Kannan Soundararajan

Affiliation: Stanford University

Email: ksound@stanford.edu

TALK 1 ( FOURTH RAMANUJAN COLLOQUIUM ) :
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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