 = L^1 H^1 + L^2 \sqrt{\gamma’} L^2$

We then build spacetime function spaces using the $U^2, V^2$ spaces (defined in S. Herr’s talk) based on these structures and the cubic dispersion relation….and not the linearized equation for the $v$ equation…..chalk coming too fast for me to write down…..ack.

Nonlinear Modulation ↩

$\psi (x,t) = Q_{c(t)} (x - y(t)) + w(x,t)$

$ \partial_t w + \partial_x (\partial_x^2 w + 4 Q^3 w) = \frac{\dot{c}}{c} {\tilde{Q}}(x-y) + ({\dot{y}} - c) (Q_c)’ (x-y) - \partial_x ( O(w^2)). $

Usually, we choose w $\perp Q, Q’$ through choice of $c, y$.

$ \frac{ {\dot{c}}}{c} \langle (Q_c) , (\tilde{Q}c ) \rangle = \langle w, (Qc ) \rangle.$

$ (\dot{y} - c^2) \langle (Q_c)’, (Q_c)’ \rangle = - \kappa < w, (Q_c)’>$

We then calculate: $$ \frac{d}{dt} \langle w, Q \rangle + \langle w, Q \rangle = O (w^2), $$

$$ \frac{d}{dt} \langle w, Q’ \rangle + \kappa \langle w, Q’ \rangle + O (w^2) = 0. $$

With this structure and the formalism of Tao, and some careful work, we can put it all together.

Postlude ↩

I had a nice follow-up conversation with Raphaël Côte. I wondered whether there were similar small data and remainder-atop-soliton scattering results for low power KdV equations. He pointed out that “clean” scattering does not hold in the small data case for the low power gKdV equations. Instead, there are modified scattering statements for data satisfying certain weighted conditions proved by Hayashi and Naumkin It is perhaps reasonable to expect corresponding statements about the error term in the asymptotic stability results around (multi)solitons. However, this is open for study.

Sijue Wu: Global and almost global wellposedness of the two and three dimensional full water wave equations ↩

arXiv: GWP of full 3D water wave problem

Introduction ↩

We are looking at the middle of the ocean. Let’s imagine infinite depth and no boundary. We have gravity pointing odwn and the density of the air is 0 and the density of the water is 1. We assume the water is inviscid, incompressible, irroational, surface tension is zero. The interface is called $\Sigma (t)$.

The motion of the fluid is described by the Euler equation ${v_t} - v \cdot \nabla v = (0, -1) - \nabla P$ in the interior $\Omega(t)$. We also have $div v = 0, curl v =0$, ….ack slide changed.

G.I. Taylor (1949) linearized about the flat interface and found that air above water is stable but water above air is unstable.

LWP ↩

LWP for arbitrary data [S. Wu 1997 (2d) 1999 (3d)]: Local existence in Sobolev spaces under the right Taylor stability condition.

Earlier Results:

Beal, How, Lowegrub 1992 formulated the Taylor sign condition: $ -\frac{\partial P}{\partial n} \geq c_0 > 0#.
Nalimov 1974 infinte depth
Yoshihara 1982.

The work has been extended in many directions. Iguch 2001, Ogawa and Tani 2002, Ambrose and Masmoudi 2005, Lannees 2005, Christodoulu and LIndblad 2003, Lindblad 2005, Coutand and Skholler 2005, Zhang and Zhang, Shatah and Zhang.

Global-in-time behavior ↩

What is the global in time behavior of the solution of the water wave equation?

We will focus on small and smooth data. This is reasonable since it is known that 90% of the waves on the ocean are smaller than 2m? I’d like to know the reference for this 90% claim. Maybe this is done using satellite data? Perhaps this remark motivates a probabilistic Cauchy theory which explains the infrequency of rogue waves?

….slides are changing fast….I can’t keep up so I will listen and make remarks wehn I can.

Quadratic interaction is too strong so the key idea is to use a change of variable which recasts the problem with a cubic nonlinearity.

A natural setting for studying 3D water wave is the Clifford Algebra and use Clifford analysis. The difficulties in 3D are that there is no Riemann mapping, the Clifford Algebra is noncommutative, products of analytic functions in 3D are not analytic. We find that in the 3D problem there is also a special structure allowing us to recast the problem so that quadratic problems disappear and the nonlinearity is cubic and higher orders in nature. It is not purely cubic, there are some quadratic terms but we can handle those as though they are cubic.

Statements ↩

Theorem: (2D) Assume initial wave is of small height, initial velocity is also $\epsilon$ small. Assume we have finitely many derivatives of f and g are in $L^2$. Then, there is a unique solution on a time interval $[0,e^{c/\epsilon}]$. During this time, the solution remains smooth and small.

Theorem: (3D) We assume less here. Suppose initial condition given as a graph. For data with small steepness (no smallness condition on the height) and possibly with infinite energy but also with small velocity on the interface, then the solution is uniquely defined and global-in-time, remains smooth and small.

It seems like we have a better result in 3D. But, in my opinion, these two results are equivalent, they are of equal strength: equally good/equally bad. We can view the 2D case inside the 3D problem and in that view we have an infinite energy 3D case. Maybe we can prove the 2D result under the small steepness condition.

Famous picture of Rogue wave with a ship in foreground.

Rogue waves are vastly massive waves (30m). Often appear in perfectly clear weather, wtithout warning. It’s exact causes are still unknown. Possible causes? Diffractive focusing (effect from caostline)? Focusing of currents? Nonlinear effects? We are avoiding wind and boundaries so we want to understand whether nonlinear effects can be explained as the source of rogue waves.

I am confused. The 3D result says that initial waves given as a graph over the bottom with small steepness remain small and smooth forever. So, this result does not explain or speak to the rogue wave phenomenon. Of course, it suggests that large initial steepness is required for a rogue wave to form within this model of the ocean. Again, this situation seems ripe to me for a probabilistic study of the Cauchy problem?

“Once you get the algebra part right, the analysis part just goes through without complication.”

We only need to know the fluid motion on the fluid interface. We therefore try to reduce the Euler equation to an equation on the fluid interface. This removes the difficulty of the free boundary.

Normal Forms Discussion ↩

The technical discussion seems to revolve around making a bilinear change of dependent variable with the goal of killing off the cubic terms. It doesn’t work….but when working in the right coordinate system with the right quantities, the nonlinearity of the 2D water wave equation is cubic and higher orders.

Nickolay Tzvetkov: On random data nonlinear wave equations ↩

Background References

(joint work with Nicolas Burq)

Framework ↩

Let $(M,g)$ be a Riemannian manifold of dimension $d=3$ with $\partial M = \phi$. We consider the cubic wave equation $$ (\partial_t^2 - \Delta_g) u + u^3 = 0 $$ with initial data $(u0, u1) \in H^s \times H^{s-1}$.

$H^{1/2}(M)$ is the critical space for this problem. He sometimes denotes the problem with (*).

Theorem (deterministic theory):

The problem (*) is locally well-posed in $H^s \times H^{s-1}, ~ s \geq 1/2$ and globally for $s \geq 1$.
The problem (*) is ill-posed in $H^s \times H^{s-1}, ~ s \in (0, 1/2).$
1. For example, $\exists ~ (u_n (t))$ sequence of smooth solutions of (*) such that the initial data goes to zero in $H^s \times H^{s-1}$. But, $ \| (u_n(t), \partial_t u_n (t)) \|{L^\inftyT_ ; H^s \times H^{s-1}} = + \infty, ~\forall T>0.$ (inspired by work of Christ-Colliander-Tao)
2. Moreover, $\exists$ a single data $(u0, u1) \in H^s \times H^{s-1}$ such that $\forall ~T>0$, (*) has no solution in $L^\infty ([0,T]; H^s \times H^{s-1})$ satisfying the finite propagation speed. (instantaneous blowup inspired by work of Lebeau)

On $R^3$, there are refined global results for $s \in [3/4, 1]$ are due to Kenig-Ponce-Vega, Gallagher-Planchon, Bahouri-Chemin, Roy, …. Probably this can be transported to the torus (using finite propagation speed) but this is not written. OPEN QUESTION

Question: Can one still prove some form of well-posedness for $s< \frac{1}{2}$?

Idea: Yes, by randomizing the data.

We have a general method to do this locally in time Burq-Tzvetkov 2008.
A very particular method for globally in time [Burq-Tzvetkov 2008]((http://arxiv.org/abs/0707.1448 “Random data Cauchy theory for supercritical wave equations II : A global existence result”)), exploiting invariant measures a la Bourgain.

Goal for today: General method for globally in time. We can skip this invariant measure business. But, if we are only PDE people, there is a method which allows us to globalize without relying upon the invariant measure aspects.

