Group Representations and Applications

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This proposal focuses on several important problems in representation theory of finite groups and its applications. It ties together different areas of mathematics, such as finite groups and algebraic groups, finite permutation group theory, group cohomology, combinatorics and finite geometry, algebraic geometry, and string theory, with the main unifying ingredient being the representation theory. Many of the problems addressed in the proposal come up naturally -- some long-standing and play a central role -- in the group representation theory, and others are motivated by various important applications. The PI will study several problems along the lines of the local-global principle, including Brauer's height zero conjecture and some conjectures concerning rationality and divisibility properties of complex and Brauer characters of finite groups. The PI will also continue his long-term project to classify cross characteristic representations of finite groups of Lie type of low dimension. He will then apply his results to seek significant progress on a number of applications, including the Kollar-Larsen Problem on linear groups with elements of bounded age (with applications in algebraic geometry and string theory), the Ore Conjecture on commutators in simple groups, a strengthening of Holt's Conjecture on the dimension of the second cohomology group for finite groups and their presentations, and representations of finite quasisimple groups with special properties (with application in the subgroup structure of finite simple groups).The main area of research in this proposal is the group representation theory. The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or science are encoded by a group, and this group carries a lot of important information about the structure of the object itself. The representation theory allows one to study groups via their actions on vector spaces which model the ways they arise in the real world. It has fascinated mathematicians for more than a century and had many important applications in physics and chemistry, particularly in quantum mechanics and in the theory of elementary particles. Finite groups and their representations have already proved valuable in coding theory and cryptography, and are expected to continue to play an important role in the modern world of computers and digital communications. The investigator's research will lead to important advances in understanding the representation theory of finite groups and help achieve significant progress in a number of its applications.