## Quantum unique ergodicity and number theory

Abstract: 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 equi-distributed. 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.