Lie group actions on symplectic manifolds

Reyer Sjamaar提出研究辛流形的拓扑和几何方面，特别是拟Hamilton李群作用，辛Hodge理论，辛流形的真实的形式，以及它们在某些特征值问题上的应用，该项目有望提高辛几何和矩阵分析领域的知识。 所提出的方法是从微分和代数拓扑学，指数理论和几何不变量理论，他们建立在成果concerningcohomological和凸性质的动量映射获得在以前的NSF资助的项目。根据数学物理的一个重要原则，被称为Noether的原则，对称性的力学系统产生守恒定律，如著名的法律的能量和动量守恒。 现代微分几何公式化的诺特原理是基于动量映射的概念。 该项目的资金将用于调查有关动量映射的几个问题，并应用于矩阵分析，这是对大型线性方程组的研究，以及模空间的拓扑结构，这是非线性方程（如杨-米尔斯方程）的解的参数空间。

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.