Organizing Committee
- Patrick Gerard
Paris-Sud University, Orsay - Pierre Germain
NYU - Courant Institute - Alex Kiselev
Duke University - Andrea Nahmod
University of Massachusetts Amherst
Abstract
The large-time behavior of (generic) solutions of nonlinear dispersive equations set on bounded domains is almost completely open as far as rigorous analysis goes, and fairly mysterious, even from a less rigorous viewpoint. Under the assumption of weak nonlinearity, physicists and applied mathematicians have devised a theory to approach this question, known as weak turbulence, a branch of statistical physics. Weak turbulence theory predicts that the equation will enter a chaotic regime, where the exchange of energy in phase space is governed by the so-called kinetic wave equation. Justifying the derivation of the kinetic wave equation is a fascinating mathematical task, for which some results are already known, but whose solution will likely require input from nonlinear PDEs, but also probability theory. Intimately related questions are the question of Sobolev growth (how much can or does, the Sobolev norm of a nonlinear dispersive equation grow over time), as well as the analysis of nonlinear dispersive equations with random data. These questions have already generated a lot of interest, but much remains to be discovered: our understanding is extremely lacunar.
Confirmed Speakers & Participants
Talks will be presented virtually or in-person as indicated in the schedule below.
- Speaker
- Poster Presenter
- Attendee
- Virtual Attendee
-
Yvonne Alama Bronsard
Sorbonne Université
-
Patricia Alonso Ruiz
Texas A&M University
-
Ioakeim Ampatzoglou
Courant Institute of Mathematical Sciences, New York University
-
Xinliang An
National University of Singapore
-
Aidan Backus
Brown University
-
Byeong-Ho Bahn
University of Massachusetts Amherst
-
Hajer Bahouri
Université Paris-Est - Créteil
-
Weizhu Bao
National University of Singapore
-
Marius Beceanu
University at Albany SUNY
-
Jacob Bedrossian
University of Maryland
-
Massimiliano Berti
SISSA
-
Roberta Bianchini
Italian National Research Council, CNR
-
Lydia Bieri
University of Michigan
-
Piotr Bizon
Jagiellonian University
-
Bjoern Bringmann
Institute for Advanced Study
-
Nicolas Burq
University Paris-Sud
-
Nicolas Camps
Université Paris Saclay
-
Esteban Cardenas
University of Texas at Austin
-
Jehanzeb Chaudhary
University of New Mexico
-
Gong Chen
Fields institute
-
Charles Collot
Cergy-Paris Université
-
Diego Cordoba
ICMAT
-
Stefan Czimek
Brown University (ICERM)
-
Magdalena Czubak
University of Colorado at Boulder
-
Joel Dahne
Uppsala University
-
Anne-Sophie de Suzzoni
Ecole Polytechnique
-
Yu Deng
University of Southern California
-
Yu Deng
University of Southern California
-
Giuseppe Di Fazio
University of Catania
-
Michele Dolce
Imperial College London
-
Hongjie Dong
Brown University
-
Jinqiao Duan
Illinois Institute of Technology
-
Daniel Eceizabarrena
University of Massachusetts Amherst
-
Chenjie Fan
Academy of Mathematics and Systems Science, CAS
-
Anxo Fariña Biasi
Jagiellonian University
-
Serena Federico
Ghent University
-
Patrick Flynn
Brown
-
Luigi Forcella
Heriot-Watt University
-
Claudia García
Universitat de Barcelona
-
Eduardo Garcia-Juarez
Universitat de Barcelona
-
Louise Gassot
Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay
-
Patrick Gerard
Paris-Sud University, Orsay
-
Tainara Gobetti Borges
Brown University
-
Javier Gomez Serrano
Princeton University
-
Ricardo Grande Izquierdo
University of Michigan
-
Benoît Grébert
University of Nantes
-
Sandrine Grellier
Université d'Orléans
-
Marcel Guardia
Universitat Politècnica de Catalunya
-
Zaher Hani
University of Michigan
-
Benjamin Harrop-Griffiths
University of California, Los Angeles
-
Susanna Haziot
Brown University Mathematics
-
Siming He
Duke University
-
Justin Holmer
Brown University
-
Slim IBRAHIM
University of Victoria
-
Alexandru Ionescu
Princeton University
-
Pranava Jayanti
University Of Maryland College Park
-
Istvan Kadar
University of Cambridge
-
Adilbek Kairzhan
University of Toronto
-
Thomas Kappeler
Universität Zürich
-
Dean Katsaros
UMass amherst
-
Panayotis Kevrekidis
University of Massachusetts Amherst
-
Alex Kiselev
Duke University
-
Friedrich Klaus
Karlsruhe Institute of Technology
-
Haram Ko
Brown University
-
Herbert Koch
University of Bonn
-
