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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.

泊松几何位于数学物理学和几何学的交叉点。它的起源可以追溯到经典力学和量子力学的数学公式，在那里出现了泊松括号的概念。在更近的时代，研究配备有这些括号的空间，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，我们可以在高能物理学的场论中的动力学公式中找到泊松括号，以及在生物学中的各种种群和进化动力学模型中找到泊松括号。理解这些空间的全局性质是一个具有挑战性的问题，因为一些不寻常的数学方面的收敛：人们发现在某些方向上的特殊类型的几何，使得泊松流形中的某些方向与其他方向不同;以及，空间中的某些点拥有丰富的局部对称性，而这些局部对称性在其他位置不存在。这个项目旨在研究泊松流形的整体几何和拓扑性质，可以说是现代泊松几何中最核心的问题。该项目包括与欧洲和南美洲的泊松几何研究人员的合作，旨在通过UIUC研讨会和一系列泊松几何区域会议，促进数学家，物理学家和相关领域工作的不同观点团体之间的互动。在该项目中，泊松结构和相关几何结构的全球方面进行了研究，主要从李群胚理论的角度，并借鉴了叶理理论，等变几何，辛和积分仿射几何的思想和技术。这些新的想法，连同在过去十年中开发的结果和技术，应导致新的方法来攻击一些长期存在的基本问题，在泊松几何，如存在的定期泊松结构，分类的泊松流形的紧凑型，并存在正常形式周围辛叶。该项目还旨在通过推进与其他数学领域的新的相互作用，如外部微分系统，可积系统和几何堆栈理论，打破泊松几何的当前边界。