This proposal focuses on several important problems in representation theory of finite groups and its applications. Many of these problems come up naturally -- some long-standing and playing a central role -- in group representation theory, and others are motivated by various applications. The proposal ties together different areas of mathematics, such as finite groups and algebraic groups, finite permutation group theory, group cohomology, combinatorics, operator algebras, and algebraic geometry, with the main unifying ingredient being the representation theory. The PI will study several problems along the lines of the local-global principle, including the Alperin weight conjecture, Brauer's height zero conjecture, and some further 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 modular representations of finite quasisimple groups of low dimension. He will then apply his results to achieve significant progress on a number of applications, including Waring-type problems for quasisimple groups, Aschbacher's conjecture on subgroup lattices, the Kollar-Larsen problem on exterior powers (with application in algebraic geometry), and the Guralnick-Holt conjecture on second cohomology groups 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 group representation theory. The concept of a group in mathematics grew out ofthe 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 has 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.
