# Mathematics Colloquium

#### Quantum Unique Ergodicity and Number Theory

**Speaker:**
Kannan Soundararajan, Stanford

**Location:**
Warren Weaver Hall 1302

**Date:**
Monday, October 12, 2009, 3:45 p.m.

**Synopsis:**

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper halfâplane by a discrete “arithmetic” subgroup of \(SL_2(R)\) (for example, \(SL_2(Z)\), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equidistributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.