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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.

本课题旨在研究流形上泊松结构的性质，以及相关几何结构如狄拉克结构、李代数群或广义复杂结构的性质。该项目主要关注全局方面，借鉴叶理理论、等变几何、积分仿射几何和辛几何的思想和技术。这些想法，加上李群样理论在过去十年中发展起来的最新成果和技术，将导致新的方法来解决泊松几何中一些长期存在的基本问题，如规则泊松结构的存在，“紧型”泊松流形的分类，或叶周围超越线性化的正规形式的存在。该项目还旨在通过提出与外微分系统理论和可积系统理论的新相互作用，超越泊松几何的现有边界。泊松几何是数学物理和几何的交叉学科。它起源于经典力学的数学表述，作为量子力学的半经典极限。近20年来，由于与微分几何和李论、量子化、非交换几何、表示论和量子群、几何力学和可积系统等数学和数学物理领域的联系，该领域得到了迅速发展。这个项目分享了泊松几何的味道，不仅旨在更深入地理解泊松括号的几何性质，而且还旨在开发其他数学领域的新应用。