### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM

SEPTEMBER 6, 2019

**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.

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491