Kannan Soundararajan

Affiliation: Stanford University

Email: ksound@stanford.edu

Quantum Unique Ergodicity and Number Theory

Wednesday, March 24 in FAB 103 at 4:00pm

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 SL2(R) (for example, SL2(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. I will not assume any familiarity with these topics, and the talk should be accessible to graduate students.

TALK 2: Mean values of multiplicative functions and applications

Thursday, March 25 in LIT 339 at 10:40am

Abstract: The pioneering results of Wirsing and Halasz describe the situations when the mean-value of a multiplicative function can be large. Understanding the structure of such multiplicative functions has proved useful in applications to the Polya-Vinogradov inequality, weak subconvexity bounds for L-functions, and to the quantum unique ergodicity problem. I will try and explain some of these results and applications.

TALK 3: The distribution of values of zeta and L-functions

Friday, March 26 in LIT 339 at 10:40am

Abstract: I will survey what is known and expected about the distribution of values of L-functions. In particular, I will try to explain probabilistic models that describe the behavior of these functions. At the center of the critical strip, the probabilistic models arise from random matrix theory, and I will discuss the Keating-Snaith conjectures for moments of L-functions, and recent progress.

Last update made Wed Mar 17 12:02:49 EDT 2010.
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