Wednesday, September 8, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Thursday, September 9, 2021

• 9:00 - 10:30 am EDT

  Local well-posedness for dispersive equations

  Tutorial - 11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University
  • Benoit Pausader, Brown University

  Abstract

  This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.

  • Slides
• 11:00 am - 12:30 pm EDT

  Ergodicity of Markov processes: theory and computation

  Tutorial - 11th Floor Lecture Hall

  • Yao Li, University of Massachusetts Amherst

  Abstract

  In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).
• 1:30 - 2:00 pm EDT

  ICERM Welcome

  Welcome - 11th Floor Lecture Hall

  • Brendan Hassett, ICERM/Brown University
• 2:00 - 3:00 pm EDT

  Grad Student/ PostDoc Introductions

  Introductions - 11th Floor Lecture Hall

  • Yvonne Alama Bronsard, Sorbonne Université
  • Nicolas Camps, Université Paris Saclay
  • Patrick Flynn, Brown
  • Louise Gassot, Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay
  • Dean Katsaros, UMass amherst
  • Sudipta Kolay, ICERM
  • Kyle Liss, University of Maryland, College Park
  • Jaemin Park, Universitat de Barcelona
  • Nancy Scherich, University of Toronto
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, September 10, 2021

• 9:00 - 10:30 am EDT

  Local well-posedness for dispersive equations

  Tutorial - 11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University
  • Benoit Pausader, Brown University

  Abstract

  This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.

  • Slides
• 11:00 am - 12:30 pm EDT

  Ergodicity of Markov processes: theory and computation

  Tutorial - 11th Floor Lecture Hall

  • Yao Li, University of Massachusetts Amherst

  Abstract

  In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).
• 2:00 - 3:30 pm EDT

  Grad Student/ PostDoc Introductions

  Introductions - 11th Floor Lecture Hall

  • Bjoern Bringmann, Institute for Advanced Study
  • Stefan Czimek, Brown University (ICERM)
  • Daniel Eceizabarrena, University of Massachusetts Amherst
  • Eduardo Garcia-Juarez, Universitat de Barcelona
  • Claudia García, Universitat de Barcelona
  • Susanna Haziot, Brown University Mathematics
  • Anastassiya Semenova, ICERM, Brown University
  • Annalaura Stingo, University of California Davis
  • Jiaqi Yang, ICERM
  • Xueying Yu, University of Washington
  • Haitian Yue, University of Southern California
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Monday, September 13, 2021

• 9:00 - 10:30 am EDT

  Computer-assisted proofs in PDEs

  Tutorial - 11th Floor Lecture Hall

  • Javier Gomez Serrano, Princeton University

  Abstract

  In this minicourse we will present some recent results concerning computer-assisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).

  • Video
• 11:00 am - 12:30 pm EDT

  Ergodicity of Markov processes: theory and computation

  Tutorial - 11th Floor Lecture Hall

  • Yao Li, University of Massachusetts Amherst

  Abstract

  In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).
• 3:00 - 4:30 pm EDT

  Welcoming Reception

  Reception - Hemenway's patio

Tuesday, September 14, 2021

• 9:00 - 10:30 am EDT

  Local well-posedness for dispersive equations

  Tutorial - 11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University
  • Benoit Pausader, Brown University

  Abstract

  This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.
• 11:00 am - 12:30 pm EDT

  Computer-assisted proofs in PDEs

  Tutorial - 11th Floor Lecture Hall

  • Javier Gomez Serrano, Princeton University

  Abstract

  In this minicourse we will present some recent results concerning computer-assisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).

  • Video
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, September 15, 2021

• 9:00 - 10:30 am EDT

  Local well-posedness for dispersive equations

  Tutorial - 11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University
  • Benoit Pausader, Brown University

  Abstract

  This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.
• 11:00 am - 12:30 pm EDT

  Computer-assisted proofs in PDEs

  Tutorial - 11th Floor Lecture Hall

  • Javier Gomez Serrano, Princeton University

  Abstract

  In this minicourse we will present some recent results concerning computer-assisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 3:15 - 4:15 pm EDT

  Grads/Postdocs Meet with ICERM Directorate

  11th Floor Lecture Hall

  • Brendan Hassett, ICERM/Brown University
  • Misha Kilmer, Tufts University

Thursday, September 16, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, September 17, 2021

• 9:00 - 10:30 am EDT

  Local well-posedness for dispersive equations

  Tutorial - 11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University
  • Benoit Pausader, Brown University

  Abstract

  This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.
• 11:00 am - 12:30 pm EDT

  Computer-assisted proofs in PDEs

  Tutorial - 11th Floor Lecture Hall

  • Javier Gomez Serrano, Princeton University

  Abstract

  In this minicourse we will present some recent results concerning computer-assisted proofs in partial differential equations, starting from the basics of interval arithmetics. Particular emphasis will be put on the techniques, as opposed to the results themselves. There will be focus both on theory (lectures) and implementation (tutorial by Joel Dahne).
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Monday, September 20, 2021

• 9:50 - 10:00 am EDT

  Welcome

  11th Floor Lecture Hall

  • Brendan Hassett, ICERM/Brown University
• 10:00 - 10:40 am EDT

  Quasilinear Diffusion of magnetized fast electrons in a mean field of quasi-particle waves packets

  11th Floor Lecture Hall

  • Speaker
  • Irene Gamba, University of Texas at Austin

  • Session Chair
  • Luis Vega, Basque Center for Applied Mathematics (BCAM)

  Abstract

  Quasi-linear diffusion of magnetized fast electrons in momentum space results from stimulated emission and absorption of waves packets via wave-particle resonances. Such model consists in solving the dynamics of a system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics waves (quasi-particles) in a quantum process of their resonant interaction. Such description results in a Mean Field model where diffusion coefficients are determined by the local spectral energy density of excited waves whose perturbations depend on flux averages of the electron pdf.
  We will discuss the model and a mean field iteration scheme that simulates the dynamics of the space average model, where the energy spectrum of the excited wave time dynamics is calculated with a coefficient that depends on the electron pdf flux at a previous time step; while the time dynamics of the quasilinear model for the electron pdf is calculated by the spectral average of the quasi-particle wave under a classical resonant condition where the plasma wave frequencies couples the spectral energy to the momentum variable of the electron pdf. Recent numerical simulations will be presented showing a strong hot tail anisotropy formation and stabilization for the iteration in a 3 dimensional cylindrical model.
  This is work in collaboration with Kun Huang, Michael Abdelmalik at UT Austin.

  • Video
• 10:55 - 11:15 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 11:15 am - 12:00 pm EDT

  Modeling inviscid water waves

  11th Floor Lecture Hall

  • Virtual Speaker
  • Christophe Lacave, Universite Grenoble Alpes

  • Session Chair
  • Luis Vega, Basque Center for Applied Mathematics (BCAM)

  Abstract

  We consider numerical strategies to handle two-dimensional water waves in a fully non-linear regime. The free-surface is discretized via lagrangian tracers and the numerical strategy is constructed carefully to include desingularizations, but no artificial regularizations. We approach the formation of singularities in the wave breaking problem and also model solitary waves and the effect of an abruptly changing bottom. We present a rigorous analysis of the singular kernel operators involved in these methods.

