Dvid Nadler

Title: Bounding Cohomology for Finite and Algebraic Groups

Abstract: Let k be an algebraically closed field of characteristic p > 0, G be a finite group,
and V be a faithful irreducible representation for G over k. For many years, it was believed
that there may exist a universal bound C for all such G and V such that dim H^{1}(G,V)
(first cohomology group with coefficients in V) is bounded by C. At an AIM workshop in
2012 examples of large cohomology groups were produced by Lubeck which provided
negative evidence towards this conjecture.

In this talk I will survey known results in this area and present new results which were obtained
at this AIM workshop. These new results involve bounding cohomology for reductive algebraic
groups using detailed calculations due to Andersen, Bendel, Nakano, Pillen, Rian and Wright.
It will be shown how our results can be applied to obtain effective bounds on the first cohomology
of the symmetric group, and how, for finite Chevalley groups, our methods permit significant
improvements over previous estimates obtained by Guralnick, Kantor, Kassabov, and Lubotsky
for the second cohomology of an arbitrary finite quasi-simple group.

The results from AIM are joint work with Christopher Bendel,
Brian Boe, Christopher Drupieski, Brian Parshall, Cornelius Pillen, and Caroline Wright.