Workshop

The aim of these talks will be to provide an overview of results, ideas and techniques which have been developed in recent years in the study of energy critical geometric nonlinear dispersive pde's. The class of problems to be discussed includes wave maps, Maxwell-Klein Gordon, Yang Mills evolutions on the wave side, and Schr\"odinger maps on the Schr\"odinger side

After a presentation of white noise and stochastic calculus in infinite dimension, I will explain how to solve classical SPDEs with white noise. I will focus on the stochastic Burgers and  reaction-diffusion equations which will be first solved with spatially  smooth noise and then with space time white noise. The case of the reaction-diffusion equation in dimension 2 is already not so obvious since the solutions are not expected to be function valued processes.  The case of dimension 3 is much more difficult and has been solved  only recently by Martin Hairer. I will not explain in details his theory  on regularity structure, this would take too much time. However, I will  explain why the problem is so difficult and give few hints on this difficult  theory. No prerequisite on stochastic calculus is expected from the audience.

I will begin by discussing invariant measures for finite dimensional Markov Processes. I will consider some classical conditions for existence and uniqueness in the finite dimensional setting. Then I will show how the situation becomes more complicated in the infinite dimensional setting of and SPDE. I will mainly concentrate on dissipative SDPEs. Time permitting I will discuss Malliavin calculus and some ideas of Hypo-ellipticity in infinite dimensions

SPDEs with white noise. I will focus on the stochastic Burgers and reaction-diffusion equations which will be first solved with spatially smooth noise and then with space time white noise. The case of the reaction-diffusion equation in dimension 2 is already not so obvious since the solutions are not expected to be function valued processes. The case of dimension 3 is much more difficult and has been solved only recently by Martin Hairer. I will not explain in details his theory on regularity structure, this would take too much time. However, I will explain why the problem is so difficult and give few hints on this difficult theory. No prerequisite on stochastic calculus is expected from the audience.

The aim of these talks will be to provide an overview of results, ideas and techniques which have been developed in recent years in the study of energy critical geometric nonlinear dispersive pde's. The class of problems to be discussed includes wave maps, Maxwell-Klein Gordon, Yang Mills evolutions on the wave side, and Schr\"odinger maps on the Schr\"odinger side

I will begin by discussing invariant measures for finite dimensional Markov Processes. I will consider some classical conditions for existence and uniqueness in the finite dimensional setting. Then I will show how the situation becomes more complicated in the infinite dimensional setting of and SPDE. I will mainly concentrate on dissipative SDPEs. Time permitting I will discuss Malliavin calculus and some ideas of Hypo-ellipticity in infinite dimensions

The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) from many body quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE.

In these lectures we will discuss the process of going from a quantum many body system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP, which was originally obtained by Erd\"os-Schlein-Yau. A key ingredient in their proof is a powerful combinatorial method that resolves the problem of the factorial growth of number of terms in iterated Duhamel expansions. In the lectures we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach of Klainerman-Machedon and the approach that we developed with Chen-Hainzl-Seiringer based on the quantum de Finetti's theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy and quantum many body systems, following results that we obtained with Chen, Chen-Tzirakis and Chen-Hainzl-Seiringer.

Solutions of nonlinear dispersive equations exhibit various space-time behavior, such as blow-up, soliton, and scattering, due to competition between the dispersion and the nonlinearity. Drastic changes are also possible along the evolution. It is hence an important and challenging problem to predict the behavior in all the future and the past from the initial data. Combining variational, dispersive, and spectral analysis, it has become possible to describe the structure of solutions and of initial data in some simple cases. In this talk I will mostly focus on the nonlinear Schrodinger equations with or without potential, which have stable and/or unstable solitons. The main goal is to obtain a global dynamical picture which contains deformation between different types of solitons

In these lectures we will go over the construction and invariance of  Gibbs and other weighted Wiener measures associated to certain nonlinear PDE. If time permits we will discuss applications, further properties and open questions

The objective of these lectures is to demonstrate a unifying role played by the property of absolute continuity in understanding the behaviour of, and construction of effective algorithms to explore, probability measures in high and infinite dimensional spaces.  Furthermore links to continuo time processes, and SPDEs in particular, will be made. These attached  notes outline the structure of the lectures, and give various references to the literature that I will not, for reasons of brevity, give in full during the lectures themselves. These also include references to related material that I will not have time to cover in the lectures at all. 

Solutions of nonlinear dispersive equations exhibit various space-time behavior, such as blow-up, soliton, and scattering, due to competition between the dispersion and the nonlinearity. Drastic changes are also possible along the evolution. It is hence an important and challenging problem to predict the behavior in all the future and the past from the initial data. Combining variational, dispersive, and spectral analysis, it has become possible to describe the structure of solutions and of initial data in some simple cases. In this talk I will mostly focus on the nonlinear Schrodinger equations with or without potential, which have stable and/or unstable solitons. The main goal is to obtain a global dynamical picture which contains deformation between different types of solitons