Poisson Structures and Lie Theory

在这个项目中，PI将继续发展泊松莫尔斯理论，其主题是使用泊松结构来研究底层流形的拓扑结构。她还将研究一类与半简单对称空间紧密相连的泊松结构的模空间及其紧化。泊松结构最早出现在19世纪的经典力学研究中。目前泊松几何受到弦理论等理论物理的影响并有其应用。本着同样的精神，这一建议是关于泊松几何和其他数学和数学物理领域之间的丰富联系，如拓扑，李论，霍普夫代数和可积系统。它在控制理论和机器人技术中也有应用。

Abstract for Proposal No. 0105195 by Jiang-Hua LuIn this project, the PI will continue to develop Poisson Morse theory, the theme of which is touse Poisson structures to study the topology of the underlyingmanifolds. She will also study a moduli space of a class of Poisson structures, which is closely connected to semi-simple symmetric spaces and their compactifications. Poisson structures first arose in the 19'th century in the study of classical mechanics. Nowadays Poisson geometry is being influenced by and has applications in theoretical physics such as string theory. In the same spirit,this proposal is about the rich connections between Poisson geometry and various other fields of mathematics and mathematical physics, suchas topology, Lie theory, Hopf algebras, and integrable systems. It also has applications in control theory and robotics.