# Institute for Computational and Experimental Research in Mathematics (ICERM)

September 8, 2021 - December 10, 2021
##### Wednesday, September 8, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Thursday, September 9, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 9:00 - 10:30 am EDT
Local well-posedness for dispersive equations
Tutorial - 11th Floor Lecture Hall
• Alexandru Ionescu, Princeton 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
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
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 9:00 - 10:30 am EDT
Local well-posedness for dispersive equations
Tutorial - 11th Floor Lecture Hall
• Alexandru Ionescu, Princeton 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
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
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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).
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 9:00 - 10:30 am EDT
Local well-posedness for dispersive equations
Tutorial - 11th Floor Lecture Hall
• Alexandru Ionescu, Princeton 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
##### Wednesday, September 15, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 9:00 - 10:30 am EDT
Local well-posedness for dispersive equations
Tutorial - 11th Floor Lecture Hall
• Alexandru Ionescu, Princeton 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
11th Floor Lecture Hall
• Brendan Hassett, ICERM/Brown University
• Misha Kilmer, Tufts University
##### Thursday, September 16, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Friday, September 17, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 9:00 - 10:30 am EDT
Local well-posedness for dispersive equations
Tutorial - 11th Floor Lecture Hall
• Alexandru Ionescu, Princeton 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
###### Numerics, Modeling, and Experiments in Wave Phenomena
• 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.
• 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.
• 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.
• 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.
• 3:00 - 4:30 pm EDT
Reception
Hemenway's Patio (weather permitting)
##### Tuesday, September 21, 2021
###### Numerics, Modeling, and Experiments in Wave Phenomena
• 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.
• 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
• 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.
• 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
###### Numerics, Modeling, and Experiments in Wave Phenomena
• 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.
• 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.
• 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.
• 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
###### 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.
##### Thursday, September 23, 2021
###### Numerics, Modeling, and Experiments in Wave Phenomena
• 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.
• 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.
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
###### Numerics, Modeling, and Experiments in Wave Phenomena
• 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.
• 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
• 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.
##### Monday, September 27, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Tuesday, September 28, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
##### Wednesday, September 29, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EDT
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Monday, October 4, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Tuesday, October 5, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EDT
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EDT
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Monday, October 18, 2021
###### Generic Behavior of Dispersive Solutions and Wave Turbulence
• 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
• 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.
• 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.
• 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.
• 3:00 - 4:30 pm EDT
Welcome Reception
Reception - Hemenway's Patio
##### Tuesday, October 19, 2021
###### Generic Behavior of Dispersive Solutions and Wave Turbulence
• 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.
• 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
###### Generic Behavior of Dispersive Solutions and Wave Turbulence
• 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).
• 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
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.
• 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).
##### Thursday, October 21, 2021
###### Generic Behavior of Dispersive Solutions and Wave Turbulence
• 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.
• 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.
• 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.
• 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.
##### Friday, October 22, 2021
###### Generic Behavior of Dispersive Solutions and Wave Turbulence
• 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.
• 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.
• 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.
##### Monday, October 25, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Tuesday, October 26, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EDT
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Monday, November 1, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Tuesday, November 2, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EDT
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EDT
Coffee Break
11th Floor Collaborative Space
##### Monday, November 8, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Tuesday, November 9, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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 - 11th Floor Lecture Hall
• 12:30 - 12:45 pm EST
Group Photo - 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Friday, November 12, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Monday, November 15, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Tuesday, November 16, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EST
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Monday, November 22, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Tuesday, November 23, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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 - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Monday, November 29, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 11:00 am - 12:30 pm EST
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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 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
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Thursday, December 2, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Friday, December 3, 2021
###### Hamiltonian Methods in Dispersive and Wave Evolution Equations
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
##### Monday, December 6, 2021
###### Hamiltonian Methods and Asymptotic Dynamics
• 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.
• 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.
• 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.
• 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).
• 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.
• 4:30 - 6:00 pm EST
Welcome Reception
Reception - 11th Floor Collaborative Space
##### Tuesday, December 7, 2021
###### Hamiltonian Methods and Asymptotic Dynamics
• 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).
• 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.
• 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.
• 3:00 - 4:30 pm EST
Lightning Talks followed by Coffee Break and discussions
Lightning Talks - 11th Floor Lecture Hall
##### Wednesday, December 8, 2021
###### Hamiltonian Methods and Asymptotic Dynamics
• 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.
• 12:30 - 12:40 pm EST
Group Photo
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).
• 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.
##### Thursday, December 9, 2021
###### Hamiltonian Methods and Asymptotic Dynamics
• 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.
• 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.
• 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.
• 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.
• 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.
##### Friday, December 10, 2021
###### Hamiltonian Methods and Asymptotic Dynamics
• 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.
• 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).
• 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.
• 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).
• 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).
• 4:00 - 4:30 pm EST
Coffee Break
11th Floor Collaborative Space

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).