Mathematical Sciences: Poisson Geometry and Quantization

这个项目属于数学物理的一般领域。研究者的目标是应用辛几何和泊松几何，李群和李代数的方法，来研究流形上的算子和更一般的量化问题。更具体地说，该项目将开发适用于广义标志流形Bruhat分解的微局部分析，并寻求傅立叶积分算子的指标定理的证明，并制定泊松模块理论，并将其应用于线性偏微分方程系统。辛几何是一种数学工具，用于建立包含大部分经典力学的理论框架。辛几何和泊松几何的思想可以用一种优雅的方式来解释和预测各种机械现象-例如，运动产生和运动控制。

9625122 Weinstein This project lies in the general area of mathematical physics. The investigator aims to apply methods of symplectic and Poisson geometry, and Lie groupoids and Lie algebroids, to the study of operators on manifolds and more generally quantization problems. More specifically, this project will develop microlocal analysis adapted to the Bruhat decomposition of generalized flag manifolds, and pursue a proof of an index theorem for Fourier integral operators and formulate a theory of Poisson modules, with applications to systems of linear partial differential equations. Symplectic geometry is a mathematical tool used to lay a theoretical framework encompassing large parts of classical mechanics. Ideas from symplectic geometry and Poisson geometry can be used to explain and predict various mechanical phenomena - e.g., locomotion generation and motion control - in an elegant manner.