# Overview

The central focus of the winter semester 2008 will be 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 will be considered include (1) 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 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. Among these we find the proofs of the Poincaré conjecture and of Thurston's geometrization conjecture for three manifolds. Also in Hamiltonian PDE, the analytic methods of Hamiltonian mechanics are making an influence on the study of the evolution of many of the principal nonlinear evolution equations of mathematical physics. These advances have had a broad impact on recent progress in geometry and topology, and they also shed light on the basic physical processes that are modeled by ordinary and partial differential equations.

The purpose of this program semester is to bring together the diverse international community of researchers who have an interest in these topics, to give a series of advanced-level courses on the subject matter so as to make the material available to new researchers in the field, and to discuss the perspectives and general directions for the next advances and directions of progress in the area.

The four principal topics of the semester are diverse areas of analysis and geometry, with the common theme that they involve the interplay between classical dynamical systems and the qualitative properties of solutions of partial differential equations. This is much like the quantum correspondence principle between classical mechanics and the behavior of quantum particles. It also entails the more modern perspective of partial differential equations as dynamical systems of evolution, posed in an infinite-dimensional function space for its phase space.