Quantum Unique Ergodicity and Number Theory
Quantum Unique Ergodicity and Number Theory

K. Soundararajan, Stanford University
Fine Hall 314
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 halfplane 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.