Randomized data on $T^3$ ↩

Starting from $(u0, u1) \in H^s \times H^{s-1}$ we form their Fourier series $$ u0 = \sum_{n \in Z^3} c^0_n e^{i n \cdot x}$$ (same for u1) and we define $$ u_0^\omega = \sum_n g_n^0 (\omega) c_n^0 e^{i n \cdot x} $$ with natural hypotheses on the random variables to ensure the data stays real valued. He also decomposes the $g(\omega)$ in real and imaginary parts is a system of i.i.d. random variables with a joint distribution $\mu$ satisfying $\exists ~c>0, ~ \forall ~ \gamma >0, \int_{-\infty}^{\infty} e^{\gamma x} d\mu (x) \leq e^{c \gamma^2}$.

Examples:

Gaussians: $d \mu (x) = e^{-x^2/2} \frac{dx}{2\pi}$
Bernoulli: $d \mu (x) = \frac{1}{2}( \delta_{-1} + \delta_1)$

The gaussians generate a dense set in $H^s$. Bernoulli does not but leaves the data on the same sphere in $H^s$.

Theorem: Let $M = T^3, (u0, u1) \in H^s \times H^{s-1}, ~ s \in [0,1]$. Then (*), with data $(u_0^\omega, u_1^\omega)$ is globally well-posed almost surely in $\omega$.

Consider the probability measure $\rho$ on $H^s \times H^{s-1}$ defined by the map: $ \omega \rightarrow $(u0^\omega, u1^\omega)$. Every function gives a different measure, so I have many measures.

Theorem (again): There exists a set $\Sigma$ such that $\rho (\Sigma) =1$ and such that $\forall ~ (v_0, v_1) \in \Sigma$ there is a unique global solution of (*) with data $(v0, v1)$ such that $$ (u, u_t) \in [Free ~Evolution ~of~ (v_0, v_1)] + C(R; H^1 \times L^2). $$ In addition, the solution satisfies the finite propagation speed and, moreover, if we denote by $\Phi(t)$ the constructed flow we have the following properties:

$\Phi (t) (\Sigma) = \Sigma$
$\forall (v_0, v_1) \in \Sigma$, $\| \Phi(t)(v_0, v-1)\|_{H^s \times H^{s-1}} \lesssim \langle t \rangle^{1-s/s +}, s>0.$ (Remark: The implicit constant here is a random variable.)
Measure same thing in $L^2 \times H^{-1}$ and we get the bound $e^{c t^2}$.

Steps in the proof ↩

Global existence step. (inspired by Paley and Zygmund)
Construction of the set $\Sigma$. (inspired by the invariant measure consideration by Bourgain)
Control on the flow for $s>0$. (inspired by the high/low frequency decompositon a la Gallagher-Planchon and by recent work by Colliander-Oh)
Control on the flow for $s=0$. Here the analysis degenerates. (inspired by the work of Yudovich on the Euler equations) “We can say that we have developed a probabilistic version of the Yudovich argument.”

On the proof of the Global existence step for $s>0$ ↩

Large deviation estimates. Consider $\square_g u_{lin}^\omega$ with the randomized data $(u0^\omega, u1^\omega)$. For $s>0, ~\delta > 0, ~\exists c>0, ~ \forall \lambda \geq 0$ we have the large deviation estimate $$ p ( \omega: \| \langle t \rangle^{-\delta} u_{lin}^\omega \|_{L^\infty (R \times T^3)} > lambda ) \leq \frac{1}{c}e^{-c \lambda^2}. $$ Of course, this is much better than what we can get from Strichartz.

We look for solutions as $u = u_{lin}^\omega + v$ and we study $\square_g v + (v + u_{lin}^\omega)^3 = 0$ with zero initial data. We have the energy $E(v) = \frac{1}{2} \int |\nabla v|^2 + |v_t|^2 + \frac{1}{4}\int v^4 dx. We then calculate $\frac{d}{dt} E(v) = \int \partial_t v (v^3 - (v + u_{lin}^\omega)^3).$ We are lucky because the $v^3$ terms cancel and by Gronwall we have global existence for $\omega$’s of big probability. Then, we make some intersections and do some measure theory to finish.

This argument gives exponential control. We revisit the analysis using the high/low frequency truncation ideas to improve to polynomial control.

Remark: We can prove similar results for ANY manifold by using a randomization due to Lebeau.

Questions ↩

Schlein: How is the set $\Sigma$ invariant?

Tzvetkov: The set \Sigma is of the form random orbit of the data plus smooth functions. Since the smooth functions have zero measure, we can throw them into \Sigma.

Ionescu: How do you see in the analysis that you are studying the defocusing question?

Tzvetkov: In the Gronwall business, we used the sign.

Pierre Germain: Global existence for coupled Klein-Gordon equations with different speeds ↩

Background References:

General Problem: Understand global existence and scattering for nonlinear dispersive equations with very nice data. ↩

We will assume the Cauchy data are small, smooth and localized. We will further restrict the problem to semilinear wave and Klein-Gordon equations in dimension 3.

NLW, $d=3$ ↩

$\square u = |u|^{p-1}u$.
- Above the Strauss exponent $ p > 1 + \sqrt{2}$.
- At the Strauss exponenent, finite time blowup was shown by [John-Schaeffer]
$\square u = |u_t^2 - |\nabla u|^2$.
- Null form structure observed by Christodoulu and Klianerman gives global existence.
$ \square u = |u_t|^2.$
- finite time blowup [John]
$\partial_t^2 u^i - c_i \Delta u^i = \sum Q^i_{jk} (Du^j, Du^k)$
- Global existnce if $Q^i_{jk}$ is a null form. [Yokoyama, Ohta, Katayama, Sogge, Metcalfe, ….]

NLKG ↩

$\partial_t^2 u - Delta u + u = |u|^{p-1}u.
- For $p>2$ (the Strauss exponenet), you have global existence [Strauss].
$\partial_t^2 u - Delta u + u = Q(u,u)$ or $Q(Du, Du)$.
- global existence [Klainerman], [Shatah]
What about different propagation speeds? $\partial_t^2 u^i - c_i \Delta u^i + u^i = \sum Q^i_{jk} (u^j, u^k)$
- This case has some difficulties and my new result addresses this issue.

All the results I quoted have been provd using the vector field method. How does this work? You find a bunch of vector field $(\Gamma_i)$ which commute with the linear part of the equation. Then you estimate $\Gamma^\alpha u.$ The method does not apply to KG with different speeds. You don’t have sufficiently many commuting vector fields to treat the multiple speed KG case.

There were some other methods used for these problems. In particular, Shatah used a normal forms method. Christodoulu used a change of variables method but most of the theory has been built on the vector field method.

NLKG with different speeds is a toy model for Euler-Maxwell, provided you restrict to high frequencies and ignore certain things.

Statement ↩

Theorem: $$ \partial_t^2 u - \Delta u + u = Q(u,v), ~ \partial_t^2 v - c^2 \Delta v + u = P(u,v) $$ with some initial data for the two equations. (No derivatives in the quadratic nonlinearities.) Assume that the data has some $L^2 $ weighted (power 1 ) control and is small enough and we also have $H^N$ smallness with a big enough N. Then there eists a global solution which furthermore scatters in $H^N \times H^{N-1}$ which means the nonlinear evolution converges to a linear solution as time goes to infinity.

The vector field method does not apply. Instead, we use a spacetime resonances method which we have applied to the water wave problem and to the NLS equation. This is a new instance where we can apply this method. The method was developed in collaboration with Shatah and Masmoudi.

Spacetime resonance method ↩

For the sake of exposition, consider $i \partial_t u + P(D) u =u^2$ emerging from data $u0$. Let $f(t) = e^{-it P(D)} u(t)$ and consider this new unknown function instead of $u$. Write the Duhamel formula for $\hat{f}$. What you find is that $$ \hat{f} (t, \xi) = \hat{u_0} (\xi) + \int_0^t \int e^{i s [P(\xi + \eta) - P(\xi) - P(\eta)]} \hat{f} (\eta, s) \hat{f} (\xi - \eta, s) d\eta ds. $$ We have a problem if the phase is stationary either in s. What can save us is the oscillations. This is what we call time resonances. Or, if the phase is stationary in $\eta$ and this is what we call space resonances. Of course, the worst situation is when we have stationarity in both senses and this is what we call spacetime resonances.

Method

If the phase factor (redenoted as) $\phi \neq 0$ an integration by parts in $s$ and push the nonlinearity to cubic. This is just the normal forms method seen on the Fourier side.
If $\partial_\eta \phi \neq 0$ you can integrate by parts in $\eta$ and you gain an $s$ in the denominator which is “always pleasant when you are trying to prove global eistence.” This is the vector field method seen in Fourier space.

