Representation Theory of Groups and Applications

Cornell University, Ithaca NY

Investigators

Abstract

Groups are the mathematical abstraction of the notion of symmetry. Since the beginning of the study of groups 150 years ago, they have proved to be of fundamental importance in an extraordinarily large number of contexts in mathematics and other disciplines such as mathematical physics, crystallography, and cryptography. Because of the extraordinary diversity of possible group structures, we still have much to learn about groups. This research project concerns new approaches to studying groups, including the construction of previously unknown groups and investigation of the detailed structure of some well-known but highly complex groups. It is anticipated that project will lead to new examples of expander graphs, highly-connected sparse graphs widely used in computer science in areas ranging from parallel computation to error-correcting codes to cryptography. This research project concerns topics in the representation theory of groups, with special emphasis on problems related to the cohomology of the automorphism group of the free group. Another emphasis is on questions related to the Kazhdan property T. The investigator aims to use algebraic, combinatorial, geometric, and probabilistic tools to reduce the problems under study to questions in combinatorics and theory of random walks and to apply the results to important open problems in graph theory. The main aims of the project are to study the expansion properties of Cayley graphs and to understand representation theoretic properties of pro-finite and discrete groups. The project involves studying objects central to geometric group theory, including automorphism groups of free groups and mapping class groups. The investigator plans to pursue projects centered on expanders, a field of study that has undergone explosive growth in the past decade. The work will involve mutually beneficial interactions among arithmetic, group theory, and combinatorics.