Thierry Laurens
University of California, Los Angeles
-
Gyu Eun Lee
University of Edinburgh
-
Tristan Leger
Princeton University
-
Yao Li
University of Massachusetts Amherst
-
Guopeng Li
University of Edinburgh
-
Felipe Linares
IMPA
-
Kyle Liss
University of Maryland, College Park
-
Ruoyuan Liu
University of Edinburgh
-
Jonas Luhrmann
Texas A&M University
-
Jeremy Marzuola
University of North Carolina
-
Nader Masmoudi
Courant Institute of Mathematical Sciences at NYU
-
Alberto Maspero
Scuola Internazionale Superiore di Studi Avanzati (SISSA)
-
Jonathan Mattingly
Duke University
-
Joseph Miller
University of Texas at Austin
-
Lin Mu
University of Georgia
-
Jason Murphy
Missouri University of Science and Technology
-
Andrea Nahmod
University of Massachusetts Amherst
-
Kenji Nakanishi
Research Institute for Mathematical Sciences, Kyoto University
-
Ludivine Oruba
Sorbonne Universite
-
José Palacios
Institut Denis Poisson, Université de Tours
-
Yulin Pan
University of Michigan, Ann Arbor
-
Jaemin Park
Universitat de Barcelona
-
Benoit Pausader
Brown University
-
Nataša Pavlović
University of Texas at Austin
-
Galina Perelman
LAMA
-
Samuel Punshon-Smith
Institute for Advanced Study
-
Oscar Riano
Florida International University
-
Tristan Robert
Université de Lorraine
-
Matthew Rosenzweig
Massachusetts Institute of Technology
-
Svetlana Roudenko
Florida International University
-
Frédéric Rousset
Département de Mathématiques d’Orsay
-
Diaraf Seck
University Cheikh Anta Diop of Dakar
-
Anastassiya Semenova
ICERM, Brown University
-
Chengyang Shao
Massachusetts Institute of Technology
-
Jalal Shatah
New York University
-
Gigliola Staffilani
Massachusetts Institute of Technology
-
Annalaura Stingo
University of California Davis
-
Walter Strauss
Brown University
-
Catherine Sulem
University of Toronto
-
Ruoci Sun
Karlsruhe Institute of Technology
-
changzhen Sun
University of Paris-Saclay
-
Chenmin Sun
CY Cergy-Paris Université
-
Mouhamadou Sy
Imperial College London
-
Tomoyuki Tanaka
Nagoya University
-
Maja Taskovic
Emory University
-
Minh-Binh Tran
Southern Methodist University
-
Nikolay Tzvetkov
University of Cergy-Pontoise
-
Tim Van Hoose
Missouri University of Science and Technology
-
Luis Vega
Basque Center for Applied Mathematics (BCAM)
-
Paolo Ventura
SISSA
-
Monica Visan
University of California, Los Angeles
-
Xuecheng Wang
Tsinghua University
-
Weinan Wang
University of Arizona
-
Billy Warner
University of Texas at Austin
-
Klaus Widmayer
EPFL, Switzerland
-
Lei Wu
Lehigh University
-
Zhuolun Yang
Brown University
-
Jiaqi Yang
ICERM
-
Xueying Yu
University of Washington
-
Lei Yu
Tongji University
-
Haitian Yue
University of Southern California
-
Zhiyuan Zhang
New York University
-
Guangqu Zheng
University of Edinburgh
-
SHIJUN ZHENG
Georgia Southern University
-
Hui Zhu
University of Michigan
Workshop Schedule
Monday, October 18, 2021
-
8:45 - 9:00 am EDTWelcome11th Floor Lecture Hall
- Brendan Hassett, ICERM/Brown University
-
9:00 - 9:50 am EDTMicrolocal analysis of singular measures11th Floor Lecture Hall
- Virtual Speaker
- Nicolas Burq, University Paris-Sud
-
10:00 - 10:30 am EDTCoffee Break11th Floor Collaborative Space
-
10:30 - 11:20 am EDTFull description of Benjamin-Feir instability of Stokes waves in deep water11th Floor Lecture Hall
- Virtual Speaker
- Alberto Maspero, Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Abstract
Small-amplitude, traveling, space periodic solutions -- called Stokes waves -- of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquet exponent, in full agreement with numerical simulations. This is a joint work with M. Berti and P. Ventura.
-
11:30 am - 1:00 pm EDTLunch/Free Time
-
1:00 - 1:50 pm EDTBreakdown of small amplitude breathers for the nonlinear Klein-Gordon equation11th Floor Lecture Hall
- Virtual Speaker
- Marcel Guardia, Universitat Politècnica de Catalunya
Abstract
Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arise from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study the nonlinear Klein-Gordon equation and show that small amplitude breathers cannot exist (under certain conditions). We also construct generalized breathers, these are solutions which are periodic in time and in space are localized up to exponentially small (with respect to the amplitude) tails. This is a joint work with O. Gomide, T. Seara and C. Zeng.