He draws two graphs where $\phi$ and where $\partial_\eta \phi$ vanish on the $\xi, \eta$ Cartesian product. We use pseudo-product operators Coifman-Meyer to decompose in the $(\xi, \eta)$ space. $$ \mathcal{F} ( B_{m(\eta, \xi)}) (f,g) (\xi)= \int m(\eta, \xi) \hat{f} (\eta) \hat{g}(\xi - \eta) d \eta. $$

Physical meaning

Time resonances are “standard resonances” in the dynamical systems sense.
Space resonances are when waves of different frequency move with the same group velocity (….not really explained)

Application to our problem ↩

You get a lot of different phase functions:

$\phi (\xi, \eta) = \langle \xi \rangle_l \pm \langle \eta \rangle_m \pm \langle \xi - \eta \rangle_n$ where $\langle x \rangle_\alpha = \sqrt{1 + \alpha^2 x^2}$ and $l,m,n$ are chosen among the two possibilities: 1 and $c$.
Look at the place where both $\phi$ and where $\partial_\eta \phi$ vanish.
- Sometimes this set is empty.
- Sometimes this set has the form $[ |\xi | = R, \eta = \lambda \xi]$ for real numbers $R, \lambda$.
- Actually, such a set is generic for interactions between waves with a dispersion relation $p(|\xi|)$ which depends only on the frequency size. Thus, the method can be applied to other settings.

He redraws the graph of the zero level sets for $\phi$ and $\partial_\eta \phi$. He then excises around the point where these sets intersect using a cutoff using a pseudo-product operator with a symbol $m$ which is increasingly singular along the set of simultaneous vanishing. This is a bit annoying because there are no general estimates for such pseudo-products. We would need to estimate the boundedness of $B_m: L^p \times L^p \rightarrow L^r$ where $m$ is singular along $[ |\xi | = R, \xi = \lambda \eta]$ for real parameters $R, \lambda$. The Coifmann-Meyer calculus requires nicer properties on $m$. In contrast, there is work by Lacey-Thiele on the bilinear hilbert transform which does have a singularity in the 2-multiplier but does not apply to our case.

We use that you are at the Strauss exponent so that rough estimates are enough to succeed.

In the theorem, we need to assume that resonances are separated. Look at the spacetime resonance set $\cal{R} = [\phi = 0] \cap [\partial_\eta \phi = 0]$ and project onto $\xi$, which I call “outcome frequencies”. If you project onto $\eta, \xi - \eta$ you get what I call “source frequencies”. We need to assume that $[outcome] \cap [source] = \phi$. This is generically true for different speeds $c$. In particular, we have this property true for all but a discrete set of speeds $c$.

There is alast point wihich is a bit problematic: Spacetime resonaces at $\infty$. $\phi, \partial_\eta \phi \rightarrow 0$ at $\infty$. To overcome this difficulty, we rely upon the high regularity $H^N$ hypothesis using Strichartz estimates. We then separate the analysis into low and high frequencies.

Questions ↩

Koch: Gain from modulation versus gain from bilinear estimate. Dualize the argument and you can recast as a condition on the nonvanishing of $\partial_\xi \phi$.

Postlude: ↩

After the talk, I learned from Pierre that he had written an expository article on the spacetime resonances method. T

Oana Ivanovici: Dispersive Estimates on convex domains ↩

(joint work with Fabrice Planchon)

Introduction ↩

Consider a domain $\Omega$ of dimension $d \geq 2$. We consider the wave equation $\partial_t^2 u - \Delta u = 0$ with initial data and vanishing on $\partial \Omega$.

Consider, for point of reference versus later statements, the situation where $\Omega = R^d$. Take $u_0 = \delta_a$ and $u_1 = 0$. Then the solution is given by the Green’s function $$ u_{a, R^d} (t,x) = \int \cos (t |\xi|) e^{i \xi \cdot (x-a)} d \xi. $$

Dispersive Estimates: $$ | \psi (h D_t) u_{a, R^d} |_{L^\infty} \leq C(d) h^{-d} \min (1, (\frac{h}{|t|})^{\frac{d-1}{2}}). $$

We are interested in the case where $\partial \Omega \neq 0$. We must confront reflected waves, glancing rays and waves which travel along the boundary.

Let $\Omega$ be a strictly convex domain. In particular, we will consider $\Omega$ to be the Friedlander domain. $\Omega = [(x,y) \in R^d: x>0, y \in R^{d-1}]$ with the associated Laplacian $\Delta = \partial_x^2 + (1+x) \Delta_y$. This is very close to the laplacian on the disk. Then, she draws the half space and describes the bicharacteristics as a bunch of circles bouncing along the floor.

Theorem: Take $a>0$ small so that $(a,0) \in \Omega$ (in the interior but close to $\partial \Omega$). $\exists ~T>0$ such that $\exists ~ C>0$ such that $\forall ~ h \in (0,1]$ we have $$ | \psi (h D_t) u (t, x, y) |_{L^\infty (\Omega)} \leq C(d) h^{-d} \min (1, (\frac{h}{|t|})^{\frac{d-2}{2} + \frac{1}{4}}). $$ The way to study this is to consider the set of points you can reach from the point $a$ upon traveling for time $T$. The method of proof involves a decomposition of the data in terms of wave packets which hit the boundary a certain number of times. The worst packets are localized in small cones that are almost parallel to the boundary.

…rapid discussion of some frequency localzations…lots of glancing rays pictures….subsequent reflections are denoted by $u_j$. Each reflection involves a loss of 1/6 derivative and there can be manyreflections ccumulating until a total loss of 1/4 derivative. After that there will be no more regularity loses.

Applications ↩

$\implies$ Spectral projector and Strichartz estimates. Smith and Sogge studied similar problems using a reflection across the boundary idea. For dimensions $d \geq 3, the reflection method does not have a chance to get optimal regularity losses. First, you don’t see the dispersion tangential to the boundary. Also, their study only captures the loss from one reflection but does not resolve the accumulated losses. The loss of 1/4 derivative happens at a special time after many reflections.

Works by Blair-Smith-Sogge are improved in this work. She draws some Strichartz diagrams and shows that her new dispersive estimate implies a wider range of valid Strichartz exponents.

We will soon see that the only possible losses are 1/6 or 1/4.

Cusp solutions hugging the boundary ↩

This result was announced at a conference in Evian by G. Lebeau. Lebeau explained the geometrical features of the argument but the analytical details were not written down. Fabrice and I are writing those down….

To demonstrate the loss, she writes the boundary and draws data that looks like a cusp. $$ u_0 (x,y) = \int e^{i \frac{\eta}{h} 9\frac{\xi^3}{3} + (x-a)\cdot \xi + y} \psi (\xi) \phi (\eta) d\xi d\eta. $$ The wave starts localized within $a$ of the boundary. AFter some time $t \sim 2 \sqrt{a}$ the cusp is upside down wrt boundary and then at the time $t = 4 \sqrt{a}$ the cusp reappears and the singularities only appear at these specific locations near the boundary. The situations is studied with $a \thicksim h^{1/2}$. For $a$ smaller than this power, we would not be able to repeat the construction for many reflections. It will degenerate. The caustic in this case is the line sliding along the boundary passing through the cusps. along the caustic, the intensity of light is much brighter. At points along the caustic, oscillatory integrals don’t enjoy good bounds.

Proof ↩

$$ u_h (z) = \frac{1}{h^{1/2}}\int e^{\frac{i}{h} \phi (z, \xi)} \sigma (z, \xi, h) d\xi, \xi \in R $$ Everyone knows that the number and degeneracy of the critical points of the phase function control the asymptotics of this guy as $h \rightarrow 0$.

Degenerate critical points: Let $k$ denote the *order of the caustic of u_h$ be defined by $\inf_{k’} [k’: | u_h | \sim O (h^{-k’})]$

Example 1: Let $\phi_F (z, \xi) = \frac{\xi^3}{3} + z_1 \xi + z_2.$ Here $z_1 = -\xi^2, ~z_2 = - \frac{2}{3} \xi^3$. So we have a fold. This type of phase function corresponds to $k = 1/6$. She draws a sideways parabola and projects it down onto a cusp.

Cusp type integral: $\phi_C (z,\xi) = \frac{\xi^4}{4} + z_1 \frac{\xi^2}{2} + z_2 \xi + z_3.$ (This has order 1/4) (Pearcy-type integral)

$\partial \phi:~ z_2 + 2 z_1 \xi + \xi^3 = 0
$\partial^2 \phi: ~ 2 z_1 + 3 \xi^2 = 0
$\partial_\eta (\eta \phi_c ): ~ z_3 _ \xi z_1 + z_2 \frac{\xi^2}{2} + \frac{\xi^4}{4}=0.

* Swallowtail:* $\phi_s (z, \xi) = \frac{\xi^5}{5} + z_1 \frac{\xi^3}{3} + z_2 \frac{\xi^2}{2} + z_3 \xi + z_4.$

We have a degenerate critical point of order 4…..ack….I am running out of battery and this is really nice stuff…

Axel Grünrock: Cauchy Problem for higher order KdV and mKdV equations ↩

I am interested in the question of optimal local well-posedness.

