Poisson Manifolds of Compact Types and Geometric Structures on Stacks

泊松几何是数学物理和几何学的交叉点。它的起源可以追溯到经典和量子力学的数学公式，在那里出现了泊松括号的概念。在近代，对配备这种括号的空间的研究，称为泊松流形，发展成为几何学的一个分支，在数学的其他领域以及其他领域有重要的应用。例如，人们可以在高能物理的场论中的动力学公式中找到泊松括号，在生物学中的种群和进化动力学的各种模型中也可以找到泊松括号。由于一些不寻常的数学方面的收敛，理解这些空间的全局性质是一个具有挑战性的问题：人们在某些方向上找到一种特殊类型的几何，从而使Poisson流形中的某些方向区别于其他方向；此外，空间中的某些点具有其他位置所不存在的丰富的局部对称性。这个项目的目的是研究泊松流形的全局几何和拓扑性质，这是现代泊松几何中最核心的问题。这个项目包括与在欧洲和南美洲工作的泊松几何的不同研究人员的合作，旨在通过UIUC研讨会和一系列关于泊松几何的地区会议，促进数学家、物理学家和在相关领域工作的不同观点的团体之间的互动。在这个项目中，主要从李群胚理论的角度，从叶化理论、等变几何、辛几何和积分仿射几何的观点和技术来研究泊松结构和相关几何结构的全局方面。这些新的想法，加上过去十年发展的结果和技巧，应该会导致新的方法来解决Poisson几何中一些长期存在的基本问题，如正则Poisson结构的存在，紧凑型Poisson流形的分类，辛叶周围正规形的存在等。该项目还旨在通过推进与其他数学领域的新互动，如外微分系统、可积系统和几何堆叠理论，打破泊松几何的现有界限。

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.