Lie Group Actions on Symplectic Manifolds

摘要奖：DMS-0071625主要研究人员：雷耶·斯贾马尔主要研究哈密顿李群作用的不变量，主要研究方法包括微分拓扑学、李理论和几何不变量理论。期望的应用包括：矩阵分析中经典不等式的推广和更强形式，以及抛物向量丛的模空间上的新结果。将使用的工具包括处理哈密顿行为(去奇异、量子化)不可避免地出现的奇点的方法，这些方法是在之前的NSF资助的项目中开发的。该项目涉及研究生参与，并与其他美国和加拿大机构的研究人员合作。我试图回答的问题类型是：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·韦尔，他在上个世纪初找到了一个部分答案，并由此得到了他对拉普拉斯本征值的著名估计。这个问题不仅与这些经典谱问题有关，而且还与力学(经典和量子)和李群的表示理论有关。这个问题的两个分支，我特别想研究的是非紧群和称为Kac-Moody群的无限维群。

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.