  • Video
• 12:15 - 1:15 pm EDT

  Lunch/Free Time
• 1:15 - 1:55 pm EDT

  Anomalous conduction in one dimensional chains: a wave turbulence approach.

  11th Floor Lecture Hall

  • Miguel Onorato, Università di Torino

  Abstract

  Heat conduction in 3D macroscopic solids is in general well described by the Fourier's law. However, low dimensional systems, like for example nanotubes, may be characterized by a conductivity that is size-dependent. This phenomena, known as anomalous conduction, has been widely studied in one dimensional chains like FPUT, mostly using deterministic simulations of the microscopic model. Here, I will present a mesoscopic approach based on the wave turbulence theory and give the evidence, through extensive numerical simulations and theoretical arguments, that the anomalous conduction is the result of the presence of long waves that rapidly propagate from one thermostat to the other without interacting with other modes. I will also show that the scaling of the conductivity with the length of the chain obtained from the mesoscopic approach is consistent with the one obtained from microscopic simulations.

  • Video
• 2:10 - 2:50 pm EDT

  On the asymptotic stability of shear flows and vortices

  11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University

  Abstract

  I will talk about some recent work on the global linear and nonlinear asymptotic stability of two families of solutions of the 2D Euler equations: shear flows on bounded channels and vortices in the plane. This is joint work with Hao Jia.

  • Video
• 3:00 - 4:30 pm EDT

  Reception

  Hemenway's Patio (weather permitting)

Tuesday, September 21, 2021

• 10:00 - 10:40 am EDT

  Small scale formations in the incompressible porous media equation

  11th Floor Conference Room

  • Virtual Speaker
  • Yao Yao, Georgia Tech

  Abstract

  The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finite-time blow-up remains open for smooth initial data, although numerical evidence suggests that small scale formation can happen as time goes to infinity. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infinite-in-time growth of Sobolev norms, provided that they remain globally smooth in time. As an application, this allows us to obtain nonlinear instability of certain stratified steady states of IPM. This is a joint work with Alexander Kiselev.

  • Video
• 10:55 - 11:15 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 11:15 am - 12:00 pm EDT

  PINNs and DeepOnets for Wave Systems

  11th Floor Lecture Hall

  • Virtual Speaker
  • George Karniadakis, Brown University

  • Video
• 12:15 - 1:45 pm EDT

  Lunch/Free Time
• 1:45 - 2:25 pm EDT

  The second boundary value problem for a discrete Monge-Ampere equation

  11th Floor Conference Room

  • Gerard Awanou, University of Illinois, Chicago

  Abstract

  In this work we propose a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport. It is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

  • Video
• 2:40 - 3:45 pm EDT

  Lightning Talks

  11th Floor Lecture Hall
• 3:45 - 4:15 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, September 22, 2021

• 10:00 - 10:40 am EDT

  Efficient and accurate structure preserving schemes for complex nonlinear systems

  11th Floor Conference Room

  • Jie Shen, Purdue University

  Abstract

  Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme. I will present some recent advances on using the scalar auxiliary variable (SAV) approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.

  • Video
• 10:55 - 11:15 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 11:15 am - 12:00 pm EDT

  Energy growth for the Schrödinger map and the binormal flow

  11th Floor Lecture Hall

  • Valeria Banica, Sorbonne Université

  Abstract

  In this talk I shall present a result of blow up of a density energy associated to the Schrödinger map and the binormal flow, a classical model for the dynamics of vortex filaments in Euler equations. This is a joint work with Luis Vega.

  • Video
• 12:15 - 1:45 pm EDT

  Lunch/Free Time
• 1:45 - 2:25 pm EDT

  Water Waves with Background Flow over Obstacles and Topography

  11th Floor Lecture Hall

  • Virtual Speaker
  • Jon Wilkening, University of California, Berkeley

  Abstract

  We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface. We also propose a new algorithm to dynamically vary the spacing of gridpoints on the free surface to efficiently resolve regions of high curvature as they develop. We study singularity formation and capillary effects and compare our numerical results with lab experiments.

  • Video
• 2:45 - 3:15 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 3:15 - 3:55 pm EDT

  Topological Origin of Certain Fluid and Plasma Waves

  11th Floor Lecture Hall

  • Brad Marston, Brown University

  Abstract

  Symmetries and topology play central roles in our understanding of physical systems. Topology, for instance, explains the precise quantization of the Hall effect and the protection of surface states in topological insulators against scattering from disorder or bumps. However discrete symmetries and topology have so far played little role in thinking about the fluid dynamics of oceans and atmospheres. In this talk I show that, as a consequence of the rotation of the Earth that breaks time reversal symmetry, equatorially trapped Kelvin and Yanai waves emerge as topologically protected edge modes. The non-trivial structure of the bulk Poincare ́ waves encoded through the first Chern number of value 2 guarantees the existence of these waves. Thus the oceans and atmosphere of Earth naturally share basic physics with topological insulators. As equatorially trapped Kelvin waves in the Pacific ocean are an important component of El Niño Southern Oscillation and other climate oscillations, these new results demonstrate that topology plays a surprising role in Earth’s climate system. We also predict that waves of topological origin will arise in magnetized plasmas. A planned experiment at UCLA’s Basic Plasma Science Facility to look for the waves is described.

  • Video

Thursday, September 23, 2021

• 10:00 - 10:40 am EDT

  Resonances as a Computational Tool

  11th Floor Lecture Hall

  • Katharina Schratz, Heriot-Watt University
• 10:55 - 11:15 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 11:15 am - 12:00 pm EDT

  Efficient time-stepping methods for the rotating shallow water equations

  11th Floor Lecture Hall

  • Virtual Speaker
  • Thi- Thao Phuong- Hoang, Auburn University

  Abstract

  Numerical modeling of geophysical flows is challenging due to the presence of various coupled processes that occur at different spatial and temporal scales. It is critical for the numerical schemes to capture such a wide range of scales in both space and time to produce accurate and robust simulations over long time horizons.
  In this talk, we will discuss efficient time-stepping methods for the rotating shallow water equations discretized on spatial meshes with variable resolutions. Two different approaches will be considered: the first approach is a fully explicit local time-stepping algorithm based on the strong stability preserving Runge-Kutta schemes, which allows different time step sizes in different regions of the computational domain. The second approach, namely the localized exponential time differencing method, is based on spatial domain decomposition and exponential time integrators, which makes possible the use of much larger time step sizes compared to explicit schemes and avoids solving nonlinear systems. Numerical results on various test cases will be presented to demonstrate the performance of the proposed methods.

