Mathematical Sciences: Representations and Cohomology of Groups

BENSON教授项目的第一部分涉及发展有限群的无限生成模块的复杂性和变异理论，以及大量无限离散群的相应理论。与他的学生Gnacadja一起，他正在研究表征之间的幻影地图的想法，这一想法借鉴了拓扑学。项目的第二部分涉及到通过模的变异理论研究群行为，以及研究分类空间的同伦分解。第三部分概括了他与乔恩·卡尔森在庞加莱问题上的研究。有限群的对偶性。这个理论已经被推广到紧李群和虚对偶群在John Greenlees的共同工作中，他们目前正致力于推广到p进解析群，基于Lazard关于无扭转情况的工作。第四部分涉及使用John Cannon的MAGMA程序对射影分辨率和群上同调进行显式计算机计算。表示理论研究的是如何将抽象群表示为矩阵群。上同调研究矩阵表示如何粘合在一起。Benson教授的研究主要围绕有限、无限和紧李群的上同调和表示理论，分类空间的代数拓扑，有限群的不变量理论。这门学科处于代数和拓扑学之间的边缘，这似乎是纯数学中相互作用的一个非常富有成效的领域。

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.