Cohomology of Groups and Representation Theory

Benson教授提出的研究涉及群的表示理论和上同调，以及与交换代数和代数拓扑的相互作用。该提案由几个部分组成。第一部分的主题是交换代数方法在群上同调中的应用。这项工作以几个猜想为中心。第一个猜想说，对于有限群，有限群上同环的Castelnuovo-Mumford正则性总是零。这个引人注目的猜想应该是本森和卡尔森大约15年前发现的对偶现象的一种表现。这个猜想有望推广到紧李群，其中正则性被看作是作为流形的群的负维数；对于虚对偶群，其正则性是虚上同维。另一个猜想认为，有限群的上同环的深度应该等于关联素数的最小维数。本研究的第二部分与此密切相关，涉及一个关于有限群无限维模的两种不同构造的猜想。这个猜想可以看作是稳定模范畴的一种格罗滕迪克对偶。Benson和Greenlees最近的一次合作，利用Dwyer， Greenlees和Iyengar最近的机器，提出了一种将这个问题转化为代数拓扑语言的方法。看起来他们应该能够解决那里的问题，尽管其中涉及到巨大的技术困难。该项目的第三部分涉及将有限群的主块核的概念推广到非主块。本森提出了一个定义，但它的工作方式类似于主块的工作方式的说法仍然是一个猜想。提案的第四部分也是最后一部分是与Kathryn Lesh的一个联合项目，试图理解奇素数对称群的上同调。这涉及到重新整理Ben Mann的工作（这绝不是一个完整的处理），并将其与代数拓扑中的Dyer-Lashof代数的结构联系起来。表示理论研究的是如何将抽象群表示为矩阵群。群上同调理论提供了一个描述矩阵表示如何可以组合在一起形成更大的表示。关于群如何作用于流形和更一般的拓扑空间的大量信息可以从群的表示理论和上同调中收集到。这些思想是大多数纯数学和数学物理的核心。Benson教授的研究主要围绕群上同调和表示理论，以及它与交换代数和代数拓扑的相互作用。这些学科为纯数学的其他领域提供了富有成效的互动。主要的应用领域是代数拓扑、群作用和代数数论。

Professor Benson's proposed research concerns the representation theory and cohomology of groups, and interactions with commutative algebra and algebraic topology. The proposal comes in several parts. The theme of the first part is the application of methods of commutative algebra to the cohomology of groups. This work centers on several conjectures. The first conjecture says that for a finite group, the Castelnuovo-Mumford regularity of the cohomology ring of a finite group is always zero. This remarkable conjecture should be a manifestation of a duality discovered by Benson and Carlson about fifteen years ago. This conjecture is expected to generalize to compact Lie groups, where the regularity is taken as minus the dimension of the group as a manifold; and to virtual duality groups, where the regularity is the virtual cohomogical dimension. Another conjecture says that the depth of the cohomology ring of a finite group should equal the minimal dimension of an associated prime. The second part of the proposed research is closely connected with this, and concerns a conjecture relating two different constructions of infinite dimensional modules for a finite group. The conjecture can be viewed as a sort of Grothendieck duality for the stable module category. A recent collaboration of Benson and Greenlees has come up with a way to translate this problem into the language of algebraic topology, using recent machinery of Dwyer, Greenlees and Iyengar. It looks as though they should be able to solve the problem there, although there are formidable technical difficulties involved. The third part of the project involves generalizing the concept of the nucleus of the principal block of a finite group to nonprincipal blocks. Benson has proposed a definition, but the statement that it works analogously to the way the principal block works is still a conjecture. The fourth and final part of the proposal is a joint project with Kathryn Lesh to try to understand the cohomology of the symmetric groups at odd primes. This involves rehashing the work of Ben Mann (which by no means gives a complete treatment), and relating it to the structure of the Dyer-Lashof algebra from algebraic topology.Representation theory is the study of how to represent abstract groups as groups of matrices. Group cohomology theory provides a description of how matrix representations can fit together to form larger representations. A great deal of information about how groups can act on manifolds and on more general topological spaces can be gleaned from the representation theory and cohomology of the group. These ideas are central to most of pure mathematics and mathematical physics. Professor Benson's research centers around group cohomology and representation theory, and its interactions with commutative algebra and algebraic topology. These subjects provide fruitful interactions for other areas within pure mathematics. The main areas of application are algebraic topology, group actions, and algebraic number theory.