Cohomology and Representations of Finite and Algebraic Groups with Applications

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This project will involve the study of finite and algebraic groups and in particular their actions on linear spaces and varieties. Groups are one of the fundamental tools in mathematics and arise in many areas including analysis, geometry and number theory as well as in the study of symmetries in chemistry and physics. The classification of finite simple groups was completed in 2006 and has led to a revolution in using group theory to study other fields. The classification basically says that the finite simple groups are analogs of the simple Lie groups and so to understand them, one must study simple Lie and algebraic groups. The best way to understand and use group theory is to study the action of groups on different objects. One aspect of this project is to understand groups acting on Riemann surfaces (and their analog over finite fields). This will lead to a new fundamental understanding of basic objects including rational functions and should lead to advances in cryptography and fundamental problems in number theory. The utility of group theory has also been greatly expanded due to advances in computation. Another aspect of this project is to find useful presentations of the finite simple groups which will lead to more computational efficiency. A third important problem addressed in this project is to greatly generalize what is called the Tits alternative. This will lead to results showing the existence (and construction) of expander graphs. These are graphs that are highly connected relative to the number of edges in them. This has been of great importance in computer science. Graduate students will be trained through research. In particular, we plan to study the problem of producing strongly dense subgroups of semisimple algebraic groups and proving a generalization of the Tits alternative. This will give some new results about superstrong approximation in number theory and results on expander graphs. Earlier results of the PI, with Breuillard, Green, and Tao, will be generalized using new stronger methods. We also want to prove the conjecture that every finite simple group has a presentation with two generators and at most four relations. This should lead to advances in computational number theory. 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. Another goal of the project is to completely classify monodromy groups of coverings of low genus Riemann surfaces leading to fundamental breakthroughs in number theory and also to classify monodromy groups of mappings from generic Riemann surfaces (first studied in Zariski's thesis). Finally, we want to classify generic stabilizers for simple algebraic groups in irreducible linear representations. This has been done in characteristic zero but new ideas are required in positive characteristic. This will have consequences for essential dimension and some special cases will fit into the program of Bhargava to solve interesting classification problems of algebraic families.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.