Group Representations and Applications

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The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or of a theoretical construct can be encoded by a group, which itself carries important information about the structure of the object. Representation theory allows one to study groups in a uniform way via their actions on vector spaces, which model common ways groups act in applications. Representation theory has been a central topic in mathematics for more than a century and has important applications in physics and chemistry, particularly in quantum mechanics and the theory of elementary particles. Finite groups and their representations have important applications in coding theory and cryptography, and play an important role in modern computation and digital communications. This project aims to advance understanding in the representation theory of finite groups and in a number of applications. In more detail, this project focuses on several important questions in group representation theory and its applications. It ties together different areas of mathematics, such as finite groups and algebraic groups, permutation groups, probabilistic group theory, group cohomology, combinatorics, vertex operator algebras, and algebraic geometry, with representation theory as the main unifying ingredient. Many of the questions addressed in the project come up naturally in the study of group representations, and others are motivated by applications. The investigator will study several problems along the lines of the global-local principle, including conjectures of McKay, Alperin, Brauer, and others that extend and generalize classical results in the representation theory of finite groups. The investigator also intends to study classification of modular representations of low dimension and to develop a theory of character level, with the goal of establishing strong bounds on character values for finite quasisimple groups. It is anticipated that the results can be applied to study Aschbacher's conjecture on subgroup lattices and the Kollar-Larsen problem on exterior powers, Miyamoto's problem, problems on random walks and anti-concentration, representation varieties of Fuchsian groups, word map distributions and Waring-type problems for quasisimple groups, refinements of Holt's conjecture on second cohomology groups, and representations of quasisimple groups with special properties.