ABSTRACT

The proposer will study some basic problems about finite and algebraic groups related to presentations, linear and permutation representations and cohomology with applications to the problems in arithmetic algebraic geometry -- particularly questions related to polynomials, rational functions and coverings of curves. In some recent work with Kantor, Kassabov and Lubotzky, it was shown that every finite simple group, with the possible of one family, have short and/or bounded presentations. This was a very surprising result and much better than conjectured. It does lead to some important and basic problems including removing the exception, understanding the differences (if any) between discrete presentations and profinite presentations and getting a good bound on cohomology. More generally, the proposer will consider linear and permutation representations of finite and algebraic groups and use these results to study problems in number theory and algebraic geometry. One wants to obtain fairly precise information about finite groups acting on curves and higher dimensional varieties. In the case of curves, one wants to know about possibilities for ramification groups. In higher dimensions, the non-irreducibility of certain modules leads to some interesting properties of varieties of families of curves as shown by earlier work of the proposer with Tiep answering questions of Katz and Kollar.

Since the classification of finite simple groups was completed, many problems about finite groups and related questions in other areas of mathematics have been solved that would have been unimaginable without this classification. The classification manifests itself in two ways. First of all, it is a list and so to prove properties about all finite groups, one can often reduce it to questions about simple groups and then attack the families. For example, one can show that the group of all permutations of a set of a finite set has a presentation with 4 generators and 10 relations. It seems likely that 2 generations and 3 relations suffice (that would be the best possible answer). The standard presentations, known for a century, have a linear number of generators and quadratic number of relations (in the size of the set). Secondly, the classification also gives information about the subgroup structure of the simple groups. This is closely related to understanding how simple groups can act on sets as permutations and on vectors by linear transformations. Rather surprisingly, such information can lead to breakthroughs in seemingly unrelated topics. For example, one aspect of the proposal is to classify exceptional polynomials over a finite field. These are polynomials which are bijective (with degree large enough compared to the size of the field). Using knowledge of simple groups and their subgroups has lead to a complete classification aside from the case where the degree is a power of the characteristic of the field. The proposer expects to complete this classification. These polynomials have been studied since the late 1800's but it has been only in the last 15 years that there has been major progress. The proposer will study related
problems in number theory and geometry and translate them to questions in group theory and then apply this powerful theory.

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