Geometric Structures in Poisson Geometry and Applications

泊松几何中的几何结构及其应用

• 批准号: 0306665
• 负责人: Ping Xu
• 金额: $ 19.68万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306665&HistoricalAwards=false
• 关键词: Geometric; Structures; Poisson; Geometry; Applications

DMS-0306665Ping XuThis project involves the study of Poisson geometry, with the goals ofunderstanding various geometric structures in connection with Poissonmanifolds, and studying their applications in analysis, integrablesystems, quantization and other related areas in mathematical physics.One of the main tools is the theory of Lie groupoids and Lie algebroids.In particular, the investigator will apply his previously developedtheory of Morita equivalence to investigate a geometric model of unifiedmomentum map theory. He will also study stacks and gerbes from theviewpoint of differentiable geometry, and investigate their relationshipto Lie groupoids. He plans to continue his study of twisted Poissonstructures, and also the universal lifting conjecture. The latterimplies many non-trivial results in Poisson geometry including theKarasev-Weinstein symplectic realization theorem and the integrationtheorem for Lie bialgebroids of Mackenzie and the investigator. Thisproject also involves the study of deformation quantization. Theinvestigator will continue to study quantization of classical dynamicalr-matrices using his previously developed deformation quantizationtechniques. Also, he will study quantization of Dubrovin Poissonstructures.Poisson geometry is largely motivated by physics, and is in fact amathematical tool used to give a theoretical framework encompassinglarge parts of classical mechanics. Lie groupoids are useful tools instudying the symmetry of various geometric problems in Poisson geometry.Quantization is developed in order to gain a better understanding of therelationship between classical mechanics and quantum mechanics. Atpresent, there are various applications of Poisson geometry includingcontrol theory, machining automation, and robotic manipulation.

DMS-0306665Ping Xu该项目涉及泊松几何的研究，目的是了解与泊松流形相关的各种几何结构，并研究它们在数学物理分析、可积系统、量化和其他相关领域的应用。主要工具之一是李群胚和李代数胚理论。特别是，研究者将应用他之前开发的 Morita 理论 等价研究统一动量映射理论的几何模型。他还将从可微几何的角度研究堆栈和非洲菊，并研究它们与李群形的关系。他计划继续研究扭曲泊松结构以及通用提升猜想。后者意味着泊松几何中的许多重要结果，包括卡拉塞夫-韦恩斯坦辛实现定理以及 Mackenzie 和研究者的李双代数胚的积分定理。该项目还涉及变形量化的研究。研究人员将继续使用他之前开发的变形量化技术来研究经典动态矩阵的量化。此外，他还将研究杜布罗文泊松结构的量子化。泊松几何很大程度上受到物理学的推动，实际上是一种数学工具，用于给出涵盖经典力学大部分内容的理论框架。李群群是研究泊松几何中各种几何问题的对称性的有用工具。量子化的发展是为了更好地理解经典力学和量子力学之间的关系。目前，泊松几何有多种应用，包括控制理论、加工自动化和机器人操纵。