Mathematical Sciences: Geometry and Representation Theory

数学科学：几何与表示论

• 批准号: 9203660
• 负责人: Shrawan Kumar
• 金额: $ 8.77万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-06-01 至 1995-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203660&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Representation; Theory

Kumar will continue his work on equivariant cohomology with generalized coefficients. He will study a new G-equivariant cohomology theory of a G-space, where G is a real Lie group. This new cohomology theory is defined as the cohomology of the complex which is obtained from the polynomial functions on the Lie algebras of G by generalized functions on the Lie algebra. This cohomology is encountered in studying the index theory of transversally elliptic operators on G-manifolds. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.

Kumar 将继续他在广义系数等变上同调方面的工作。 他将研究 G 空间的新 G 等变上同调理论，其中 G 是实李群。 这种新的上同调理论被定义为由G的李代数上的多项式函数通过李代数上的广义函数得到的复形的上同调。 这种上同调是在研究 G 流形上横向椭圆算子的指数理论时遇到的。 李群理论以挪威数学家索弗斯·李的名字命名，一直是二十世纪数学的主要主题之一。 作为利用系统固有对称性的数学工具，李群的表示论对数学本身（特别是分析和数论）以及理论物理学（特别是量子力学和基本粒子物理学）产生了深远的影响。