Cohomology of Groups and Representation Theory

本森教授的研究方向是群的表示论和上同调，以及与交换代数和代数拓扑的相互作用。该提案分为几个部分。第一部分的主题是交换代数的方法在群的上同调中的应用。这项工作以几个方面为中心。第一个猜想说，对于有限群，有限群的上同调环的Castelnuovo-Mumford正则性总是零。这个非凡的猜想应该是一个二元性的表现形式发现本森和卡尔森约15年前。这一猜想有望推广到紧李群，其中正则性被取为减去作为流形的群的维数;以及虚对偶群，其中正则性是虚上同调维数。另一个猜想说，有限群的上同调环的深度应该等于相关素数的最小维数。第二部分提出的研究与此密切相关，并涉及一个猜想有关的两个不同的建设无限维模块的有限群。这个猜想可以看作是稳定模范畴的一种Grothendieck对偶。最近的合作本森和Greenlees提出了一种方法来翻译这个问题的语言代数拓扑，使用最近的机器德怀尔，Greenlees和Iyengar。看起来他们应该能够在那里解决这个问题，尽管这涉及到巨大的技术困难。该项目的第三部分涉及推广的概念的核心的主块的有限集团nonprincipal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。该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。这涉及到对本·曼的工作进行简化（这绝不是一个完整的处理），并将其与代数拓扑中的戴尔-拉肖夫代数的结构联系起来。表示论研究如何将抽象群表示为矩阵群。群上同调理论描述了矩阵表示如何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。本森教授的研究中心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学、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.