Background References

Grünrock: Paper described in this talk

Equations ↩

KdV hierarchy

Lax 1968 introduced the hierarchy of higher order KdV equations. $$ \partial_t u + \partial_x G_j (u) = 0 $$ We will refer to this as (hoKdV-j), the higher order KdV equation. $$ \langle Gj (u), v \rangle = \frac{d}{d\epsilon} Hj (u+\epsilon v)|_{\epsilon = 0} $$

where

$$ Hj (u) = \int P_j (u, \partial_x u, \dots, \partial_x^j u) dx. $$ These are the Hamiltonians of KdV.

$P_{-1} (u) = u$
$P_0 = - \frac{1}{2} u^2$
$P_1 (u) = - \frac{1}{2} u_x^2 - u^3$

The iteration procedure then defines the hierarchy:

$G_1 (u) = u_{xx} - 3 u^2 \implies u_t + \partial_x^3 u = 6 u u_x$
$ u_t + \partial_x^5 + 5 \partial_x ( \partial_x^2 u^2 - (\partial_x u)^2 - 3u^3)=0$
$ u_t + \partial_x^7 u - 7 \partial_x (\partial_x^4 u^2 - 2 \partial_x^2 (\partial_xu )^2 _ (\partial_x^2 u)^2 - 10 u \partial_x (u \partial_x u + 5 u^4) = 0.$
….

We can thus define some general structure of the higher order KdV equations based on rank properties where $rank_{KdV} = degree + \frac{1}{2}~ derivatives ~in~ x = j+2$. We find that $|\rho| = 2 (j-k) + 3.$

For all the equations in the hierarchy, we have the same scaling critical regularity of $s_c = - \frac{3}{2}$.

There is a second shared property for all the equations in the hierarchy. The Hamiltonians in the KdV hierarchy are all in involution with respect to the Poisson bracket: $$ { H_k, H_l } := \langle G_k (u), \partial_x G_l (u) \rangle, ~\forall k, l \geq -1.$$ We can therefore calculate that $$ \frac{d}{dt} H_k (u) = \langle G_k (u), \partial_t u \rangle = - \langle G_k (u), \partial_x G_l (u) \rangle = 0. $$

mKdV hierarchy

A similar tower or hierarchy of equations may be built around the mKdV equation using the Miura map: $v \rightarrow v_x + v^2$.

Sequence of ${\tilde{H}}j (v) = H(j-1) (v_x + v^2)$. This spawns ${\tilde{G_j}}(v)$ by writing $$ \partial_t v + \partial_x {\tilde{G}}_j (v) = 0 $$ which we denote by (homKdV-j). What can we say about the structure of the nonlinear terms in the mKdV hierarchy of equations.

The rank condition for KdV hierarchy is transferred via the Miura map into a rank condition for the mKdV hierarchy.

nonlinear terms in mKdV hierarchy are all odd in $v$, so no quadratic terms.
$|l| = 2 (j-k) + 1$
We thus find that the mKdV hierarchy enjoys a joint scaling invariance corresponding to $s_c = - \frac{1}{2}$.

Earlier Results ↩

(Incomplete)

1979 Saut: Existence of persistent solutions of hoKdV-j and homKdV-j in $H^j$ using the energy method which works equally well in the periodic or nonperiodic setting.
1993 Ponce: hoKdV-2, LWP in $H^s (R)$ provided that $s> \frac{7}{2}$ and, combining the LWP result with conservation laws, he obtained GWP for $s \geq 4$.
2008 Kwon: LWP for hoKdV-2 for $s> \frac{5}{2}$ and GWP for $s\geq 3$ using a refined Energy method developed by Koch-Tzvetkov for treating Benjamin-Ono.
1993/4 Kenig-Ponce-Vega: $\exists ~s_0 = s_0 (j)$ and $m - m(j)$ such that $\forall ~ s \geq s_0$, hoKdV-j is LWP in $H^s (R) \cap L^2 (|x|^m dx)$
- Corresponding results for homKdV-j. It was remarked there that the weights are not necessary for treating the cubic and higher power cases.
1995 Linares: homKdV-2 is GWP in $H^s (R)$ provided $s \geq 2$.
2008 Kwon: LWP improved down to $s\geq - 3/4$ and thus GWP in $H^1$.
2008 Pilod: Without the weights in the data spaces, one has ill-posedness in the hoKdV-j hierarchy, ~$\forall ~j \geq 2$. In particular, he showed that the flow map can not be $C^2, ~ \forall s\in R$. The argument involves an interaction between high and very low frequencies. Higher order Sobolev regularity is not beneficial at all.

Killip: Is there a contradiction here with the positive result of Kwong vs. Pilod?

Grünrock: Kwon uses energy methods so obtains continuous dependence, not $C^2$ dependence of the flow map.

New Results ↩

Data spaces: $\| f \|{\hat{H}s^r} =\| \langle \xi \rangle^s \hat{f} \|{L^{r’}\xi}, ~ \frac{1}{r} + \frac{1}{r’} = 1.$ Here $1 < r \leq 2. We have $H^{s,r} \subset {\hat{H}}_s^r.

Spacetime spaces: $ | u |{X{s,b}^{r,p}} = | \langle \xi \rangle^s \langle \tau - \phi (\xi) \rangle^b \hat u|{L^{r’}\xi (L^{p’}\tau)}.$ Here we have $\phi (\xi) \sim \xi^{2j + 1}$.

What are the crucial estimate we need that will lead to local well-posedness?

Ingredients (tools)

Smoothing estimates
- linear: $\| D_x^{\frac{2j-1}{3r} u \|{L^r{tx}} \lesssim \| u \|{X^r{0b}}$ if $b > \frac{1}{r}, ~ \frac{4}{3}< r \leq 2$ ~(fails for $r \leq \frac{4}{3}$.)
- triliner estimates with the same gain order (up to $\epsilon$).
- bilinear refinement: For $b > \frac{1}{p}, ~ 1 < r \leq r_{1,2} \leq p \leq 2, ~ \frac{1}{r} + \frac{1}{p} = \frac{1}{r_1} + \frac{1}{r_2},$ $$ \| M_{j,p} (u,v) \|{ {\hat{L}^rx {\hat{L}}^p_x}} \lesssim \| u \|{X{0b}^{r_1, p}} \| u \|{X{0b}^{r_2, p}$$ We have an increasing gain of regularity with these estimates of gain $D_x^{\frac{2j}{p}}$ in the parameter $\frac{1}{r}$ or $\frac{1}{p’}$, respectively.
Resonance relation $(k=2)$. $$ \sum_{i=0}^2 \langle \tau_i - \xi_k^{2j+1} \rangle \gtrsim |\xi \xi_1 \xi_2| \times ( \xi_1^{2(j-1) } + \xi_2^{2(j-1)})$$. We have a gain: $D_x^{\frac{2j+1}{p’} -}$ since $\langle \tau_0 - \xi_0^{2j+1} \rangle^{b - 1 - \epsilon = \frac{1}{p’}}$. This gain is decreasing in $\frac{1}{r}$ or $\frac{1}{p’}$, respectively.

Statements

homKdV-j: He expresses the LWP results in the $(\frac{1}{r}, s)$ as lines leaving the vertical $s$ axis and all passing through the point $(1,0)$.

The results on $H^s$-scale for $j \geq 3$ are new. We obtain GWP in $H^s$ for $s \geq [\frac{j+1}{2}]$ (integer part). Thus, the use of $\hat{H}^{s,r}$ spaces lead to new insights.

Moreover, the results converge toward a nice statement which identifies a common joint space $\hat{L}^1$ which contains finite measures and which contains $L^1$. Unfortunately, the result at that endpoint is not yet established.

For KdV, he draws a similar picture. The lines do not appear to converge. we are far away from finding a joint space.

Questions ↩

Tataru: C^2 vs. mereley continuous dependence properties?

Staffilani: Periodic case?

Grünrock: No, I don’t have results there.

Colliander: $NLS_3$ in $\hat{L}^1$?

Postlude ↩

For me, fantastically interesting conversations with Koch, Grünrock, Tataru and Vega.

OPEN: Is there a space of functions wherein each equation in the mKdV hierarchy is GWP?
OPEN: The space ${\hat{L}}^1$ appears to be a natural candidate given the visual description Axel gave of his results.
Corresponding questions about cubic NLS in one space dimension? L. Vega points out that ${\hat{L}}^1$ can not do the job because of nonuniqueness results for NLS evolution emerging from the Dirac mass.
NLS has galilean invariance; mKdV does not so perhaps there is some hope for mKdV in ${\hat{L}}^1$?
I will ask Boris Khesin about whether the integrable hierarchy of equations containing cubic NLS is exposed nicely somewhere. It might be interesting to try and carry out an analogous study of the NLS hierarcy.