  • Video
• 12:15 - 1:45 pm EDT

  Lunch/Free Time
• 1:45 - 2:25 pm EDT

  Dynamics in particle suspension flow

  11th Floor Conference Room

  • Li Wang, University of Minnesota

  Abstract

  In this talk, I will consider two set up of particle suspension flow. One is a gravity driven flow down an incline, and the other is a pressure driven flow in a Hele-Shaw cell. In the former case, the interesting phenomena is the formation of singular shock that appears in the high particle concentration case that relates to the particle-rich ridge observed in the experiments. We analyze the formation of the singular shock as well as its local structure. In the latter case, we rationalize a self-similar accumulation of particles at the interface between suspension and air. Our results demonstrate that the combination of the shear- induced migration, the advancing fluid-fluid interface, and Taylor dispersion yield the self-similar and gradual accumulation of particles.

  • Video
• 2:45 - 3:15 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 3:15 - 3:55 pm EDT

  Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

  11th Floor Conference Room

  • Debbie Eeltink, MIT

  Abstract

  The nonlinear Schrödinger equation (NLSE) is a workhorse for many different fields (e.g. optical fibers, Bose-Einstein condensates, water waves). It describes the evolution of the envelope of a field in time or space, taking into account the nonlinear interaction of the components of the spectrum of the envelope. While the NLSE is well-studied in its conservative form, a relevant question to ask is how does it respond to damping and forcing? Limiting the spectrum to only three components allows one to construct a phase-space for the NLSE, spanned by the relative phase of the sidebands, and the energy fraction in the sidebands. Using wave-tank measurements, we show that forcing and damping the NLSE induces separatrix crossing: switching from one solution-type to the other in the phase-space. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We, therefore, extend our three-wave analysis to this system. However, our conclusions are general as the dynamics are driven by the leading order terms. To our knowledge, it is the first phase evolution extraction from water-wave measurements. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.

Friday, September 24, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

• 10:00 - 10:40 am EDT

  Extreme Wave Events in Reflective Environments

  11th Floor Lecture Hall

  • Amin Chabchoub, University of Sydney

  Abstract

  Waves dynamics in coastal zones is known to comprise incident and reflective wave motion. We report an experimental study in which several incident JONSWAP wave trains have been generated in a uni-directional water wave tank while the artificial beach inclination and its permeability have been varied to allow a variety of reflective wave conditions. Key statistical features obtained from an adaptive coupled nonlinear Schrödinger model simulations show an excellent agreement with the laboratory data collected near the beach.

  • Video
• 10:55 - 11:15 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 11:15 - 11:55 am EDT

  Symmetry in stationary and uniformly rotating solutions of the Euler equations

  11th Floor Lecture Hall

  • Javier Gomez Serrano, Princeton University

  • Video
• 12:10 - 12:50 pm EDT

  High order strong stability preserving multi-derivative implicit and IMEX Runge--Kutta methods with asymptotic preserving properties

  11th Floor Lecture Hall

  • Sigal Gottlieb, University of Massachusetts Dartmouth

  Abstract

  In this talk we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$, and IMEX Runge--Kutta schemes of order up to $p=3$. For the multi-derivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation.

  • Video

Monday, September 27, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Tuesday, September 28, 2021

• 10:00 - 11:00 am EDT

  RIEMANN’S NON-DIFFERENTIABLE FUNCTION AND THE BINORMAL CURVATURE FLOW (Joint work with Valeria Banica)

  11th Floor Lecture Hall

  • Luis Vega, Basque Center for Applied Mathematics (BCAM)

  Abstract

  We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious non- linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.
• 11:30 am - 12:30 pm EDT

  ALMOST-GLOBAL WELL-POSEDNESS FOR 2D STRONGLY-COUPLED WAVE-KLEIN-GORDON SYSTEMS

  11th Floor Lecture Hall

  • Annalaura Stingo, University of California Davis

  Abstract

  (Joint with Mihaela Ifrim) In this talk we discuss the almost-global well-posedness of a wide class of coupled Wave-Klein-Gordon equations in 2+1 space-time dimensions, when initial data are small and localized. The Wave-Klein-Gordon systems arise from several physical models especially related to General Relativity but few results are known at present in lower space-time dimensions. Compared with prior related results, we here consider a strong quadratic quasilinear coupling between the wave and the Klein-Gordon equation and no restriction is made on the support of the initial data which only have a mild decay at infinity and very limited regularity.
• 2:30 - 3:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 3:00 - 4:00 pm EDT

  Semilinear Dispersive Equations

  11th Floor Lecture Hall

  • Alexandru Ionescu, Princeton University
  • Benoit Pausader, Brown University

Wednesday, September 29, 2021

• 11:30 am - 12:15 pm EDT

  Gender Diversity and the Mathematical Community

  11th Floor Lecture Hall

  • Andrea Nahmod, University of Massachusetts Amherst
  • Gigliola Staffilani, Massachusetts Institute of Technology

  Abstract

  Followed by a “brown-bag lunch” (provided by ICERM) to be taken outside, where the discussion is expected to continue among smaller groups. RSVP required.
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Thursday, September 30, 2021

• 11:00 am - 12:30 pm EDT

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall

  • Daniel Eceizabarrena, University of Massachusetts Amherst
  • Claudia García, Universitat de Barcelona
  • Haitian Yue, University of Southern California
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, October 1, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Monday, October 4, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Tuesday, October 5, 2021

• 10:00 - 11:00 am EDT

  The characteristic gluing problem of general relativity

  11th Floor Lecture Hall

  • Stefan Czimek, Brown University (ICERM)

  Abstract

  In this talk we introduce and solve the characteristic gluing problem for the Einstein vacuum equations. We show that obstructions to characteristic gluing come from an infinite-dimensional space of conservation laws along null hypersurfaces for the linearized equations at Minkowski. We prove that this space splits into an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges. We identify the 10 gauge-invariant charges to be related to the energy, linear momentum, angular momentum and center-of-mass of the spacetime. Based on this identification, we explain how to characteristically glue a given spacetime to a suitably chosen Kerr spacetime. As corollary we get an alternative proof of the Corvino-Schoen spacelike gluing to Kerr. Moreover, we apply our characteristic gluing method to localise characteristic initial data along null hypersurfaces. In particular, this yields a new proof of the Carlotto-Schoen spacelike localization where our method yields no loss of decay, thus resolving an open problem. We also outline further applications. This is joint work with S. Aretakis (Toronto) and I. Rodnianski (Princeton).
• 11:30 am - 12:30 pm EDT

  The role of hyperbolicity in deciding uniqueness for minimizers of an energy with linear growth

  11th Floor Lecture Hall

  • Gilles Francfort, Universite Paris 13
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, October 6, 2021

• 9:00 - 10:00 am EDT

  Professional Development: Ethics I

  Professional Development - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Thursday, October 7, 2021

• 11:00 am - 12:30 pm EDT

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 2:00 - 4:00 pm EDT

  Probabilistic well-posedness for nonlinear Schrödinger equation (I)

  11th Floor Lecture Hall

  • Haitian Yue, University of Southern California

  Abstract

  In this mini-course, we will introduce the probabilistic well-posedness theory in the background of the nonlinear Schrödinger equation and in particular will focus on local (in time) dynamics with the random initial data. The following topics will be covered: 1) the basic settings of random data Cauchy theory; 2) Bourgain's re-centering method; 3) the random averaging operator method.
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, October 8, 2021

