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Organizing Committee
ICMAT
INRIA Rennes
Paris-Sud University, Orsay
NYU - Courant Institute
Princeton University
Duke University
University of Massachusetts Amherst
Research Institute for Mathematical Sciences, Kyoto University
Brown University
Massachusetts Institute of Technology
Abstract
Dispersive equations are ubiquitous in nature. They govern the motion of waves in plasmas, ferromagnets, and elastic bodies, the propagation of light in optical fibers and of water in canals. They are relevant from the ocean scale down to atom condensates. There has been much recent progress in different directions, in particular in the exploration of the phase space of solutions of semilinear equations, advances towards a soliton resolution conjecture, the study of asymptotic stability of physical systems, the theoretical and numerical study of weak turbulence and transfer of energy in systems out of equilibrium, the introduction of tools from probability and the recent incorporation of computer assisted proofs. This semester aims to bring together these new developments and to explore their possible interconnection.
Dispersive phenomena appear in physical situations, where some energy is conserved, and are naturally related to Hamiltonian systems. This semester proposes to explore this link further by bringing together experimentalists, scientists, computational scientists and mathematicians with a common interest in exploring the various aspects of dispersive equations, from their analysis to their applications, and developing tools to facilitate experimentation. One key focus will be on global approaches, either in the sense of analyzing the overall landscape of the phase space, or in the study of generic solutions (e.g. of properties “almost surely true” in an appropriate sense). Another key focus will be experimental, in the sense of developing and analyzing instructive toy-models, implementing numerical experiments, and in some cases, simply of looking at interesting special cases.
The main events will be centered around three workshops
One of the main objectives of this semester will be to integrate researchers from different horizons, and therefore special attention will be devoted to foster interdisciplinary interactions. There will be an additional introductory workshop at the beginning of the semester, and various events held in preparation of each workshop as well as in-depth follow-up discussions.
Dispersive phenomena appear in physical situations, where some energy is conserved, and are naturally related to Hamiltonian systems. This semester proposes to explore this link further by bringing together experimentalists, scientists, computational scientists and mathematicians with a common interest in exploring the various aspects of dispersive equations, from their analysis to their applications, and developing tools to facilitate experimentation. One key focus will be on global approaches, either in the sense of analyzing the overall landscape of the phase space, or in the study of generic solutions (e.g. of properties “almost surely true” in an appropriate sense). Another key focus will be experimental, in the sense of developing and analyzing instructive toy-models, implementing numerical experiments, and in some cases, simply of looking at interesting special cases.
The main events will be centered around three workshops
- one workshop on numerics, modeling and experiments in wave phenomena
- one workshop on generic behavior of dispersive solutions and wave turbulence
- one workshop on Hamiltonian methods and asymptotic dynamics
One of the main objectives of this semester will be to integrate researchers from different horizons, and therefore special attention will be devoted to foster interdisciplinary interactions. There will be an additional introductory workshop at the beginning of the semester, and various events held in preparation of each workshop as well as in-depth follow-up discussions.
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Semester Schedule
Wednesday, September 08, 2021
Hamiltonian Methods in Dispersive and Wave Evolution Equations
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
Thursday, September 09, 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
Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.
11:00 AM - 12:30 PM EDT
Ergodicity of Markov processes: theory and computation
Tutorial - 11th Floor Lecture Hall
Yao Li, University of Massachusetts Amherst
Abstract
In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).
1:30 - 2:00 PM EDT
ICERM Welcome
Welcome - 11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
2:00 - 3:00 PM EDT
Grad Student/ PostDoc Introductions
Introductions - 11th Floor Lecture Hall
Yvonne Alama Bronsard, Sorbonne Université
Nicolas Camps, Université Paris Saclay
Patrick Flynn, Brown
Louise Gassot, Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay
Dean Katsaros, UMass amherst
Sudipta Kolay, ICERM
Kyle Liss, University of Maryland, College Park
Jaemin Park, Universitat de Barcelona
Nancy Scherich, University of Toronto
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
Friday, September 10, 2021
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
Benoit Pausader, Brown University
Abstract
This is a short introduction to the classical techniques for dispersive equations. We will present various equations and some methods to obtain local and global well-posedness and study the asymptotics.
11:00 AM - 12:30 PM EDT
Ergodicity of Markov processes: theory and computation
Tutorial - 11th Floor Lecture Hall
Yao Li, University of Massachusetts Amherst
Abstract
In this short course, I’ll cover the ergodicity of Markov processes on measurable state spaces. Both theoretical results and computational methods are based on the coupling technique. The following topics will be covered. 1, Markov process, transition kernel, and coupling. 2, Renewal theory with focusing on simultaneous renewal time. 3, Lyapunov criterion for geometric/polynomial ergodicity. 4, How to construct a Lyapunov function? 5, Numerical estimation of geometric/polynomial ergodicity. 6, Numerical estimation of invariant probability measure (if time permits).
2:00 - 3:30 PM EDT
Grad Student/ PostDoc Introductions
Introductions - 11th Floor Lecture Hall
Bjoern Bringmann, Institute for Advanced Study
Stefan Czimek, Brown University (ICERM)
Daniel Eceizabarrena, University of Massachusetts Amherst
Claudia García, Universitat de Barcelona
Eduardo Garcia-Juarez, 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