Selberg: Global existence for the Maxwell-Dirac system in two space dimensions ↩

(joint work with Piero d’Ancona)

The Maxwell-Dirac system (MD): $$ (-i \alpha^\mu \partial_\mu + M \beta) \psi = A_\mu \alpha^\mu \psi$$ $$ \square A_\mu = -\alpha \langle \alpha_\mu \psi , \psi \rangle$$

$B = \nabla \times A, ~ E = \nabla A_0 - \partial_t A.$ We are interested in evolution starting from data $\psi_0, E_0, B_0$ satisfying the constraints $\nabla \cdot E_0 = |\psi_0|^2, ~ \nabla \cdot B_0 = 0.$

We are using the Lorenz gauge condition: $\partial_\mu A^\mu = 0$.

2d: $\alpha^0 = I, ~ \alpha^1 = \sigma^1, ~ \alpha^2 = \sigma^2, ~\beta = \sigma^3$ where the $\sigma$’s are the Pauli matrices and the $\alpha$’s are called the Dirac matrices.

He decomposes the electric field into divergence free and curl free parts. We can then write $E_0 = E_0^{df} + \Delta^{-1} \nabla (|\psi_0|^2)$. We are restricting the motion to take place in the $x^1, x^2$ plane so the magnetic field must be perpindicular to that plane. All fields are independent of $x^3$. $A = (A_1, A_2, 0)$ and $B = (0, 0, \partial_1 A_2 - \partial_2 A_1)$. Given the initial consitions on the $E, B$ fields and the Lorenz gauge condition, we can specify the initial data for the potential $A$.

Maxwell-Dirac and Dirac-Klein-Gordon ↩

DKG: $(-i \alpha^\mu \partial_\mu + M \beta) \psi = \phi \beta \psi$….ack too fast. Best reference for this is the paper of Glassey-Strauss 1979.

Energy - no sign
Charge: $\int |\psi (t,x)|^2 dx = const.$
Scale invariant regularity: $\psi_0 \in {\dot{H}}^{d-3/2}, ~ E_0, ~ B_0 \in {\dot{H}}^{d-2/2}$.
- MD is critical is charge critical in 3d
- charge subcritical in 2d and 1d.

Of course, we would like to prove global regularity. A natural strategy is to prove low regularity LWP and then exploit conservation laws. However, this is not so clear yet…..

Results ↩

Global:

1d MD GWP: 1973 Chadam
3d MD global regularity for small data: 1993 Georgiev
3d MD stationary solutions: Esteban, Georgiev, Séré 1996 EGS
2d DKG GWP: Grunrock and Pecher
2d MD GWP: [d’Ancona and Selberg 2010]((http://arxiv.org/abs/1004.1715 “Global well-posedness of the Maxwell-Dirac system in two space dimensions”)))

Are there stationary solutions for 2d MD? Are there other obstructions to decay/scattering? Are there size thresholds for the 3d MD setting. I should study the [EGS] works….

Local theory in 3d:

Gross 1966
Bournaveas 1996
Masmoudi and Nakanishi 2004
d’Ancona, Foschi, Selberg: Complete null structure of DKG 2007 and MD 2010 and almost optimal LWP.

2d DKG ↩

Charge class data and $(\phi, \phi_t) \in H^{1/2} \times H^{-1/2} (R^2)$.

LWP known for such data.
To get the global result, we need to control $D(t)$, which is his notation for the $H^{1/2} \times H^{-1/2}$ size of the evolving solution $(u(t), \partial_t u(t))$.

Theorem (Grunrock-Pecher 2010): 2d DKG is LWP up to time $T>0$ s.t $T^{1/2} [1 + D(0)] \leq \epsilon$. Moreover, $\sup_{|t| \leq T } D(t) \leq D(0) + C T^{1/2}$ with $C$ dependent on charge constant.

The globalizing procedure follows a general argument introduced by Colliander, Holmer and Tzirakis 2008.

How does it go?

$T^{1/2} [1 + D(0)] = \epsilon/2$
$T^{1/2} \sim \frac{1}{D(0)}$
You iterate $n$ steps and accumlate errors until you grow until $n C T^{1/2} \sim D(0)$. This develops the solution onto a time interval of size $nT \sim 1$ so you have advanced the solution to a local interval whose length only depends upon the charge. Therefore, you can iterate this process to make it go global.

We want to apply this procedure to do the same for MD.

Theorem (d’Ancona and Selberg 2010): 2d MD is LWP up to time $T>0$ s.t. $T^{1/2} [1 + D_T (0] \leq \epsilon$ where $\epsilon$ depends upon the chage constant. Moreover, $$\sup_{|t| \leq T} D_T (t) \leq D_T (0) + C T^{1/2} \log (\frac{1}{T}).$$

Corollary: 2d MD is GWP.

The iteration procedure is more involved than the CHTz scheme due to a logarithmic loss. There is an intermediate iteration which reduces matters to a harmonic series! This was exposed nicely so I watched it without typing…..

What lies behind the proof? ↩

LWP, Subcriticality
Growth estimate for EM field

Key Points

Null structore of nonlinear terms
Refined bilinear estimates needed to exploit the structure
Subcriticality is crucial

He went on to describe the null structure and ideas in extracting the required quantitative slack in the local theory to run the globalization scheme.

Jason Metcalfe: Long time existence for nonlinear wave equations in exterior domains ↩

(many years of collaboration with Sogge and Nakamura)

At the quadratic level, all that work only with derivative terms and not terms involving the solution itself.

Let $K$ be a compact obstacle with $C^\infty$ boundary. We want to solve, in dimensions 3 and 4,

Problem $S$: $$\square u = |u|^p$$ in the exterior of $K$ with vanishing neumann condition and small initial data. Let’s call this problem $S$. Let’s assume here that $K$ is nontrappling

Problem $Q$: $$ \square u = Q(u, u’, u”) $$ with vanishing Dirichlet boundary conditions and with $K$ starshaped.

There are issues that make it difficult to work with the Klainerman vector field, especially the boosts and the scaling vector fields.

In this work, we will only work with $Z = [\partial_i, \Omega_{jk}= x_j \partial_k - x_k\partial_j]$.

Localized Energy Estimate: $$ [\log(2+T)]^{-1/2} | \langle x \rangle^{-1/2} u’ |{L^2{tx}} \lesssim | u’(0)|{2} + \int0^T | \box u(s, \cdot)|_2 ds. $$

A weighted Sobolev inequality $$ R^{(n-1)/2} | h |{L^\infty (\frac{R}{2}< |x| < R) \lesssim | Z^{\leq \frac{n+2}{2}} h |{L^2 (\frac{R}{4}< |x| < 2R)}. $$

KSS: $$ [\log(2+T)]^{-1/2} | \langle x \rangle^{-1/2} Z^{\leq 10} u’ |{L^2{tx}} \lesssim \epsilon + \int_0^T | Z^{\leq 10} (\partial_t u)^2|{L^2} dx \lesssim \epslion + | \langle x \rangle^{-1/2} Z^{\leq 10}u |^2{L^2_{tx}}. $$

Problem $S$: ↩

$p> p_c$ where $p_c > 0$ solves $(n-1) p_c^2 - (n+1) p_c - 2 = 0$:

$n=3 \implies p_c = 1 + \sqrt{2}$
$n=4 \implies p_c =2$.

Theorem (Hidono-Metcalfe-Smith-Sogge-Zhou): $n=3,4; ~p_c < p < \frac{n+3}{n-1}, \gamma = \frac{n}{2}- \frac{2}{p-1}.$ $$\sum_{|\alpha | \leq 2} ( \| Z^\alpha u(0, \cdot)\|{\dot{H}}^\gamma + \| Z^\alpha \partial_t u(0, \cdot)\|{\dot{H}}^{\gamma -1} ) < \epsilon $$ $\implies global existence.

…

Problem $Q$: ↩

No boundary case
- $n=3: \frac{c}{\epsilon^2}$ is the life-span (Lindblad and Hörmander)
- $n=4: exp(\frac{C/\epsilon})$ is the life-span (Lindblad and Hörmander)
$(\partial^2_u Q)(0,0,0)$ (this kills $u^2$ terms, but we are considering here the startshaped boundary.)
- $n=3:$ (in progress with a student John Helms)
- $n=4: \infty$ is the lifetime (Metcalfe-Sogge)

Scipio Cuccagna: The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states ↩

Reference

arXiv:0910.3797v5 Abstract: In this paper we prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M.Weinstein, are also asymptotically stable, for seemingly generic equations. Here we assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.

(This talk relates to the talk of Schlag.)

It seems to me this talk is also closely related to the talks of Marzuola and Muñoz.

We study the nonlinear Schrödinger equation:

$$ i u_t = -\Delta u + V(x)u + \beta (|u|^2) u, ~ in R^3$$

Results for this work do not work for $ i u_t = -\Delta u -|u|^{p-1} u$ with $p < 1 + \frac{4}{n}$.