• 1:30 - 3:00 pm EDT

  Probabilistic well-posedness for nonlinear Schrödinger equation (I)

  11th Floor Lecture Hall

  • Haitian Yue, University of Southern California

  Abstract

  In this mini-course, we will introduce the probabilistic well-posedness theory in the background of the nonlinear Schrödinger equation and in particular will focus on local (in time) dynamics with the random initial data. The following topics will be covered: 1) the basic settings of random data Cauchy theory; 2) Bourgain's re-centering method; 3) the random averaging operator method.
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Tuesday, October 12, 2021

• 10:00 - 11:00 am EDT

  Superharmonic Instability of Stokes Waves

  11th Floor Lecture Hall

  • Anastassiya Semenova, ICERM, Brown University

  Abstract

  We consider the classical problem of water waves on the surface of an ideal fluid in 2D. This work offers an investigation of dynamics and stability of nonlinear waves. We provide new insight into the stability of the Stokes waves by identifying previously inaccessible branches of instability in the equations of motion for fluid. The eigenvalues of the linearized problem that become unstable follow a selfsimilar law as they approach instability threshold, and a power law is suggested for unstable eigenvalues in the immediate vicinity of the limiting wave. Future direction of work is to study superharmonic instability of Stokes waves in finite depth.
• 11:30 am - 12:30 pm EDT

  Revisit singularity formation for the inviscid primitive equation

  11th Floor Lecture Hall

  • Slim IBRAHIM, University of Victoria

  Abstract

  The primitive equation is an important model for large scale fluid model including oceans and atmosphere. While solutions to the viscous model enjoy global regularity, inviscid solutions may develop singularities in finite time. In this talk, I will review the methods to show blowup, and share more recent progress on qualitative properties of the singularity formation.
• 2:00 - 4:30 pm EDT

  Probabilistic well-posedness for nonlinear Schrödinger equation (II)

  11th Floor Lecture Hall

  • Yu Deng, University of Southern California
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, October 13, 2021

• 9:00 - 10:00 am EDT

  Professional Development: Ethics II

  Professional Development - 11th Floor Lecture Hall
• 2:00 - 4:30 pm EDT

  Probabilistic well-posedness for nonlinear Schrödinger equation (II)

  11th Floor Lecture Hall

  • Yu Deng, University of Southern California
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Thursday, October 14, 2021

• 11:00 am - 12:30 pm EDT

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, October 15, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Monday, October 18, 2021

• 8:45 - 9:00 am EDT

  Welcome

  11th Floor Lecture Hall

  • Brendan Hassett, ICERM/Brown University
• 9:00 - 9:50 am EDT

  Microlocal analysis of singular measures

  11th Floor Lecture Hall

  • Virtual Speaker
  • Nicolas Burq, University Paris-Sud

  • Video
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:20 am EDT

  Full description of Benjamin-Feir instability of Stokes waves in deep water

  11th 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.

  • Video
  • Slides
• 11:30 am - 1:00 pm EDT

  Lunch/Free Time
• 1:00 - 1:50 pm EDT

  Breakdown of small amplitude breathers for the nonlinear Klein-Gordon equation

  11th 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.

  • Video
  • Slides
• 2:00 - 2:50 pm EDT

  Fluctuations of \delta-moments for the free Schrödinger Equation

  11th 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.

  • Video
  • Slides
• 3:00 - 4:30 pm EDT

  Welcome Reception

  Reception - Hemenway's Patio

Tuesday, October 19, 2021

• 9:00 - 9:50 am EDT

  Mathematical 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 EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:20 am EDT

  Mathematical 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 EDT

  Lunch/Free Time
• 1:00 - 1:50 pm EDT

  Energy 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.

  • Video
• 2:00 - 2:50 pm EDT

  Determinants, commuting flows, and recent progress on completely integrable systems

  11th 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 EDT

  Lightning Talks

  11th Floor Lecture Hall
• 4:00 - 4:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, October 20, 2021

• 9:00 - 9:50 am EDT

  On the derivation of the Kinetic Wave Equation in the inhomogeneous setting

  11th 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).

  • Video
  • Slides
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:20 am EDT

  3 Problems in Wave Turbulence

  11th Floor Lecture Hall

  • Jalal Shatah, New York University
• 11:30 - 11:40 am EDT

  Group Photo (Immediately After Talk)

  11th Floor Lecture Hall
• 11:40 am - 1:00 pm EDT

  Lunch/Free Time
• 1:00 - 1:50 pm EDT

  Constructing global solutions for energy supercritical NLS equations

  11th 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.

  • Video
• 2:00 - 2:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 2:30 - 3:20 pm EDT

  Invariant Gibbs measures for NLS and Hartree equations

  11th 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).

  • Video
  • Slides

Thursday, October 21, 2021

• 9:00 - 9:50 am EDT

  Singularities in the weak turbulence regime

  11th 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.

  • Video
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:20 am EDT

  A New model for the stochasticly perturbed 2d navier-stokes equations

  11th 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.

  • Video
• 11:30 am - 1:00 pm EDT

  Lunch/Free Time
• 1:00 - 1:50 pm EDT

  Global well-posedness for the fractional NLS on the unit disk

  11th 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.

  • Video
• 2:00 - 2:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 2:30 - 3:20 pm EDT

  The wave maps equation and Brownian paths

  11th 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.

  • Video

Friday, October 22, 2021

• 9:00 - 9:50 am EDT

  Positive Lyapunov exponents for the Galerkin-Navier-Stokes equations with stochastic forcing

  11th 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.

  • Video
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:20 am EDT

  A tale of two generalizations of Boltzmann equation

  11th 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.

  • Video Available Soon
  • Slides
• 11:30 am - 12:20 pm EDT

  Some Recent Results On Wave Turbulence: Derivation, Analysis, Numerics and Physical Application

  11th 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.

  • Video Available Soon
  • Slides

Monday, October 25, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Tuesday, October 26, 2021

• 10:00 - 11:00 am EDT

  Weak universality for the fractional \Phi_2^4 under the wave dynamics

  11th Floor Lecture Hall

  • Chenmin Sun, CY Cergy-Paris Université

  Abstract

  Parabolic \Phi^4 models, given by singular stochastic heat equation are believed to be the scaling limit for some physical relevant models. This motivated many recent results in the field of singular SPDE. In the context of dispersive equations, it is also of interest to investigate the limit of certain dispersive \Phi^4 models perturbed by higher-order potentials with correct scalings. In this talk, we consider the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional \Phi_2^4, we first establish a sufficient and almost necessary criteria for the convergence of invariant measures to the fractional \Phi_2^4. Then we prove the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. The main difficulty is that we do not have a good local Cauchy-theory for the highly supercritical nonlinearities. To prove the dynamical convergence, we rely on probabilistic ideas exploiting independence of different scales of frequencies. This is a joint work with Nikolay Tzvetkov and Weijun Xu.
• 11:30 am - 12:30 pm EDT

