Lie Group Actions on Symplectic Manifolds

AbstractAward：DMS-0071625首席研究员：Reyer Sjamaar首席研究员计划使用微分拓扑学，李群理论和几何不变量理论的方法来研究Hamilton李群作用的不变量。 预期应用包括：矩阵分析中经典不等式的推广和更强形式，抛物向量丛模空间的新结果。 所使用的工具包括处理不可避免地从哈密顿作用（去奇异化，量子化）中产生的奇点的方法，这些方法是在以前的NSF资助项目中开发的。这个项目涉及研究生的参与以及与美国和哥伦比亚其他机构的研究人员的合作。 我想回答的问题是：给定两个矩阵（大小相同）的奇异值，它们和的可能奇异值是什么？ 在这种形式下，这个问题至少可以追溯到H。外尔，谁在上个世纪初找到了部分答案，并从这获得了他著名的估计特征值的拉普拉斯。 这个问题不仅与经典的谱问题有关，而且与力学（经典的和量子的）和李群的表示论有关。 两个分支的问题，我特别希望追求的版本为非紧群体和无限维群体称为卡茨-穆迪集团。

AbstractAward: DMS-0071625Principal Investigator: Reyer SjamaarThe principal investigator plans to study invariants ofHamiltonian Lie group actions using methods from differentialtopology, Lie theory and geometric invariant theory. Expectedapplications include: extensions and stronger versions ofclassical inequalities in matrix analysis, and new results onmoduli spaces of parabolic vector bundles. The tools to be usedinclude methods for handling the singularities that inevitablyarise from Hamiltonian actions (desingularization, quantization),which were developed in a previous NSF-funded project.This project involves graduate student participation andcollaboration with researchers at other US and Canadianinstitutions. The type of question I seek to answer is: giventhe singular values of two matrices (of the same size), what arethe possible singular values of their sum? In this form thequestion goes back at least to H. Weyl, who early in the lastcentury found a partial answer and from this obtained his famousestimates for the eigenvalues of a Laplacian. The question isnot only related to such classical spectral problems, but also tomechanics (classical and quantum) and to the representationtheory of Lie groups. Two ramifications of the problem which Iparticularly wish to pursue are versions for noncompact groupsand for the infinite-dimensional groups known as Kac-Moodygroups.