Global Problems in Poisson Geometry and Related Structures

泊松几何及相关结构中的全局问题

基本信息

批准号：
1308472
负责人：
Rui Loja Fernandes
金额：
$ 21.73万
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：
2013
资助国家：
美国
起止时间：
2013-06-15 至 2017-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308472&HistoricalAwards=false
关键词：
Global Problems Poisson Geometry Related

项目摘要

This project aims to study properties of Poisson structures on manifolds as well as of related geometric structures such as Dirac structures, Lie algebroids or generalized complex structures. The project focus primarily on global aspects, drawing on ideas and techniques from Foliation Theory, Equivariant Geometry, Integral Affine Geometry and Symplectic Geometry. These ideas, together with recent results and techniques in Lie groupoid theory developed in the last 10 years, will 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", or the existence of normal forms around leaves that go beyond linearization. The project also aims at going beyond the current boundaries of Poisson Geometry by proposing new interactions with the theory of Exterior Differential Systems and with the theory of Integrable Systems.Poisson Geometry lies on the intersection of Mathematical Physics and Geometry. It originates in the mathematical formulation of classical mechanics as the semiclassical limit of quantum mechanics. The field developed rapidly in the last 20 years, stimulated by the connections with a large number of areas in mathematics and mathematical physics, including differential geometry and Lie theory, quantization, noncommutative geometry, representation theory and quantum groups, geometric mechanics and integrable systems. This project shares this flavor of Poisson geometry, aiming not only at a deeper understanding of geometric properties of Poisson brackets, but also at developing new applications in other fields of Mathematics.

该项目旨在研究泊松结构对流形以及相关几何结构（例如狄拉克结构，谎言代数或广义复合结构）的性质。该项目主要集中在全球方面，借鉴叶面理论，模棱两可的几何形状，整体仿射几何和符号几何形状的思想和技术。这些想法以及在过去十年中开发的lie groupoid理论中的最新结果和技术将导致新的方法来攻击泊松几何学中一些长期存在的基本问题，例如常规泊松结构的存在，Poisson歧管的分类“紧凑型类型”，或叶子周围的正常形式的存在，这些形式超出了线性化。该项目还旨在通过提出与外部差异系统理论和可集成系统理论的新相互作用来超越泊松几何形状的当前边界。Poisson几何形状在于数学物理学与几何学的交集。它起源于经典力学作为量子力学的半经典极限的数学公式。该领域在过去20年中迅速发展，这是由于数学和数学物理学领域的大量领域的联系，包括差异几何学和谎言理论，量化，非共同的几何形状，表示理论和量子群，几何学机制和综合系统。该项目具有这种泊松几何形状的风味，不仅旨在更深入地了解泊松支架的几何特性，还旨在在其他数学领域开发新应用。