  Global axisymmetric Euler flows with rotation

  11th Floor Lecture Hall

  • Klaus Widmayer, EPFL, Switzerland

  Abstract

  We discuss the construction of a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform "rigid body" rotation. These solutions are axisymmetric, of Sobolev regularity and have non-vanishing swirl. At the heart of this result is the dispersive effect due to rotation, which is captured in our new "method of partial symmetries". This approach is adapted to maximally exploit the symmetries of this anisotropic problem, both for the linear and nonlinear analysis, and allows to globally propagate sharp decay estimates. This is joint work with Y. Guo and B. Pausader.
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, October 27, 2021

• 9:00 - 10:00 am EDT

  Professional Development: Job Applications in Academia

  Professional Development - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Thursday, October 28, 2021

• 11:00 am - 12:30 pm EDT

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, October 29, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Monday, November 1, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Tuesday, November 2, 2021

• 10:00 - 11:00 am EDT

  Global solutions of aggregation equations and other flows with random diffusion

  11th Floor Lecture Hall

  • Matthew Rosenzweig, Massachusetts Institute of Technology

  Abstract

  Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. In this talk, we investigate whether random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability. This talk is based on joint work with Gigliola Staffilani.
• 11:15 am - 12:15 pm EDT

  Numerical computations of invariant probability measures and their ergodicity

  11th Floor Lecture Hall

  • Yao Li, University of Massachusetts Amherst

  Abstract

  Consider a stochastic process (such as a stochastic differential equation) that admits an invariant probability measure. We are interested in many things like the landscape, the sensitivity, and the speed of convergence of the invariant probability measure. Since it is difficult to obtain sharp estimates rigorously, some numerical computations are usually necessary. In this talk I will introduce a few novel data-driven computational methods, including both classical and deep learning approaches, to solve these problems for a class of stochastic processes. Generally those data-driven methods are less affected by the curse-of-dimensionality than classical grid-based methods. I will demonstrate a few high (up to 100) dimensional examples in my talk.
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Wednesday, November 3, 2021

• 9:00 - 10:00 am EDT

  Professional Development: Hiring Process

  Professional Development - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Thursday, November 4, 2021

• 11:00 am - 12:30 pm EDT

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Friday, November 5, 2021

• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

Monday, November 8, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Tuesday, November 9, 2021

• 10:00 - 11:00 am EST

  Large-time dynamics of NLS in 3D with random data

  11th Floor Lecture Hall

  • Nicolas Camps, Université Paris Saclay

  Abstract

  In this talk, we investigate the large time behavior of a probabilistic flow, solution to the defocusing cubic Schr ̈odinger equation in the Euclidean space of dimension 3. First, we consider the stability of small ground states which emerge from the presence of an external short-range potential. By performing a “critical-weighted” strategy, we prove that these ground states are asymp- totically stable below the energy space. At supercritical regularity, this result holds true almost-surely, when the data are randomized according to the Wiener decomposition. We then address the scattering issue, outside the small data regime. In order to do so, we propose a way to revisit in a probabilistic setting the I-method with a Morawetz bootstrap.
• 11:15 am - 12:15 pm EST

  Zero-dispersion limit for the Benjamin-Ono equation on the torus

  11th Floor Lecture Hall

  • Louise Gassot, Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay

  Abstract

  We discuss the zero-dispersion limit for the Benjamin-Ono equation on the torus given a single well initial data. We prove that there exist approximate initial data converging to the initial data, such that the corresponding solutions admit a weak limit as the dispersion parameter tends to zero. The weak limit is expressed in terms of the multivalued solution of the inviscid Burgers equation obtained by the method of characteristics. We construct our approximation by using the Birkhoff coordinates of the initial data, introduced by Gérard, Kappeler and Topalov. In the case of the cosine initial data, we completely justify this approximation by proving an asymptotic expansion of the Birkhoff coordinates.
• 1:00 - 5:00 pm EST

  Reserved for Maintenance

  11th Floor Lecture Hall
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Wednesday, November 10, 2021

• 9:00 - 10:00 am EST

  Professional Development: Papers and Journals

  Professional Development - 11th Floor Lecture Hall
• 11:00 am - 12:30 pm EST

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 12:30 - 12:45 pm EST

  Postdoc/ Grad Student Group Photo

  Group Photo (Immediately After Talk) - 11th Floor Lecture Hall
• 1:00 - 5:00 pm EST

  Reserved for Maintenance

  11th Floor Lecture Hall
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Thursday, November 11, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Friday, November 12, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Monday, November 15, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Tuesday, November 16, 2021

• 9:30 - 10:20 am EST

  Persistence of Invariant Manifolds under Singular Perturbations

  11th Floor Lecture Hall

  • Jiaqi Yang, ICERM

  Abstract

  We study persistence of the invariant manifolds of systems under delay-related perturbations. The perturbations are very singular no matter how small they are. Our treatment bypasses the discussion of the phase space and evolution of the perturbed systems, which are known to be debatable. More precisely, we take advantage of the parameterization method, and solve equations of functions (invariance equations) with analysis. Our results are in a-posteriori format. The constructive proofs rely on the fixed point approach. Therefore, our results provide algorithms and are suitable for computer-assisted proofs.
• 10:30 - 11:20 am EST

  Hamiltonian Dysthe equation for 3d deep-water gravity waves.

  11th Floor Lecture Hall

  • Virtual Speaker
  • Adilbek Kairzhan, University of Toronto

  Abstract

  In this talk, we consider the water wave problem in a three-dimensional domain of infinite depth and examine the modulational regime for weakly nonlinear wavetrains. Using the method of normal form transformations near the equilibrium state, we discuss how to derive the Hamiltonian Dysthe equation describing the slow evolution of the wave envelope. We also discuss how a precise calculation of the third-order normal form allows for a refined reconstruction of the free surface.
  In conclusion, we show the comparison between our approximation, direct numerical simulations of the three-dimensional Euler system and predictions from the classical Dysthe equation.
• 11:30 am - 12:20 pm EST

  Thermal convection in a penny shape cylinder: from idealized numerical models to tropical cyclones

  11th Floor Lecture Hall

  • Ludivine Oruba, Sorbonne Universite

  Abstract

  We investigate atmospherical vortices using idealized mathematical models and direct numerical simulations. We establish connection both with geophysical observations and asymptotic developments.
  We consider rotating convection in an elongated cylindrical domain, which intends to model the lower atmosphere. We examine the conditions under which the main vortex develops an eye at its core, which is reminiscent of that seen in a tropical cyclone ; that is, a region where the poloidal flow reverses and the angular momentum is low.
  We first focus on the stationary flow and highlight the key role played by the viscous bottom boundary layer in the eye formation. We show that the eye results from a non-linear instability.
  In a mixed numerical / asymptotic approach, we then study the linear inertial wave activity which develops in the time dependent problem. These inertial waves are indeed reminiscent of oscillations which have been observed near the eye of actual tropical cyclones.
  References:
  L Oruba, P.A. Davidson and E. Dormy, J. Fluid Mech. (2017), 812, 890-904.
  L Oruba, A. M. Soward and E. Dormy, J. Fluid Mech. (2020), 888, A9.
  L Oruba, A. M. Soward and E. Dormy, J. Fluid Mech. (2021), 915, A53.
  
