Poisson Manifolds of Compact Types and Geometric Structures on Stacks

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Poisson geometry lies at the intersection of mathematical physics and geometry. Its origins go back to the mathematical formulation of classical and quantum mechanics, where the notion of a Poisson bracket emerged. In more recent times, the study of spaces equipped with these brackets, called Poisson manifolds, developed into a branch of geometry, with important applications to other areas of mathematics, as well as other fields. For example, one can find Poisson brackets in the formulation of dynamics within field theory in high-energy physics, and in various models for population and evolutionary dynamics within biology. Understanding global properties of these spaces is a challenging problem due to the convergence of some unusual mathematical aspects: one finds a special type of geometry in certain directions, so that some directions in a Poisson manifold are distinguished from others; as well, some points in the space possess a rich set of local symmetries not present at other locations. This project aims to study global geometric and topological properties of Poisson manifolds, arguably the most central issue in modern day Poisson geometry. This project includes collaborations with various researchers in Poisson geometry working in Europe and South America, and aims to promote interaction between mathematicians, physicists and groups with different points of view working on related areas, through a UIUC seminar and through a series of regional conferences in Poisson geometry. In this project, global aspects of Poisson structures and related geometric structures are studied, primarily from the perspective of Lie groupoid theory and drawing on ideas and techniques from foliation theory, equivariant geometry, and from symplectic and integral affine geometry. These new ideas, together with results and techniques developed in the last decade, should lead to new methods to attack some long standing fundamental problems in Poisson geometry, such as the existence of regular Poisson structures, the classification of Poisson manifolds of compact type, and the existence of normal forms around symplectic leaves. The project also aims at breaking the current boundaries of Poisson geometry by advancing new interactions with other mathematical areas, such as exterior differential systems, integrable systems and the theory of geometric stacks.