Lie group actions on symplectic manifolds

Reyer Sjamaar建议研究辛流形的拓扑和几何方面，特别是准哈密顿李群作用，辛Hodge理论，辛流形的实形式，以及它们在某些特征值问题中的应用。这一项目有望促进辛几何和矩阵分析领域的知识。所提出的方法借鉴了微分和代数拓扑学、指数理论和几何不变量理论，它们建立在以前NSF资助的项目中关于动量映射的上同调和凸性性质的结果的基础上。根据一个重要的数学物理原理，即Noether原理，机械系统的对称性产生守恒定律，如众所周知的能量守恒定律和动量守恒定律。诺特原理的现代微分几何表述是基于动量图的概念。这个项目的资金将用于研究与动量图有关的几个问题，并将其应用于矩阵分析，即研究大型线性方程组的研究，以及模空间的拓扑，模空间是解非线性方程的参数空间，如杨-米尔斯方程。

Reyer Sjamaar proposes to investigate topological and geometricaspects of symplectic manifolds, in particular quasi-Hamiltonian Liegroup actions, symplectic Hodge theory, real forms of symplecticmanifolds, and their applications to certain eigenvalue problems.This project is expected to advance knowledge in the fields ofsymplectic geometry and matrix analysis. The proposed methods areborrowed from differential and algebraic topology, index theory andgeometric invariant theory, and they build on results concerningcohomological and convexity properties of momentum mappings obtainedin previous NSF-funded projects.According to an important principle of mathematical physics, known asNoether's principle, symmetries of mechanical systems give rise toconservation laws, such as the well-known laws of conservation ofenergy and momentum. The modern differential-geometric formulation ofNoether's principle is based on the notion of a momentum map. Thefunds for this project will be used to investigate several problemsconcerning momentum maps, with applications to matrix analysis, whichis the study of large systems of linear equations, and the topology ofmoduli spaces, which are parameter spaces for solutions of nonlinearequations such as the Yang-Mills equation.