Representation Theory of Groups and Applications

Cornell University, Ithaca NY

Investigators

Abstract

The PI plans to continue working on various topics in the representation theory of groups, with special emphasis on problems related to the Kazhdan property T or pro-finite groups. The main aims of the project are to study the expansion properties of Cayley graphs and to understand various representation theoretic properties of pro-finite and discrete groups. The project also involves studying objects central to the geometric group theory, like automorphism groups of free groups. The PI will use algebraic, combinatorial, geometric and probabilistic tools to reduce the proposed problems to questions in combinatorics and theory of random walks and apply the results to important open problems in graph theory. As a byproduct of the project, questions in these areas are likely be answered. Expanders are highly-connected sparse graphs widely used in Computer Science, in areas ranging from parallel computation to cryptography. Margulis gave the first explicit construction of expanders, following Pinsker's observation that random sparse graphs are expanders. This construction was improved in the eighties by Margulis, Lubotzky, Phillips and Sarnak who constructed Ramanujan graphs (optimal expanders) using deep number-theoretic results. In the ensuing years the scope and depth of applications of expanders has increased -- over the past decade several completely new and unexpected lines of development have emerged and the field has undergone explosive growth. The principal investigator plans to pursue projects centered on expanders and involving mutually beneficial interactions between arithmetic, group theory and combinatorics.