Cohomology of Groups and Representation Theory

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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.