We assume existence of a family of ground states. When they are gound states the look like you expect but he also had a graph involving nodes and I didn’t understand…

Notions of stability:

linear stability (i.e. Weinstein’s sufficient hypotheses for orbital stability)
- Only for ground states?
orbital stability
asymptotic stability
- $\lim_{t \rightarrow +\infty} \| u(t,x) - e^{i \theta(t)} \phi_{\omega+} (x) - e^{it\Delta} (h+)(x)\|{H^1x} = 0$
CONJECTURE: 1. $\iff$ 2. $\iff$ 3.
Theorem: 1. $\implies$ 3. generically.

Specifically, we prove nonlinear Fermi golden rule (terminology introduced by Soffer and Weinstein, Buslaev and Perlman used different terminology):

a. Some key coefficients are $\geq 0$; b. Generically they are $>0$.

One wants to prove that the remainder scatters. We have discrete and continuous modes. One wants to find a way to describe a mechanism of transfer from the discrete modes into the continuous modes. We want some way of writing the coordinates of the dynamics to reveal a damping effect in the discrete modes due to the transfer of the energy from the discrete modes into the continuous modes. The description of this transfer mechanism is the goal of the Fermi golden rule.

Asymptotic stability is analogous to showing that $u(t)$ solving an NLS-type equation is not only of the same siaze in $H^1$ for all time but also showing that the solution scatters. This is the analysis we want to do on the remainder. Eigenvalues obstruct asymptotic stability. That explains the preoccupation of Schlag with his proof of the nonexistence of eigenvalues in the gap.

Near ground states, we write the solution in a canonical way as a sum of a modulated ground state plus a remainder term. The NLS can be recast as a dynamical system of the phase and scaling parameter coupled to the (presumably dispersive) behavior. He then changes variables so that the system is expressed as a matrix equation in which the “Hamiltonian structure is obscured”. This is the standard way in the literature that the system is expressed. But somehow this way of writing it is wrong. (???)

He makes some assumptions about the absence of embedded eignevalues. He suggests this hypothesis is not necessary but is not certain….some discussion with Tataru.

He writes on the board a horizontal line and draws points at 0, and sa few eignevalues parametrized by $\omega$. He then draws wavy stuff over the right half starting at some point to the right of the eigenvalues representing the continuoys spectrum.

…slides are coming fast and they are too dense for me to type in real time….

Alexandru Ionescu: Uniquness theorems in general relativity ↩

General relativity…

Spacetimes ↩

Spacetimes $(M^4, g)$ are solutions of the Einstein vacuum equations $$R_{\mu \nu} = g^{\alpha \beta} R_{\alpha \beta \mu \nu} = 0$$

The metric is in 4 dimenions, it has 10 components. The Riemann tensor has 20 components. These are 10 equations for the 20 components.

Minkowski space: $(R^3 \times R, -dt^2 + dx^2 + dy^2 + dz^2)$.

Besides being Ricci flat, in fact this solution also has zero Riemann tensor and this condition completely characterizes the Minkowski space.

Schwarzschild spaces:

$ds^2 = -(1 - \frac{2m}{r}) dt^2 + (1 - \frac{2m}{r})^{-1} dr^2 + r^2 (d\theta^2 + (\sin \theta)^2 d\phi^2)$ where $(r, t, \theta, \phi) \in (2m, \infty)\times R \times (0, \pi) \times S^1$. It took several decades to realize that $r=2m$ is merely a coordinate singularity. This was realized with the Kruskal coordinates in which the metric may be expressed $ds^2 = F^2 (-dt^2 + (dx’)^2) + r^2 (d\theta^2 + (\sin \theta)^2 d\phi^2)$. The Kruskal picture is the region between the lobes of hyperboloid of two sheets. The region below the lobes and inside the $|y| = |x|$ cone regions containing the lobes is called the black hole. The domain of outer communication is outside the cone.

Kerr spaces:

$m$ is the mass of the black hole and $J$ is the angular momentum of the black hole. We assume $m>0, ~a = \frac{J}{m} \in [0, m)$ and let $r_+ = m+ (m^2 - a^2)^{1/2}. In Boyer-Lindquist coordinates $(r, t, \theta, \phi) \in (r_+, \infty)\times R \times (0, \pi) \times S^1 $$ -\frac{\rho^2 \Delta}{\Sigma^2} dt^2 + \frac{\Sigma^2 (\sin \theta)^2}{\rho^2 }( d\phi - \frac{2amr}{\Sigma^2}dt)^2 + \frac{\rho^2}{\Delta}(dr)^2 + \rho^2 (d\theta)^2$$ where

$\Delta -= r^2 + a^2 - 2mr$
$\rho^2 = ….$ slide changed….
$\Sigma^2 = …

For Minkowski, 20 of 20 components of the Riemann tensor vanish. For Schwarzshild 19 of the 20 componenents of the Riemann tensor vanish in the right coordinates. For Kerr, 18 of the 20 components vanish in the right coordinates.

Key properties of Kerr spacetimes: ↩

Solutions of the einstein vacuum equations $R_{\alpha \mu} = 0$;
Killing vector field $T = \partial_t$ timelike at “infinity”;
Killing vector field $Z - \partial_\phi$ wiht closed orbits;
Geometric properties: asymptotic flatness, smooth bifurcate sphere, global hyperbolicity;
Rigididty: Kerr spaces are real-analytic.

“No hair” theorems: such properties charcaterize the Kerr spaces (Carter, Robinson, Hawking-Ellis, Mazur, Bunting, Weinstein, Chrusciel-Costa). “We are trying to understand final states.”

Main Conjecture: If $(M^4, g, T)$ is a regular stationary vacuum, then the domain of outer communication of $M^4$ is isometric to the domain of outer communication of some Kerr spacetime of mass $m$ and angular momentum $ma$, $a \in [0, m)$.

What is regular in the conjecture? It took a long time to characterize what that means

There is a lot of supporting evidence.

Carter 1971: axially symmetric black holes have only 2 degrees of freedom
- Mathematically, an imprecise statement. It said there are “no bifurcations”
Robinson 1975: the uniqueness conjecture holds in the case of axially symmetric black holes
- global argument involving the whole space.
Hawking-Ellis 1973: the conjecture holds in the case of real-analytic spacetimes.

Hawking’s strategy is to define an additional Killing vector-field in the spacetime and reduce to the Carter-Robinson theorem. The assumption of real analyticity is not what you really want.

Theorem 1 (Ionescu-Klainerman): The conjecture holds provided that the scalar identity is assumed to be satisfied on the bifurcation sphere.

Theorem 2 (Alexakis-Ionescu-Klainerman): The conjecture holds proved that the spacetime is assumed to be “close” to a Kerr spacetime.

* Theorem 3 (Aliexakis-Ionescu-Klainerman):* Assume $\cal{N}, \underline{\cal{N}}$ are smooth, null, nnexpanding hypersurfaces in an Einstein vacuum $(O, g)$ which intersect transversally in a 2-sphere $Z$. Then there is an opern neighborhood $O’$ of $Z$ and a nontrivial Killing vector-field $K$ in $O’$ which is tangent to the null generators of $\cal{N}, \underline{\cal{N}}$.

This is a local version of Hawking’s Rigidity Theorem, without assuming analyticity of the spacetime.

Construct the Hawking v. K in the domoan of dependence of $ \cal{N}\cup \underline{\cal{N}}$ (Friedrich-Racz-Wald)
Extend the v.f to a full neighborhood of Z by solving a transport equation $[L, K] = cL

Key steps in our strategy:

We deine some tensors: $\pi_{\alpha \beta}, W_{\alpha \beta \mu \nu}.
Prove a system of wave/transport equations of the form:
- $\square_g W = {\cal{M}} (W, Dw, \pi, D\pi)$
- D_L pi ={\cal{M}} (W, Dw, \pi, D\pi)$
Use Carleman estimates and a unique continuation argument to conclude that $W, \pi$ vanish in a neighborhood of $Z$.

Model Theorem (I-Klainerman): Assume $\phi \in C^2 (M)$ and $A, B^l \in C^0 (M)$ for $l = 0, \dots, d$. Assume that $$ \square \phi = A \phi + \sum D^l \cdot \partial \phi…..ack slide change.

Unique Continuation: assume $\phi$ is smooth in $(O,g)$ and solves a wave equation $D^\alpha D_\alpha \phi = A \phi + B^\alpha D_\alpha \phi.$ Assume $\phi$ vanishes in the set $[h<0]$, where $h \in C^\infty (O), ~ \nabla h \neq 0.$ Does $\phi$ vanish in a neighborhood of $[h \leq 0]$?