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Wednesday, November 17, 2021

• 9:00 - 10:00 am EST

  Professional Development: Grant Proposals

  Professional Development - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Thursday, November 18, 2021

• 11:00 am - 12:30 pm EST

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Friday, November 19, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Monday, November 22, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Tuesday, November 23, 2021

• 9:30 - 10:20 am EST

  Energies and phase space for the 1-d Gross-Pitaevskii equation

  11th Floor Lecture Hall

  • Virtual Speaker
  • Herbert Koch, University of Bonn

  Abstract

  The Gross-Pitaevskii equation is essentially the defocusing cubic nonlinear Schroedinger equation, but with energies which enforce that the absolute value of solutions tends to 1 as x tends to +/- infinity. Striking solutions are black and dark solitons. I will explain nonlinear phase spaces for solutions which contain the black and dark solitons, explain a new metric on it and its smooth and topological structure. I will conclude with the construction of a continuous family of conserved energies. This is joint work with Xian Liao
• 10:30 - 11:20 am EST

  Phase transitions of the focusing Φ^p_1 measures

  11th Floor Lecture Hall

  • Virtual Speaker
  • Leonardo Tolomeo, Universität Bon

  Abstract

  We study the behaviour of the focusing Φ^p_1 measures on the one-dimensional torus, initiated by Lebowitz, Rose, and Speer (1988). Because of the focusing nature of the measure, it is necessary to introduce a mass cutoff K, and restrict the measure to the set where the mass is smaller than K. We will show the following phase transitions: - When K is smaller than a certain threshold, the measure is well defined, while for K bigger than the threshold, the measure becomes non-normalisable. We will also discuss what happens at the optimal threshold, and show that the measure is well defined in this case as well, solving a long-standing open problem. This is joint work with T. Oh (University of Edinburgh) and P. Sosoe (Cornell University).
  - When the size L of the torus is going to infinity, in the weakly non-linear regime, numeric simulations in the original paper by Lebowitz, Rose, and Speer (1988) suggest a phase transition depending on the temperature of the system. We show that this is indeed the case: if the temperature is high, the Φ^p_1 measure converges to a given gaussian measure. However, this convergence does not happen in the low-temperature regime, and instead the measure progressively concentrates around a single soliton. This is joint work with H. Weber (University of Bath).
• 11:30 am - 12:20 pm EST

  The regularity of solutions to the Muskat equation with turnover points.

  11th Floor Lecture Hall

  • Jia Shi, Princeton University

  Abstract

  The Muskat equation describes the interface of two liquids in porous media. When the heavier liquid lies under the lighter one, the equation behaves like a parabolic equation. When the heavier one is above the lighter one, it looks like a backward parabolic equation. However, the behavior has been proved to be local in time. There exist solutions that start from a graph and develop a turnover point as the time is going. I will discuss the regularity of the solutions to the Muskat problems with turnover points and show that they are analytic except at the turnover points, given that the solutions are smooth enough.
• 12:20 - 12:30 pm EST

  Semester Program Group Photo

  Group Photo (Immediately After Talk) - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Monday, November 29, 2021

• 11:00 am - 12:30 pm EST

  Post Doc/Graduate Student Seminar

  Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Tuesday, November 30, 2021

• 9:30 - 10:20 am EST

  Asymptotic stability of the sine-Gordon kink under odd perturbations.

  11th Floor Lecture Hall

  • Jonas Luhrmann, Texas A&M University

  Abstract

  The sine-Gordon model is a classical nonlinear scalar field theory that was discovered in the 1860s in the context of the study of surfaces with constant negative curvature. Its equation of motion features soliton solutions called kinks and breathers, which play an important role for the long-time dynamics. I will begin the talk with an introduction to classical 1D scalar field theories and the asymptotic stability problem for kinks. After surveying recent progress on the problem, I will present a joint work with W. Schlag on the asymptotic stability of the sine-Gordon kink under odd perturbations. Our proof is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key aspects are a super-symmetric factorization property of the linearized operator and a remarkable non-resonance property of a variable coefficient quadratic nonlinearity.
• 10:30 - 11:20 am EST

  Gagliardo-Nirenberg-Sobolev inequality in Dirichlet spaces.

  11th Floor Lecture Hall

  • Patricia Alonso Ruiz, Texas A&M University

  Abstract

  The Gagliardo-Nirenberg-Sobolev inequality in R^n is a classical Sobolev embedding with many applications in the theory of PDEs and calculus of variations. In this talk, I will present the full scale of Gagliardo-Nirenberg-Sobolev inequalities in the more general framework of Dirichlet spaces with (sub-)Gaussian heat kernel estimates. In particular, we will discover that the optimal exponent in the embedding not only depends on the Hausdorff dimension of the underlying space, but also on other invariants. To this end, I will use a recent approach to (1,p)-Sobolev spaces via heat semigroups inspired by ideas going back to work of de Giorgi and Ledoux. If time permits, I will outline some results and conjectures concerning further Sobolev embeddings in this setting. The talk is based on joint work with F. Baudoin.
• 11:30 am - 12:20 pm EST

  Birkhoff normal forms for Hamiltonian PDEs in their energy space

  11th Floor Lecture Hall

  • Benoît Grébert, University of Nantes

  Abstract

  We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, only modes with the same linear frequency will be able to exchange energy in a reasonable time. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension $d=1$ and nonlinear Schrödinger equations in dimension $d\leq 2$.
• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Wednesday, December 1, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Thursday, December 2, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Friday, December 3, 2021

• 3:00 - 3:30 pm EST

  Coffee Break

  11th Floor Collaborative Space

Monday, December 6, 2021

• 8:55 - 9:00 am EST

  Welcome

  11th Floor Lecture Hall

  • Brendan Hassett, ICERM/Brown University
• 9:00 - 10:00 am EST

  Asymptotic stability of the Sine-Gordon kink under odd perturbations via super-symmetry

  11th Floor Lecture Hall

  • Wilhelm Schlag, Yale University

  Abstract

  We will describe the recent asymptotic analysis with Jonas Luehrmann of the Sine-Gordon evolution of odd data near the kink. We do not rely on the complete integrability of the problem in a direct way, in particular we do not use the inverse scattering transform.

  • Video
  • Slides
• 10:00 - 10:30 am EST

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EST

  Time-dependent Bogoliubov-de-Gennes and Ginzburg-Landau equations

  11th Floor Lecture Hall

  • Virtual Speaker
  • Rupert Frank, LMU Munich

  Abstract

  We study the time-dependent Bogoliubov--de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg--Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time.

  • Video
  • Slides
• 11:30 am - 12:15 pm EST

  On the well-posedness of the derivative nonlinear Schr\"odinger equation

  11th Floor Lecture Hall

  • Maria Ntekoume, Rice University

  Abstract

  We consider the derivative nonlinear Schr\"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, not much is known regarding the well-posendess of the equation below $H^{\frac 12}$. In this talk, we prove that this problem is globally well-posed for initial data in the Sobolev spaces $H^s$ for $\frac 1 6\leq s<\frac 12$. The key ingredient in our argument is proving that ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. This is joint work with Rowan Killip and Monica Visan.

