Poisson Manifolds of Compact Types and Geometric Structures on Stacks

紧凑型泊松流形和堆栈上的几何结构

基本信息

批准号：
1710884
负责人：
Rui Loja Fernandes
金额：
$ 17.4万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710884&HistoricalAwards=false
关键词：
Poisson Manifolds Compact Types Geometric

项目摘要

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.

泊松几何形状在于数学物理和几何形状的交集。它的起源可以追溯到经典和量子力学的数学表述，其中出现了泊松支架的概念。最近，对配备这些括号的空间的研究（称为泊松歧管）已发展为几何分支，并在其他数学领域以及其他领域都有重要的应用。例如，人们可以在高能物理学中的田间理论中的动力学以及生物学内的种群和进化动力学模型中找到泊松括号。由于某些不寻常的数学方面的融合，了解这些空间的全局特性是一个具有挑战性的问题：一个人在某些方向上找到了一种特殊的几何形状，因此泊松歧管中的某些方向与其他方向区分开来；同样，空间中的某些点具有在其他位置不存在的丰富局部对称性。该项目旨在研究泊松歧管的全球几何和拓扑特性，这可以说是现代泊松几何学中最中心的问题。该项目包括与在欧洲和南美工作的泊松几何学研究人员的合作，旨在促进数学家，物理学家和具有不同观点的数学家之间的互动，通过UIUC研讨会在相关领域工作，通过UIUC研讨会以及一系列在Poisson几何学中的区域会议。以及叶叶理论，e象几何形状以及符号和整体仿射几何形状的技术。这些新想法以及过去十年中开发的结果和技术应该导致新方法来攻击泊松几何形状中一些长期存在的基本问题，例如常规泊松结构的存在，紧凑型类型的泊松歧管的分类以及围绕同义叶子的正常形式存在。该项目还旨在通过与其他数学领域（例如外部差异系统，可集成系统和几何堆栈理论）进行新的相互作用来打破泊松几何形状的当前边界。