Mathematical Sciences: Group Proposal for Research: Topologyand Differential Geometry

数学科学：研究小组提案：拓扑学和微分几何

• 批准号: 9401714
• 负责人: William Browder
• 金额: $ 27.49万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1998-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401714&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Group; Proposal; Research

9401714 Browder Research will be carried forward in the areas of cohomology of groups, with applications to group actions on manifolds, and various topics on symplectic manifolds. Among the areas considered are the Euler characteristic of central extensions by Z of groups, such that the resultant group is finite dimensional, the rank of symmetry of products of spheres, i.e. the calculation of the maximal rank of products of Z/p which act effectively (or freely) on products of spheres, and related problems on other manifolds. In symplectic geometry the research will focus on Hamiltonian group actions, with particular attention to questions on the cohomology of the reduced spaces, the symplectic quotient, the ring structure of this cohomology, with applications to moduli spaces of flat vector bundles. Also to be considered are obstructions to the construction of symplectic structures and questions related to the algebraic topology of symplectic manifolds. Manifolds are the generalized spaces in which so much of modern physics and engineering find their natural expression, and whose topological (i.e. continuous) properties have been found to play a crucial role in the expression of these theories. In particular, the symmetry properties or the dynamical properties of these spaces, especially the spaces of mechanics, the so called symplectic spaces, and the dynamics associated to them are basic to understanding many physical and mathematical problems. This project will make its contribution using means from algebraic topology and differential geometry. ***

9401714 Browder研究将在群的上同调领域继续进行，并应用于流形上的群作用和辛流形上的各种主题。其中考虑的领域包括群的Z的中心扩张的欧拉特征，使得结果群是有限维的，球面的乘积的对称性，即计算有效(或自由)作用于球面的乘积的Z/p的最大秩，以及其他流形上的相关问题。在辛几何中，研究将集中在哈密顿群作用上，特别关注约化空间的上同调、辛商、上同调的环结构等问题，并将其应用于平坦向量丛的模空间。还需要考虑的是构造辛结构的障碍和与辛流形的代数拓扑有关的问题。流形是许多现代物理和工程中自然表达的广义空间，其拓扑(即连续)性质被发现在这些理论的表达中起着至关重要的作用。特别是，这些空间的对称性或动力学性质，特别是力学空间，即所谓的辛空间，以及与之相关的动力学性质，是理解许多物理和数学问题的基础。本项目将利用代数拓扑学和微分几何的方法做出自己的贡献。***