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 half‐plane 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.