  • Video
• 12:30 - 2:30 pm EST

  Lunch/Free Time
• 2:30 - 3:30 pm EST

  Quantitative derivation and scattering of the 3D cubic NLS in the energy space

  11th Floor Lecture Hall

  • Justin Holmer, Brown University

  Abstract

  We consider the derivation of the {defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under $H^{1}$ regularity. The $H^{1}$ convergence rate estimate we obtain is almost optimal for $H^{1}$ datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state. This is joint work with Xuwen Chen (University of Rochester).

  • Video
• 3:30 - 4:30 pm EST

  Local smoothing estimate for the cone in R^3

  11th Floor Lecture Hall

  • Virtual Speaker
  • Hong Wang, Institute for Advanced Study (IAS)

  Abstract

  If u is a solution to the wave equation on R^n, a local smoothing inequality bounds $\|u\|_{L^p(\mathbb{R}^n\times [1,2])}$ in terms of the Sobolev norms of the initial data. We prove Sogge's local smoothing conjecture in 2+1 dimensions.
  In the proof, we introduced an approximation of the $L^4$--norm that works better for induction. Another key ingredient is an incidence estimate for points and tubes.
  This is joint work with Larry Guth and Ruixiang Zhang.

  • Video
• 4:30 - 6:00 pm EST

  Welcome Reception

  Reception - 11th Floor Collaborative Space

Tuesday, December 7, 2021

• 9:00 - 9:45 am EST

  High-Order Rogue Waves and Solitons, and Solutions Interpolating Between Them

  11th Floor Lecture Hall

  • Virtual Speaker
  • Peter Miller, University of Michigan

  Abstract

  A family of exact solutions to the focusing nonlinear Schrödinger equation is presented that contains fundamental rogue waves and multiple-pole solitons of all orders. The family is indexed with a continuous parameter representing the "order" that allows one to continuously tune between rogue waves and solitons of different integer orders. In this scheme, solitons and rogue waves of increasing integer orders alternate as the continuous order parameter increases. For example, the Peregrine solution can be viewed as a soliton of order three-halves. We show that solutions in this family exhibit certain universal features in the limit of high (continuous) order. This is joint work with Deniz Bilman (Cincinnati).

  • Video
  • Slides
• 10:00 - 10:30 am EST

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EST

  Landau law and stability of 3D shocks

  11th Floor Lecture Hall

  • Virtual Speaker
  • Igor Rodnianski, Princeton University

  Abstract

  We will discuss stability and long term behavior of 3 dimensional compressible irrotational shocks arising in the compressible Euler equations.
  In 1945, Landau argued that spherically symmetric solutions which form weak shocks will settle to a profile with 2 shocks decaying at the rate proportionate to 1/{t\sqrt{\log t}}. We address this conjecture by first identifying the asymptotic profile which exhibit 2 shocks, as a self-similar solution of a related Burgers equation, and then proving its stability and the conjectured rate of decay for general (non-spherically symmetric) perturbations. This is joint work with D. Ginsberg.
• 11:30 am - 12:15 pm EST

  Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential

  11th Floor Lecture Hall

  • Virtual Speaker
  • Dmitry Pelinovsky, McMaster University

  Abstract

  In order to prove the existence of a ground state (a positive, radially symmetric solution in the energy space), we develop the shooting method and deal with a one-parameter family of classical solutions to an initial-value problem for the stationary equation. We prove that the solution curve (the graph of the eigenvalue parameter versus the supremum norm) is oscillatory below a threshold and monotone above a threshold. Compared to the existing literature, rigorous asymptotics are derived by constructing families of solutions to the stationary equation with functional-analytic rather than geometric methods. The same analytical technique allows us to characterize the Morse index of the ground state.

  • Video
  • Slides
• 12:30 - 2:00 pm EST

  Lunch/Free Time
• 2:00 - 2:45 pm EST

  Rigidity for solutions to the quintic NLS equation at the ground state level

  11th Floor Lecture Hall

  • Benjamin Dodson, John Hopkins University

  Abstract

  In this talk, we will prove rigidity for solutions to the quintic nonlinear Schrodinger equation in one dimension, at the level of the ground state. Specifically, we show that the only solutions that fail to scatter are the solitons and the pseudoconformal transformation of the solitons.

  • Slides
• 3:00 - 4:30 pm EST

  Lightning Talks followed by Coffee Break and discussions

  Lightning Talks - 11th Floor Lecture Hall

Wednesday, December 8, 2021

• 9:00 - 9:45 am EST

  A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schrodinger Equation

  11th Floor Lecture Hall

  • Virtual Speaker
  • Nataša Pavlovic, University of Texas at Austin

  Abstract

  Many mathematical works have focused on understanding the manner in which the dynamics of the nonlinear Schrodinger equation (NLS) arises as an effective equation. By effective equation, we mean that solutions of the NLS equation approximate solutions to an underlying physical equation in some topology in a particular asymptotic regime. For example, the cubic NLS is an effective equation for a system of N bosons interacting pairwise via a delta or approximate delta potential. In this talk, we will advance a new perspective on deriving an effective equation, which focuses on structure. In particular, we will show how the Hamiltonian structure for the cubic NLS in any dimension arises from corresponding structure at the N-particle level. Also we will discuss what we have learned so far about understanding origins of integrability of the 1D cubic NLS. The talk is based on joint works with Dana Mendelson, Andrea Nahmod, Matthew Rosenzweig and Gigliola Staffilani.
• 10:00 - 10:30 am EST

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EST

  Gibbs measures, canonical stochastic quantization, and singular stochastic wave equations

  11th Floor Lecture Hall

  • Virtual Speaker
  • Tadahiro Oh, The University of Edinburgh

  Abstract

  In this talk, I will discuss the (non-)construction of the focusing Gibbs measures and the associated dynamical problems. This study was initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain (1994), Brydges-Slade (1996), and Carlen-Fröhlich-Lebowitz (2016). In the one-dimensional setting, we consider the mass-critical case, where a critical mass threshold is given by the mass of the ground state on the real line. In this case, I will show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988).
  In the three dimensional-setting, I will first discuss the construction of the $\Phi^3_3$-measure with a cubic interaction potential. This problem turns out to be critical, exhibiting a phase transition: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. Then, I will discuss the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic $\Phi^3_3$-model). As for the local theory, I will describe the paracontrolled approach to study stochastic nonlinear wave equations, introduced in my work with Gubinelli and Koch (2018). In the globalization part, I introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational formula and ideas from theory of optimal transport.
  The first part of the talk is based on a joint work with Philippe Sosoe (Cornell) and Leonardo Tolomeo (Bonn), while the second part is based on a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn).
• 11:30 am - 12:15 pm EST

  Mathematical Construction for Gravitational Collapse

  11th Floor Lecture Hall

  • Yan Guo, Brown University

  Abstract

  We will discuss recent constructions of blowup solutions for describing gravitational collapse for Euler-Poisson system.

