Mathematics Colloquium: Hearing the shape of a locally symmetric space, and arithmetic groups

Speaker: Andrei Rapinchuk, University of Virginia

Title: Hearing the shape of a locally symmetric space, and arithmetic groups

Abstract: I will discuss a new form of rigidity which is expected to hold for arbitrary Zariski-dense subgroups of simple algebraic groups. It is based on the consideration of the eigenvalues of elements of a (linear) group rather than its structure, hence has been termed "eigenvalue rigidity." This approach was motivated by the famous question of Mark Kac "Can one hear the shape of a drum?" In a joint work with G. Prasad, we were able to resolve this question for many compact locally symmetric spaces using the new notion of weak commensurability (which is a way of matching the eigenvalues of two matrices) and our detailed analysis of weakly commensurable arithmetic groups. I will spend most of the talk explaining the background, the meaning of and the challenges associated with Kac's question, and then state some of our results. Time permitting, I will also mention some problems in the theory of algebraic groups that the study of eigenvalue rigidity has led to and the recent progress in this direction achieved jointly with V. Chernousov and I. Rapinchuk.

Slides

Time: Friday, September 6, 2019, 3:30-4:20 p.m.

Place: Exploratory Hall, room 4106

Refreshments will be served at 3:00 p.m.