Jean-Morlet Chair (2104) - Integrability and Randomness in Mathematical Physics and Geometry Conference
Over the last 20 years integrability has assumed an increasingly prominent role in various fields of mathematical physics. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with origins in the seminal work of Zabusky and Kruskal about the recurrence behaviour of solutions, the discovery of the Lax pair, multi-soliton solutions and infinite number of conservation laws. In other surprising connections, integrable systems like the KdV equation and the Toda lattice were proven to appear in fundamental combinatorial models, in random matrices and the geometry of moduli spaces. Here is a short list of relevant examples: 1. The generating function of Hurwitz numbers and various classes on moduli spaces coincide with tau-functions of integrable systems. 2. Partition functions of exactly solvable statistical models like the Ising model and the six vertex model have been shown to be tau-functions of integrable equations. 3. Random particle models appearing through connections to representation theory have an integrable structure, now referred to as integrable probability. 4. Local statistics of a large collection of exactly solvable matrix models in scaling limits (either in the bulk or at spectral edges) are related to integrable operators and determinantal point processes). This relationship has recently been proven, via perturbative results, to be universal and valid for more general classes of non-integrable models like the Wigner matrices. In general, integrability provides the route to an explicit description of the answer. This statement is quite true also in the theory of nonlinear PDEs where the typical behaviour of integrable PDEs is canonical far beyond the integrable examples. Finally, the phenomenon of recurrence in solutions, as opposed to thermalisation, first observed in the FPU model and then in the Zabusky and Kruskal experiment for KdV, leads to the field of meta-stability and opens the question of typical behaviour of solutions of PDEs in the periodic setting over long times. Information about both typical behaviour and fluctuations when initial data is sampled from a suitable probability measure are generally more relevant for understanding the behaviour of real systems than the solution of a specific initial value problem. The aim of the conference is to foster interactions among researchers that work in the following fields: - integrable systems and their connections to geometry - random matrices, determinantal point processes and integrable probability - dispersive PDEs in random environment.
