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:

 

 

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)).

Zaher Hani

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:

  1. Pass to the resonant sum.
  2. Obtain good discrete to continuum error estimates.
  3. 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?
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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:

 

 

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.

 

 

 

 

(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.
NSERC Postdoctoral Fellowship Investment over time
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?

NSERC Budget Changes

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.

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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.

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