-
2:00 - 2:50 pm EDTFluctuations of \delta-moments for the free Schrödinger Equation11th Floor Lecture Hall
- Luis Vega, Basque Center for Applied Mathematics (BCAM)
Abstract
I will present recent work done with S. Kumar and F.Ponce-Vanegas.
We study the process of dispersion of low-regularity solutions to the free Schrödinger equation using fractional weights. We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound for the concentration of mass. We consider also the evolution when the initial datum is the Dirac comb in R. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits multifractality. -
3:00 - 4:30 pm EDTWelcome ReceptionReception - Hemenway's Patio
Tuesday, October 19, 2021
-
9:00 - 9:50 am EDTMathematical wave turbulence and propagation of chaos (I)11th Floor Lecture Hall
- Yu Deng, University of Southern California
Abstract
The theory of wave turbulence can be traced back to the 1920s and has played significant roles in many different areas of physics. However, for a long time the mathematical foundation of the theory has not been established. The central topics here are the wave kinetic equation, which describes the thermodynamic limit of interacting wave systems, and the propagation of chaos, which is a fundamental physical assumption in this field that lacks mathematical justification. In this first talk, I will present recent results with Zaher Hani (University of Michigan), where we provide the first rigorous derivation of the wave kinetic equation, and also justify the propagation of chaos assumption in the same setting. In part (II), we will discuss some important ideas in the proof.
-
10:00 - 10:30 am EDTCoffee Break11th Floor Collaborative Space
-
10:30 - 11:20 am EDTMathematical wave turbulence and propagation of chaos (II)11th Floor Lecture Hall
- Zaher Hani, University of Michigan
Abstract
The theory of wave turbulence can be traced back to the 1920s and has played significant roles in many different areas of physics. However, for a long time the mathematical foundation of the theory has not been established. The central topics here are the wave kinetic equation, which describes the thermodynamic limit of interacting wave systems, and the propagation of chaos, which is a fundamental physical assumption in this field that lacks mathematical justification. This talk is a continuation of that of Yu Deng (University of Southern California) who will present our recent joint results that provide the first rigorous derivation of the wave kinetic equation, and also justify the propagation of chaos assumption in the same setting. In this second part, we will discuss some important ideas in the proof.
-
11:30 am - 1:00 pm EDTLunch/Free Time
-
1:00 - 1:50 pm EDTEnergy transfer for solutions to the nonlinear Schrodinger equation on irrational tori.11th Floor Lecture Hall
- Gigliola Staffilani, Massachusetts Institute of Technology
Abstract
We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schr\"odinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.
-
2:00 - 2:50 pm EDTDeterminants, commuting flows, and recent progress on completely integrable systems11th Floor Lecture Hall
- Virtual Speaker
- Monica Visan, University of California, Los Angeles
Abstract
We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE. These include a priori bounds, the orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.
-
3:10 - 4:00 pm EDTLightning Talks11th Floor Lecture Hall
-
4:00 - 4:30 pm EDTCoffee Break11th Floor Collaborative Space
Wednesday, October 20, 2021
-
9:00 - 9:50 am EDTOn the derivation of the Kinetic Wave Equation in the inhomogeneous setting11th Floor Lecture Hall
- Virtual Speaker
- Charles Collot, Cergy-Paris Université
Abstract
The kinetic wave equation arises in weak wave turbulence theory. In this talk we are interested in its derivation as an effective equation from dispersive waves modelled with quadratic nonlinear Schrodinger equations. We focus on the space-inhomogeneous case, which had not been treated earlier. More precisely, we will consider such a dispersive equations in a weakly nonlinear regime, and for highly oscillatory random Gaussian fields with localised enveloppes as initial data. A conjecture in statistical physics is that there exists a kinetic time scale on which, statistically, the Wigner transform of the solution (a space dependent local Fourier energy spectrum) evolve according to the kinetic wave equation.