  • Video
• 12:30 - 12:40 pm EST

  Group Photo (Immediately After Talk)

  11th Floor Lecture Hall
• 12:40 - 2:30 pm EST

  Lunch/Free Time
• 2:30 - 3:15 pm EST

  Internal Modes and Radiation Damping for 3d Klein-Gordon equations

  11th Floor Lecture Hall

  • Virtual Speaker
  • Fabio Pusateri, University of Toronto

  Abstract

  We consider quadratic Klein-Gordon equations with an external potential $V$ in $3+1$ space-time dimensions. We assume that $V$ is generic and decaying, and that the operator $-\Delta + V + m^2$ has an eigenvalue $\lambda^2 < m^2$. This is a so-called ‘internal mode’ and gives rise to time-periodic localized solutions of the linear flow. We address the question of whether such solutions persist under the full nonlinear flow. Our main result shows that small nonlinear solutions slowly decay as the energy is transferred from the internal mode to the continuous spectrum, provided a natural Fermi golden rule holds. Moreover, we obtain very precise asymptotic information including sharp rates of decay and the growth of weighted norms. These results extend the seminal work of Soffer-Weinstein for cubic nonlinearities to the case of any generic perturbation. This is joint work with Tristan Léger (Princeton University).

  • Video
• 3:30 - 4:00 pm EST

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EST

  Solutions to the KdV and Related Equations With Almost Periodic Initial Data

  11th Floor Lecture Hall

  • David Damanik, Rice University

  Abstract

  We discuss recent work concerning the existence, uniqueness, and structure of solutions to the KdV equation, as well as related ones, with almost periodic initial data. The talk is based on several joint works with a variety of co-authors, including Ilia Binder, Michael Goldstein, Yong Li, Milivoje Lukic, Alexander Volberg, Fei Xu, and Peter Yuditskii.

  • Video

Thursday, December 9, 2021

• 9:00 - 9:45 am EST

  The Quartic Integrability and Long Time Existence of Steep Water Waves in 2D

  11th Floor Lecture Hall

  • Virtual Speaker
  • Sijue Wu, University of Michigan

  Abstract

  Abstract. It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non trivial resonant manifold. In this talk, I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals Ej (t), directly in the physical space, which are explicit in the Riemann mapping variable and involve material derivatives of order j of the solutions for the 2d water wave equation, so that ddtEj (t) is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than ε, then the lifespan of the solution for the 2d water wave equation is at least of order O(ε−3), and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size ε, then the lifespan of the solution is at least of order O(ε−5/2). Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.

  • Video
• 10:00 - 10:30 am EST

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EST

  Global wellposedness of the Zakharov System below the ground state

  11th Floor Lecture Hall

  • Virtual Speaker
  • Sebastian Herr, Bielefeld University

  Abstract

  We consider the Cauchy problem for the Zakharov system with a focus on the energy-critical dimension d = 4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr ̈odinger equation with potentials solving the wave equation. This is joint work with Timothy Candy and Kenji Nakanishi.

  • Video
• 11:30 am - 12:15 pm EST

  Stability of solitary waves in the generalized Benjamin-Ono equation and its 2d extensions

  11th Floor Lecture Hall

  • Svetlana Roudenko, Florida International University

  Abstract

  We discuss solitary wave solutions in the generalized Benjamin-Ono equation, including the critical and supercritical cases. We then look at its higher-dimensional generalization in 2d, the Shrira equation, and its fractional generalization, the HBO (Higher-dimensional Benjamin-Ono) equation and examine the behavior of solutions in various cases as well as the stability of solitary waves.

  • Video
• 12:30 - 2:30 pm EST

  Lunch/Free Time
• 2:30 - 3:15 pm EST

  Low regularity solutions for nonlinear waves

  11th Floor Lecture Hall

  • Daniel Tataru, University of California, Berkeley

  Abstract

  The sharp local well-posedness result for generic nonlinear wave equations was proved in my work with Smith about 20 years ago. Around the same time, it was conjectured that, for problems satisfying a suitable nonlinear null condition, the local well-posedness threshold can be improved. In this talk, I will describe the first result establishing this conjecture for a good model. This is joint work with Albert Ai and Mihaela Ifrim.

  • Video
  • Slides
• 3:30 - 4:00 pm EST

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EST

  Simple motion of stretch-limited elastic strings

  11th Floor Lecture Hall

  • Virtual Speaker
  • Casey Rodriguez, University of North Carolina

  Abstract

  Perfectly flexible strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (force produces stretching) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of force is applied the stretch of the string is maximized (becoming inextensible). In this talk, we discuss a simple model for these stretch-limited elastic strings, in what way they model ``elastic" behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

  • Video
  • Slides

Friday, December 10, 2021

• 9:00 - 9:45 am EST

  Kink stability in nonlinear Klein-Gordon equations

  11th Floor Lecture Hall

  • Pierre Germain, NYU - Courant Institute

  Abstract

  Nonlinear Klein-Gordon equations whose potential have a double well admit kink solutions (most famous examples: Phi4, Sine-Gordon). I will present joint work with Fabio Pusateri which establishes the stability of this kink under some spectral conditions on the linearized problem. The key idea of the proof is to view the problem through the distorted Fourier transform associated with the linearized problem.

  • Video
• 10:00 - 10:30 am EST

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EST

  Non-trivial self-similar blowup in energy supercritical wave equations

  11th Floor Lecture Hall

  • Birgit Schoerkhuber, University of Innsbruck, Austria

  Abstract

  Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new self-similar solutions with non-trivial profiles, which are completely explicit in all supercritical dimensions. Furthermore, we analyse their stability locally in backward light cones without symmetry assumptions. This involves a delicate spectral problem that we are able to solve rigorously only in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations in a backward light cone of the blowup point. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).

  • Video
• 11:30 am - 12:15 pm EST

  The stability of charged black holes

  11th Floor Lecture Hall

  • Virtual Speaker
  • Elena Giorgi, Columbia University

  Abstract

  Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

  • Video
• 12:30 - 2:00 pm EST

  Lunch/Free Time
• 2:00 - 2:45 pm EST

  Invariance of the Gibbs measures for the periodic generalized KdV equations

  11th Floor Lecture Hall

  • Virtual Speaker
  • Andreia Chapouto, UCLA

  Abstract

  In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the L2-based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure. Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations. This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).

  • Video
  • Slides
• 3:00 - 3:45 pm EST

  Global in x Stability of Prandtl's Boundary Layer for 2D, Stationary Navier-Stokes Flows

  11th Floor Lecture Hall

  • Virtual Speaker
  • Sameer Iyer, UC Davis

  Abstract

  In this talk, I will discuss a recent work which proves stability of Prandtl's boundary layer in the vanishing viscosity limit. The result is an asymptotic stability result of the background profile in two senses: asymptotic as the viscosity tends to zero and asymptotic as x (which acts a time variable) goes to infinity. In particular, this confirms the lack of the "boundary layer separation" in certain regimes which have been predicted to be stable. This is joint work w. Nader Masmoudi (Courant Institute, NYU).

  • Video
• 4:00 - 4:30 pm EST

  Coffee Break

  11th Floor Collaborative Space