Special meeting: Dynamical systems and evolution equations, CRM

Trustees Of Boston University, Boston

Investigators

Abstract

The focus of the thematic program semester of winter 2008 at the CRM is on dynamical systems, interpreted in a broad sense so as to include applications to fundamental problems in differential geometry as well as in mathematical physics. Topics that are considered include:(1) the interplay between dynamical systems and PDE, in particular in the context of Hamiltonian systems, (2) geometric evolution equations such as Ricci flows and extrinsic curvature flows, (3) spectral theory and its relationship to Hamiltonian dynamics, and (4) Floer theory and Hamiltonian flows. In the past several years there have been dramatic achievements in these four areas, representing progress on a number of the most basic and difficult questions in this field. These advances have had a broad impact on recent progress in geometry and topology, and they also shed light on basic physical processes, such as nonlinear wave phenomena, that are modeled by ordinary and partial differential equations. The purpose of this program semester is to bring together members of the diverse international community of researchers who have an interest in these topics, to give a series of advanced-level courses on relevant subject matter so as to make the topic accessible to new researchers in the field, and to bring into discuss the perspectives and general indications for the next advances and directions of progress in the area. The central focus of the theme semester of winter 2008 at CRM is dynamical systems. The theory of dynamical systems is concerned with the description of the evolution of systems depending on time. Such systems are fundamental, and appear very commonly in the modeling of physical, chemical and biological phenomena, as well as in geometry and many other areas of mathematics. In the past several years there have been dramatic achievements in the area of dynamical systems, including the proof of Poincaré conjecture by G. Perelman (an event known to the public through the drama of the Fields medal awards in 2006). These advances have had a broad impact on recent progress in geometry and topology. The modern theory of dynamical systems has also been fundamental in the study of many basic physical processes, and their modeling by ordinary and partial differential equations. The purpose of this program semester is to bring together representatives of the diverse international community of researchers who have an interest in these topics. Its activities will comprise (1) a series of advanced-level courses on the subject matter, so as to make the field available to students and new researchers, (2) to host discussions of the perspectives and future directions for the next advances and areas of progress in the field.