I will present a joint work with Ioakeim Ampatzoglou and Pierre Germain (Courant Institute) in which we approach the problem of the validity of this kinetic wave equation through the convergence and stability of the corresponding Dyson series. We are able to identify certain nonlinearities, dispersion relations, and regimes, and for which the convergence indeed holds almost up to the kinetic time (arbitrarily small polynomial loss). -
10:00 - 10:30 am EDTCoffee Break11th Floor Collaborative Space
-
10:30 - 11:20 am EDT3 Problems in Wave Turbulence11th Floor Lecture Hall
- Jalal Shatah, New York University
-
11:30 - 11:40 am EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall
-
11:40 am - 1:00 pm EDTLunch/Free Time
-
1:00 - 1:50 pm EDTConstructing global solutions for energy supercritical NLS equations11th Floor Lecture Hall
- Virtual Speaker
- Mouhamadou Sy, Imperial College London
Abstract
The last decades were very fruitful for the realm of dispersive PDEs. Besides several new deterministic developments in the study of the initial value problem and the behavior of solutions, probabilistic methods were introduced and made important progresses, particularly on bounded domain settings. Invariant measure are of considerable interest in these questions. However, in the context of energy supercritical equations, both the well-known Gibbs measures based strategy and the standard fluctuation-dissipation approach come across serious limitations. In this talk, we will present a new approach that combines the aforementioned ones to construct invariant measures, almost sure GWP, and strong controls on the time evolution of the solutions for the periodic NLS, with arbitrarily large power of nonlinearity and in any dimension. We will discuss the application to other contexts including non-dispersive PDEs.
-
2:00 - 2:30 pm EDTCoffee Break11th Floor Collaborative Space
-
2:30 - 3:20 pm EDTInvariant Gibbs measures for NLS and Hartree equations11th Floor Lecture Hall
- Haitian Yue, University of Southern California
Abstract
In this talk, I'll present our results about invariant Gibbs measures for the periodic nonlinear Schrödinger equation (NLS) in 2D, for any (defocusing and renormalized) odd power nonlinearity and for the periodic Hartree equation in 3D. The results are achieved by introducing a new method (we call the random averaging operators method) which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Andrea Nahmod (UMass Amherst).
Thursday, October 21, 2021
-
9:00 - 9:50 am EDTSingularities in the weak turbulence regime11th Floor Lecture Hall
- Virtual Speaker
- Anne-Sophie de Suzzoni, Ecole Polytechnique
Abstract
In this talk, we discuss the different regimes for the derivation of kinetic equations from the theory of weak turbulence for the quintic Schrödinger equation. In particular, we see that there exists a specific regime such that the correlations of the Fourier coefficients of the solution of the Schrödinger equation converge (in this regime) to a function that has an inifinite number of discontinuities.
-
10:00 - 10:30 am EDTCoffee Break11th Floor Collaborative Space
-
10:30 - 11:20 am EDTA New model for the stochasticly perturbed 2d navier-stokes equations11th Floor Lecture Hall
- Jonathan Mattingly, Duke University
Abstract
I will introduce a new model of the stochastically forced navier-stokes equation. The model will be targeted at studying the Equations forced by a large scale forcing. I give a number of properties of the model.
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11:30 am - 1:00 pm EDTLunch/Free Time
-
1:00 - 1:50 pm EDTGlobal well-posedness for the fractional NLS on the unit disk11th Floor Lecture Hall
- Xueying Yu, University of Washington
Abstract
In this talk, we discuss the cubic nonlinear Schr\"odinger equation with the fractional Laplacian on the unit disk. We show the global well-posedness for certain radial initial data below the energy space and establish a polynomial bound of the global solution. The result is proved by extending the I-method in the fractional nonlinear Schr\"odinger equation setting.
-
2:00 - 2:30 pm EDTCoffee Break11th Floor Collaborative Space
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2:30 - 3:20 pm EDTThe wave maps equation and Brownian paths11th Floor Lecture Hall
- Bjoern Bringmann, Institute for Advanced Study
Abstract
We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Riemannian manifold $\mathcal{M}$. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold $\mathcal{M}$ as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting involves analytic, geometric, and probabilistic aspects. This is joint work with J. Lührmann and G. Staffilani.
Friday, October 22, 2021
-
9:00 - 9:50 am EDTPositive Lyapunov exponents for the Galerkin-Navier-Stokes equations with stochastic forcing11th Floor Lecture Hall
- Jacob Bedrossian, University of Maryland
Abstract
In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.
-
10:00 - 10:30 am EDTCoffee Break11th Floor Collaborative Space
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10:30 - 11:20 am EDTA tale of two generalizations of Boltzmann equation11th Floor Lecture Hall
- Nataša Pavlovic, University of Texas at Austin
Abstract
In the first part of the talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. This part of the talk is based on the joint work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. In the second part of the talk we will discuss a rigorous derivation of a Boltzmann equation for mixtures of gases, which is a recent joint work with Ampatzoglou and Miller. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases.
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11:30 am - 12:20 pm EDTSome Recent Results On Wave Turbulence: Derivation, Analysis, Numerics and Physical Application11th Floor Lecture Hall
- Minh-Binh Tran, Southern Methodist University
Abstract
Wave turbulence describes the dynamics of both classical and non-classical nonlinear waves out of thermal equilibrium. In this talk, we will discuss some of our recent results on some aspects of wave turbulence, concerning the derivation and analysis of wave kinetic equations, some numerical algorithms and physical applications in Bose-Einstein Condensates.
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