Suppose we have $T(u)=0 $ in $B$. Suppose $u_1, u_2$ solutions in $B$ and $u_1 \sim u_2$ inside small set $A \subset B$. Basically, there are three possibilities:

lack of uniquneess: $u_1 = u_2$ inside $A$ but $u_1$ is far from $u_2$ in the big set $B$.
Well-posedness: If $u_1$ is close to $u_2$ in $A$ then $u_1$ is close to $u_2$ in $B$.
Unique continuation:
- If $u_1 = u_2$ in $A \implies u_1 = u_2$ in $B$.
- If $u_1$ is close to $u_2$ in $A$ we are unable to conclude that $u_1$ is close to $u_2$ in $B$

Hormander’s pseudo-convexity condition: Unique continuation holds if $X^\alpha X^\beta D_\alpha D_\beta h < 0$…ack slide change.

The method is based on Carleman Estimates.

Model theroem in Kerr spaces (I-Klainerman): Assume $W, A, B, C$ are smooth tensors in the Kerr space $K^4$, and $$ \square_{g_0} W = A \cdot W + B \cdot D W $$ $$ {\cal{L}}_T W = C \cdot W. $$ If $W=0$ on the horizon then $W=0$ everywhere outside.

T-conditional pseudoconvexity property

We would really like a tensor $\cal{S}$ (an analog of the Riemann tensor $\cal{R}$) which has the following properties:

It describes locally the Kerr spaces
It satisfies a suitable geometric equation of the form $$ \square_g \cal{S} = A \cdot W + B \cdot DW$$ $$ {\cal{L}}_T W = C \cdot W $$ We want to then uniquely continue the vanishing.

Mars-Simon found such a tensor. This is a tensor for Kerr which is analogous to the Riemann tensor for Minkowski. The Riemann tensor is a local quantity which characterizes the Minkowski space in the sense that when it vanishes, we know that we are in Minkowski space. Similarly, the Mars-Simon tensor characterizes Kerr. To go from local vanishing to conclude global vanishing, we need an analytic continuation.

More precise statement of Theorem 1: The domain of outer communication $E$ of a regular stationary vacuum $(M, g, T)$ is locally isometric to the domain of outer communication of a Kerr spacetime, provided that the identity $$ - 4 m^2 {\cal {F}}^2 = (1 - \sigma)^4 $$ holds on the bifurcation sphere $S_0$.

More precise statement of Theorem 2: The domain of outer communication $E$ of a regular stationary vacuum $(M,g,T)$ is isometric to the domain of outer communication of a Kerr spacetime, provided that the smallness condition $$ | (1 - \sigma) {\cal{S}} (T, e_\alpha, e_\beta, e_\gamma)| \leq {\overline{\epsilon}} $$ holds along a Cauchy hypersurface in $E$ for some sufficiently small ${\overline{\epsilon}}.$

Postlude ↩

I discussed with Alex whether one could (or should….) formulate a statement similar to Theorem 2 about Minkowski space using the Riemann tensor like: Suppose that the Riemann tensor is small on some (small? geometric conditions?) set $A$ inside a spacetime manifold $(M,g)$. Can one conclude that the Riemann tensor must therefore vanish on $A$ or perhaps on a bigger set $B$? One can view the [AIK] theorem 2 as a Liouville-type theorem: a smallness condition on the Mars-Simon tensor on a subset of $(M,g)$ with certain conditions implies that the Mars-Simon tensor vanishes. Is there a corresponding Liouville-type theorem where smallness of the Riemann tensor on an appropriate subset implies that the Riemann tensor actually vanishes?

Notes on Nonlinear Dispersive Equations Workshop in Istanbul

Sat, 28 Aug 2010 22:21:13 +0000

During last week’s NDE meeting in Istanbul, I experimented and took real-time notes in MultiMarkDown during the talks. If any of the speakers wish, I can post links to their presentations provided they send me a copy of their slides. –J. Colliander

James Colliander (Toronto)

Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2({\mathbb{T}})$

I spoke about recent work with Hiro Oh . Here is a link to my slides.

Some conversations after my talk about Bourgain’s high/low frequency truncation method and refined bilinear estimate are amplified in the slides of my Warwick talk.

Albert Erkip (Sabanci University)

A review of some results on a class of nonlocal nonlinear wave-type Cauchy problems

A lot of the work here is inspired by the thesis of Nilay Duruk.

Overview

Nonlocal Elasticity
Examples
Cauchy Problem
Ongoing Studies

Nonlocal nonlinear equation

$$u_{tt}=\\[\beta*(u+g(u))]_x$$

$$0≤{\hat{\beta}}(\xi)≤C(1+\xi^2)^{-r/2}$$

Examples

Take $\beta=\delta,$ the Dirac measure. The equation becomes a more standard nonlinear equation.

Different choices of $\beta$ lead to different equations.

Cauchy Problem

The results identify conditions, usually involving smoothing assumptions on $\beta$, under which they obtain local well-posedenss results.

Global results are also obtained provided there is appropriate control in $L^\infty$.

How to get the $L^\infty$ control? If the integral of the nonlinearity is bounded from below by $-k u^2$ then we have a global solution.

Ongoing Studies

Obvious generalizations….2d case and efforts to generalize to equations which are meaningful in elasticity. We have also considered the peridynamic problem. This is a nonlocal generalization of the classical problems in elasticity which allows for tears and cracks.

Scattering? Small amplitude initial data? Travelling Waves?

Postlude

OK, so I discussed this further with H. Erbay. There are some issues with the linear problem. Let

$\Delta*\beta$ be the Fourier multiplier operator given by $-ξ^2 \beta(\xi )$. This collapses to the Laplacian when $\beta=1$. For the usual wave operator we have inhomogeneous smoothing of order 1 by the usual denominator games. However, for the wave operator corresponding to $\Delta*\beta$ we have smoothing by division by $\xi \beta(\xi)$. If $\beta(\xi)$ decays like $\xi^{-r}$ with $r>2$ we lose the smoothing property and have new troubles.

In the discussion after the talk, E. Titi asked what they would do on a bounded domain. In this case, the convolution operator used to express the dynamics on the spatial side does not make sense near the boundary. Upon thinking about this a bit, it seems to me that a natural thing to do is to express the data in the basis of eigenfunctions of the Laplacian on the domain and then recast the dynamics as a multiplier operator in that basis. The issues of the domain are addressed then by the eigenfunctions and the nonlocal aspects near the boundary are resolved.

Sadet Erbay (Isik University)

The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials

(joint work with H. Erbay and A. Erkip)

Two dimensional nonlocal equations
LWP
Conservation of Energy and Global Existence
Blowup

Elasticity Motivates Study of Nonlocal Wave Equations

Deformation fields in an elastic body might be influenced by distant points. Therefore, we encounter nonlocal elasticity.

$$ w*{tt}=(\beta* F*{w*x})_x + (\beta *F*{w_y})_y)$$

$$0 \leq \hat{\beta}(\xi) \leq (1 + |\xi|^2)^{-r/2}$$

Taylor expansion in $\beta$ leads to higher order powers in $\xi$, which produces higher order derivative correction terms.

Cauchy Problem

Convert IVP into a Banach space valued ODE. Sobolev embedding. Algebra property of $H^s \cap L^\infty$. Convenient assumptions allow them to control the nonlinearity. (All this is done pointwise in time and the regularity is quite high.)

(My impression is that ideas from (Kenig-Ponce-Vega, Indiana Math Journal, 1991 vol. 40 (1) pp. 33-69) could be used to prove Strichartz-type estimates adapted to this family of problems, under more precise assumptions on the decay of $\hat{\beta}$.

Blowup Alternative expressed in terms of $| Dw |_{L^\infty}$.

Conservation of Energy and Global Existence

The energy involves a Fourier multiplier replacing the usual appearance of $\nabla$ in the kinetic energy. Under certain lower bound conditions on the potential energy, they can prove that a certain norm is bounded for all time which in turn controls the blowup alternative norm $\|Dw\|_{\infty}$.

Nilay Duruk (Sabanci University)

Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

(joint work with H. Erbay and A. Erkip, this is part of her thesis)

Nonlocal Cauchy problem
Local
Global
Blowup

Nonlocal Cauchy Problem

Nonlocal coupled system of two fields each of the flavor flavor

$$u_{tt}=[\beta*(u+g(u))]_x$$

with assumptions that $$0≤\hat{\beta}(\xi)≤C(1+\xi^2)^{-r/2}.$$

Examples include certain coupled improved Boussinesq Equations. Such systems have been studied by Godefroy 1998 and Wang, Li (2009). The system models interaction of transverse waves in an elastic medium.

GWP and Blowup for a coupled system with $\beta = e^{-|x|}$ is open.

LWP

Local Lipschitz continuity of the right hand side of the system using pointwise in time estimates coming from Sobolev control of $L^\infty$.

GWP

Define an operator $P$ which plays the role of $\nabla$ in the energy depending upon $\beta$.

The nonlinearity for the system is assumed to arise from a Lagrangian/Hamiltonian formulation. Thus, we have a conserved energy. We then prove some Sobolev style bounds adapted to $Pu$ generalizing the case of $\nabla u$. With lower bounds on the potential energy, she obtains globalizing control.

