The PI plans to 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 automorphisms and coverings of curves. Using the work of Tiep and the PI, small representations of simple groups will be studied. This work has already had applications to various questions of Katz, Kollar, and Larsen in algebraic geometry. The most critical open case is to classify all closed subgroups which are irreducible on exterior powers of a module. This is also closely related to questions about the classification of maximal subgroups of the finite simple groups. This latter questions leads to many basic problems. In particular the PI will look at problems about bounding the number of irreducible representations of bounded dimension for various families of simple groups.
The PI and his collaborators have shown recently that with the possible exception of one family, every finite simple group has a presentation with at most fifty relations. In many cases, one can do much better (for example, an infinite number of alternating groups can be presented with two generators and four relations). Allowing the number of relations to increase a bit, one can even produce short presentations (essentially best possible). The most fundamental problem is the relationship between discrete presentations and profinite presentations and cohomology (one way to think of a profinite presentation is that if one is given by generators and relations and one already knows the group is finite, then one can identify the group). This leads one to try to produce very good bounds on the size of the first and second cohomology groups of finite and algebraic groups with coefficients in a simple module. One goal is to prove that every finite simple group has a profinite presentation with two generators and at most four relations (one cannot do better). This may be even be true without the profiniteness condition but no one has any idea of how to approach this in general. Another major problem is to complete the classification of exceptional polynomials over finite fields. Exceptional polynomials are precisely the bijective polynomials assuming the field size is sufficiently large compared to the degree and have been studied seriously since the thesis of Dickson in the 1890's, as well as by Schur, Fried and others. The classification of indecomposable exceptional polynomials whose degree is not a power of the characteristic used a combination of deep group theory together with methods arithmetic algebraic geometry. The PI intends to use these methods and use new results to study these problems. This should have applications to cryptography.
Considerable progress was made on the project supported by this grant. Several 40 year old conjectures were solved. In particular in two articles (one with Maroiti and one with Malle) two 1966 conjectures of Peter Neuman regarding fixed point spaces of elements in irreducible groups acting on a finite dimensional space were given. In joint work with Kantor, Kassabov and Lubotzky, a series of papers were written which vastly improved the state of the art for finding generators and relations for finite simple groups. In almost all cases, it was shown that there is a presentation with 2 generators and at most 50 relations. Moreover, the relations could be taken to be short. This has already been used in software packages for computational group theory. Part of the project also involved solving a conjecture of Holt about the size of second cohomology groups. In joint work with Chinburg and Harbater, we generalized a conjecture of Oort about lifting covers of curves in characteristic p to characteristic 0. In an article with Malle, we answered a long standing conjecture of Bauer, Catanese and Grunewald regarding Beauville surfaces and simple groups (Beauville surfaces are quotients of a product of curves with certain nice properties). Another consequence that came out of this work is that every finite simple group can be invariably generated by two elements (more precisely, there exist two conjugacy classes so that the simple group is generated by any pair elements containing one from each class). This is quite helpful in computing Galois groups where often one only knows the conjugacy classes of elements and not the elements themselves. In work with Larsen and Tiep, we showed that there was a low degree polynomial bound (in terms of the rank) for the number of conjugacy classes of maximal subgroups of a finite simple group of Lie type (e.g, SL(n,q)). The previous bound had been worse than exponential. In work with Fulman, we continued to improve results about fixed point free elements in certain types of permutation groups. These results should be useful in computational group theory and other contexts as well. In a paper with Herzig, Taylor and Thorne, we studied adequate linear representations. This a notion introduced by Thorne generalzing and simplifying earlier of Wiles, Taylor, Clozel and others to show certain representations are automorphic. These results were used by Wiles and Taylor in proving Fermat's last theorem. These results were extended in a series of papers and have been used by Dieulefait, Gee and others in proving results in number theory. In terms of broader impact, one Ph. D. student finshed during this period and were supported by the grant. One other graduate student was partially supported by the grant. The PI also was involved in organizing several conferences including a workshop in Banff and an 80th birthday conference for John Thompson at the University of Cambridge. The PI also gave many lectures on the work supported by this grant and aimed at relaitvely general mathematical audiences at colloquia and conferences. In particular, he gave a plenary lecture at the annual AMS meeting in 2013. He gave a talk at Princeton University for graduate students there. The PI was also involved with editorial duties with five journals including spending some of this time as Managing Editor of the Transactions of the American Mathematical Society.
