Mathematical Sciences: Representations and Cohomology of Groups

数学科学：群的表示和上同调

• 批准号: 9700416
• 负责人: David Benson
• 金额: $ 16.7万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2000-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700416&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Representations; Cohomology; Groups

BENSON, 97-00416 The first part of Professor Benson's project involves developing the theory of complexity and varieties for infinitely generated modules for finite groups, and the corresponding theory for a large class of infinite discrete groups. With his student Gnacadja, he is studying the idea of a phantom map between represenations, an idea borrowed from topology. The second part of the project involves the study of group actions via the theory of varieties for modules, and the study of homotopy decompositions of classifying spaces. The third part involves generalizations of his work with Jon Carlson on Poincaré duality for finite groups. This theory has already been generalized to compact Lie groups and to virtual duality groups in joint work with John Greenlees, and they are currently working on the generalization to p-adic analytic groups, based on the work of Lazard for the torsion free case. The fourth part involves explicit computer calculations of projective resolutions and group cohomology using John Cannon's MAGMA program. Representation theory is the study of how to represent abstract groups as groups of matrices. Cohomology studies how matrix representations glue together. Professor Benson's research centers around the cohomology and representation theory of finite, infinite and compact Lie groups, the algebraic topology of classifying spaces, and invariant theory for finite groups. This subject is on the borderline between algebra and topology, which seems to be a very fruitful area of interaction within pure mathematics.

本森，97-00416 本森教授的项目的第一部分涉及发展理论的复杂性和品种的无限生成模块的有限群体，以及相应的理论为一大类的无限离散群体。 他和他的学生Gnacadja正在研究表示之间的幻影映射的想法，这是一个从拓扑学中借用的想法。的第二部分 项目涉及通过模的簇理论研究群作用，以及研究分类空间的同伦分解。 第三部分涉及概括他的工作与乔恩卡尔森庞加莱eacute;对偶有限群体。这个理论已经被推广到紧李群和虚对偶群中，他们与John Greenlees合作，目前正在进行推广工作 p-adic解析群，基于Lazard的工作的扭转自由的情况下。第四部分是利用John Cannon的MAGMA程序对射影分解和群上同调进行的显式计算机计算。 表示论是研究如何将抽象群表示为矩阵群的理论。上同调研究矩阵表示如何粘合在一起。本森教授的研究中心是上同调 和有限、无限和紧李群的表示论， 分类空间的代数拓扑，有限空间的不变量理论 组这门课介于代数和拓扑学之间， 似乎是纯数学中一个非常富有成效的互动领域。