Geometric Structures in Poisson Geometry and Applications

DMS-0306665徐平这个项目涉及Poisson几何的研究，目的是了解与Poisson流形有关的各种几何结构，并研究它们在分析、可积系统、量子化和数学物理中其他相关领域的应用。主要工具之一是李群胚和李代数体理论。特别是，研究者将应用他先前发展的Morita等价理论来研究均匀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细菌，并研究它们与李群胚的关系。他计划继续研究扭曲的泊松结构，以及万能提升猜想。后者蕴含了Poisson几何中许多不平凡的结果，包括Mackenzie和研究者关于Lie双代数体的Karasev-Weinstein辛实现定理和积分定理。该项目还涉及形变量化的研究。研究人员将继续使用他以前开发的形变量化技术来研究经典动态矩阵的量子化。此外，他还将研究杜布罗文·泊松结构的量子化。泊松几何在很大程度上是由物理学推动的，实际上是一种数学工具，用于给出一个涵盖经典力学大部分内容的理论框架。李群胚是研究泊松几何中各种几何问题对称性的有用工具。量子化是为了更好地理解经典力学和量子力学之间的关系而发展起来的。目前，泊松几何在控制理论、加工自动化、机器人操作等方面有广泛的应用。

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.