Gronwall is used to show that the energy density stays bounded….

Blowup

Adaptation of the virial identity (following Godefroy 1998) shows that negative energy solutions explode.

(This strikes me as something that could be explored further from the generalized virial identity point of view.)

Postlude

I suggested that they look at the KPV paper and to try to imitate Morawetz-type calculations using the generalized virial identity.

G. M. Muslu (Istanbul Technical University)

The Cauchy problem for the one-dimensional nonlinear peridynamic model

(joint work with H. Erbay, A. Erkip, G. Muslu)

Motivation
Peridynamic Model
LWP
GWP
Blowup

Need for a new theory of solid mechanics

For example, across a crack we have discontinuities across. We need a theory which replaces PDEs with integral equations. The peridynamic model was first proposed by Silling in 2000.

Peridynamic Model

Classical elasticity

$$\rho_0 u_tt = (f(u_x))_x$$

Peridynamic theory

$$\rho_0 u_tt = \int f(u(y,t) - u(x,t), y-x)dy$$

Newton’s third law demands that $f(η,ξ)=-f(-η,ξ)$. There are many studies on the modelling but there is relatively little mathematical analysis. Our aim is to study the nonlinear problem.

LWP

For convenience, we study $f(η,ξ)=β(ξ)g(η)$ where $\beta$ is even and $g$ is odd and $g(0)=0.$

Analysis in an appropriate space (pointwise in time tricks) leads to a LWP result. The treat the continuous and bounded case and the $C^1$ and bounded case. They also treat the $H^s \cap L^\infty$ case for all polynomial nonlinearities.

Energy Identity

$$E=\|u_t|^2+ \int \int \beta(y-x) w(u(y,t)-u(x,t)) dydx$$

Nice symmetrization tricks based on even/odd leading to energy identity.

Blowup

Concavity result. Negative energy solutions blowup.

Gonca Aki (Weierstrass Institute, Berlin)

Thermal effects in gravitational Hartree systems

(part of Ph.D thesis, joint w. Jean Dolbeaut and Christof Sparber)

Intersted in self-gravitating quantum particles. We represent this by density matrix operator. We are given a total mass of the system which is the integration of the occupation numbers over all occupation sites.

Energy is the kinetic energy of each state plus an interaction term described using the occupation number operators.

Variational problem corresponding to $H^1$ expressed in terms of the density matrices $\rho$.

The free energy is lower bounded by the kinetic energy using the Hardy-Littlewood-Sobolev inequality.

ack….the talk is dense and fast for me to keep up this way, maybe even without trying to type but I like this stuff!.

Defines a notion of maximal temperature, which could perhaps be infinity. Minimizers satisfy an EL equation so we know more about them.

Compensated compactness leads to proof of existence of minimizers. Obital stability follows. Positivity of critical temperature for all $M>0$. This extends a theorem of Lieb who showed the minimizer was a pure state when $T=0$ to the setting of $T \in [0, T_c]$. They have also found an explicit expression of the value of $T_c$

Remarks for finite maximal temperature: For $\beta (s)=s^p$ with $p∈(1,7/5)$ , the maximal termperautre is finite.

Louis Jeanjean (Université de Franche-Comté, Besancon, France)

Stability and instability results for standing waves of quasi-linear Schrödinger equations

(joint with M. Colin and M. Squassina)

Nonlinearity 23 (2010), 1353-1385

Many other issues can be studied. Lots of open problems.

$$i\phi_t+ \Delta \phi + \phi \Delta |\phi|_2+f(|\phi|_2)\phi=0$$

Cauchy Problem

We will first address the Cauchy problem. Next, we will study the traveling waves and their stability.

Poppenberg, JDE 172 (2001) proved LWP in $H^\infty$.

New energy term: $\int|\phi|^2 |\nabla|\phi|^2 dx$

Cauchy problem is based on work of M. Colin, CPDE 27 (2002), 325-354. This is based on energy methods to overcome the loss of derivatives induced by the quasi-linear term, gauge transforms, ….

OPEN: Solve the local Cauchy problem under more general conditions on the nonlinearity and on the data. Look for global existence results.

Standing Waves

Ansatz: $$\phi\\_\omega= u\\_\omega (x) e^{-i\omega t}.$$

A calculation shows that

$$-\Delta u - u \Delta(|u|^2) + \omega u = |u|^{p-1}.$$

$$<m*\omega = \inf{ E*\omega (u): u is a nontrivial weak solution of the elliptic problem.}$$

The results identified a new critical threshold. We have $1+ \frac{4}{N}$ as usual but this problem also involves $3 + \frac{4}{N}$.

Henrik Kalisch (U. Bergen, Norway)

Conservation equations for long wave models and applications to undular bores

This is basically a modeling problem. There will not be a single proof in this. (joint work with Al Fati Ali and Magnar Bjorkavag)

Surface Waves

Assume the fluid is incompressible, inviscid, two dimensional, irrotational, assumption that the wave does not overturn.

Euler equations, some boundary conditions at surface and at bottom. The LWP problem has been solved in just the last 10 years or so. Numerically, this is a difficult problem. The problem is often simplified by putting long wavelength or small amplitude assumptions.

Long wavelength gives the shallow water wave equations. Shallow water waves equation looks like a coupled system of Burger’s equations. There are an infinite number of conserved quantities in the shallow water wave equations.

Small amplitude case is known as the Airy theory. Rewrite Euler in terms of the velocity potential but you still have the boundary conditions. The pressure is removed. Linearize the Bernoulli equation on the boundary and calculate the dispersion formula by putting in plane waves. It emerges that $\omega^2=g k \tanh(h_0 k)$.

One can compare the dispersion relation for the KdV, BBM and water wave formulae. KdV is a bad model for water waves since

Internet Censorship in Turkey

Mon, 23 Aug 2010 22:14:00 +0000

Istanbul is amazing!

I am having a wonderful visit to Istanbul for a meeting on nonlinear dispersive equations.

But, I was surprised when I tried to go to youtube.com and found that it was censored. The image below is what internet censorship in Turkey looks like. I found out this is old news.

Posts | James Colliander

Digital public goods for Earth system management: U.S. Greenhouse Gas Center launches

Scientific Context

International Context

🇺🇸 U.S.A. Context

The U.S. Greenhouse Gas Center

Openness: Amplified Knowledge Sharing

References

Democracy’s Duty to Open Science

2022 NASA SMD Data and Computing Architecture Study

2022 Jack Eddy Symposium

Why did UBC spend $70M for 3 acres in Surrey?

Where did UBC buy?

What’s the business case?

Is this linked to the Broadway Subway Project?

Is this connected to the new medical school?

What about quantum?

NSERC Open Data Exploration

NSERC Open Data Exploration

TODO Set up similar analyses for SSHRC

TODO Set up similar analysis for CIHR

Overview

NSERC Budget

Accumulated Program Investments [1995 - 2016] Top 25

Accumulated Principal Investigator Investments [1995-2016] Top 25

Discovery

Evaluation Groups

Award Amounts

Number of Awards

Bin Frequency

Engage

National Scale Interactive Computing

Demand for interactive computing

Technical milestone achieved

Service level requirements

What’s next?

Notes on Academic Freedom Dialogue hosted by UBCFA

Canadians Land on Jupyter

Budget 2017, Naylor’s review, and the Mathematical Sciences in Canada

Fortier's Attack on Academic Freedom

Toward Global Science Excellence

Artificial Intelligence as a Service: Text Analysis

Experimenting with Text Analysis

Conclusion

Experiments with IBM Watson Tone Analyzer

Reagan Experiment

Clinton Experiment

Kennedy Experiment

Eisenhower Experiment

Overview Comparisons using IBM Watson Tone Analyzer

Experiments with LIWC

Experiments with Receptiviti Personality Insights

The Lost Opportunity

Hiring Gupta as President was no Fluke

Lost Serendipity: Federal Election

Lost Serindipity: Provincial Election

Lost Money, Lost Leadership, Lost Momentum

No Confidence

Calculating MRR over variable term subscriptions using Pandas

Mathematical Scientist's Guide to Innovation Funding Sources in Canada

NSERC Innovation Programs

NSERC Connect

NSERC Engage

Mitacs Innovation Programs

Mitacs Accelerate

Mitacs Elevate

Further Assistance

I am Joining PIMS and UBC Math

Ubiquity of Mathematics: Ingrid Daubechies

University of Toronto Math Department Colloquium Videos Winter 2013

Wednesday April 24, 2013

Thursday March 28, 2013

Wednesday March 20, 2013

Wednesday March 13, 2013

Wednesday March 06, 2013

Wednesday February 06, 2013

Wednesday January 30, 2013

Wednesday January 23, 2013

Ubiquity of Mathematics: Luis Seco

Ubiquity of Mathematics: Spyros Alexakis