## Mathematical Physics and Harmonic Analysis Seminar

**Date:** December 8, 2017

**Time:** 1:50PM - 2:50PM

**Location:** BLOC 628

**Speaker:** Naser Talebi Zadeh, University of Wisconsin-Madison

**Title:** *QUANTUM CHAOS ON RANDOM CAYLEY GRAPHS OF SL2 [Z/pZ]*

**Abstract:** We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs i.e. the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL2[Z/pZ] and the explicit LPS Ramanujan projective graphs of P1(Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.