Representations of Finite Groups and Applications

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The main area of research in this proposal is the representation theory of finite groups. 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, which carries a lot of important information about the structure of the object itself. Group 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 had many important applications in quantum mechanics, the theory of elementary particles, coding theory, and cryptography, and is expected to continue to play an important role in the modern world of computers and digital communications. This research will contribute to advances in the understanding and applications of representation theory of finite groups. Student involvement will be a scientifically important component of the project.This project focuses on several problems in the representation theory of finite groups and its applications. Many of these problems come up naturally in the group representation theory. Others are motivated by various applications. The PI will study several global-local problems, including the conjectures of McKay, Alperin, and Brauer, and some other conjectures which generalize classical results in representation theory of finite groups. The PI will also continue his long-term project to develop a theory of character level and to establish strong bounds on character values of finite groups of Lie type, and to classify modular representations of finite quasisimple groups of low dimension or with special properties. He will then apply the results to achieve significant progress on a number of applications, including local systems and their monodromy groups, Aschbacher's conjecture on subgroup lattices, random walks on Cayley and McKay graphs and representation varieties of Fuchsian groups, word map distributions and Thompson's conjecture for simple groups, Miyamoto's problem with applications to vertex operator algebras, and bounds on cohomology groups and presentations of finite quasisimple groups.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.