This project will involve the study of finite and algebraic groups and in particular their actions on linear spaces and varieties. One important aspect of the project is to improve bounds for the sizes of low degree cohomology groups. This should lead to profinite presentations of the finite simple groups with two generators and a very small number of relations. There will also be a study of discrete presentations of the finite simple groups building on earlier work of the proposer and others showing that with the possible exception of one family, every finite simple group has a presentation with at most 50 relations. The project will also consider the notion of adequate representations of finite groups. This is a weakening of a condition used by Taylor and Wiles to prove certain representations are automorphic (and used in the proof of Fermat's last theorem). The idea is to show that many representations do satisfy this condition (and also constructing others that do not). This notion has already been used to great effect by Clozel, Thorne, Gee, Dieulefait and others. Another part of the project will be to attempt to generalize the Tits alternative. The conjecture is that any finitely generated Zariski dense subgroup of a semisimple algebraic group contains a strongly dense free subgroup. Here strongly dense means every nonabelian subgroup is Zariski dense in the algebraic group. Finally, certain problems related to mappings of smooth algebraic curves will be studied using the group theory results.
One of the major motivations for this project has been the major advances in computational group theory. These advances are not just based on improvements in computing power but very innovative programs which use the theory in a crucial way. The presentations mentioned above are already being implemented into MAGMA (a very powerful computational algebra software package) and is used to recognize groups. The results on adequate representations have already been used in some significant advances in showing representations are automorphic and should help in the fundamental Langlands program. The results on strongly dense subgroups have already been used in proving new results on expander graphs. These are graphs that are relatively sparse but are also very highly connected. These have sparked a revolution in computer science in the last decade. The new results will have even more applications to this field. Deep results in group theory have led to major advances in basic problems about bijective polynomials over finite fields (viewed as mappings on a smooth projective curve) and has had applications to cryptography and solved problems over a century old.
