Geometric Structures in Poisson Geometry

在这个项目中，研究者旨在应用辛几何和泊松几何的方法，以及李群胚和李代数体的方法，来研究与泊松流形和量子化问题有关的各种微分结构。更具体地说，该项目涉及使用李双代数体理论研究双哈密尔顿结构和超卡勒结构。此外，这个项目还将研究拉格朗日子流形的形变量子化，辛流形上星积的特征类，泊松流形的量子化，以及它们在纽结理论中的应用。辛几何是一种数学工具，用于奠定包含牛顿经典力学大部分内容的理论框架。辛几何和泊松几何的思想可以用来解释和预测各种力学现象，例如运动生成和运动控制。纽结理论在理解不同DNA结构等生物技术领域有着深远的应用。

In this project, the investigator aims to apply methods of symplectic and Poisson geometry, and Lie groupoids and Lie algebroids, to the study of various differential structures related to Poisson manifolds and the quantization problem. More specifically, the project involves the study of bihamiltonian structures and hyperkahler structures using the theory of Lie bialgebroids. Also, this project will investigate deformation quantization of Lagrangian submanifolds, the characteristic class of star products on symplectic manifolds, quantization of Poisson manifolds, and their application in knot theory. Symplectic geometry is a mathematical tool used to lay a theoretical framework encompassing large parts of classical mechanics of Newton. Ideas from symplectic geometry and Poisson geometry can be used to explain and predict various mechanical phenomena - e.g., locomotion generation and motion control. Knot theory has profound application in biotechnology such as the understanding of different DNA structures.