Geometric Structures in Poisson Geometry and Quantization

DMS-0072171Ping Xu这个项目涉及到Poisson结构的研究，利用Lie群胚和Lie代数体的理论，特别是Poisson群胚和Lie双代数胚。Poisson群胚理论是Drinfel‘dPoisson群理论和Karasev-Westein辛群胚理论的统一。研究者的目的是将这一理论应用于研究可积系统，如Calogero-Moser系统。他还将继续从狄拉克生成算子的角度研究Courant代数体和Dirac结构，如应用于泊松几何中的对象，如矩映射和等变上同调。这个项目还涉及形变量子化的研究，特别是关于量子群胚的研究。更具体地说，它包括对Courant代数体的泛包络代数的研究，Kontsevich关于李代数体的形式型猜想，以及Hopf代数体形变的上同调理论，所有这些都是李双代数体量子化的组成部分。一个重要的应用是研究经典动力学r-矩阵的量子化。泊松几何在很大程度上是由物理学推动的，而物理学实际上是一种数学工具，用来给出一个涵盖经典力学大部分内容的理论框架。量子化的发展是为了更好地理解经典力学和量子力学。目前，泊松几何在控制理论、加工自动化和机器人操作等领域得到了广泛的应用。

DMS-0072171Ping XuThis project involves the study of Poisson structuresusing the theory of Lie groupoids and Lie algebroids, in particular, Poisson groupoids and Lie bialgebroids.The theory of Poisson groupoids was developed as a unification of both Drinfel'd's Poisson group theory and the theory of symplectic groupoids of Karasev-Weinstein.The investigator aims to apply this theory to study integrable systems such as Calogero-Moser systems. He will also continue his study on Courant algebroids and Dirac structures from the viewpoint of Dirac generating operators,as applied to objects in Poisson geometry such as moment maps and equivariant cohomology. This project also involves the study of deformation quantization, in particular on quantum groupoids. More specifically, it includes the study of universal enveloping algebras of Courant algebroids, Kontsevich's formalitytype conjecture for Lie algebroids, and cohomology theory of deformation of Hopf algebroids, all of which are components in quantization of Lie bialgebroids. An important application is to study quantization of classical dynamical r-matrices.Poisson geometry is largely motivated by physics, which isin fact a mathematical tool used to give a theoretical framework encompassing large parts of classical mechanics. Quantizationis developed in order to gain a better understanding betweenclassical mechanics and quantum mechanics. At present, Poisson geometry finds various applications including control theory, machining automation and robotic manipulation.