Analytic number theory and higher rank groups (Nice poster. )

Photos from the conference are available at Jeff Mozzochi's website. Thanks Jeff!

Titles and abstracts

Bhargava: "Class number-regulator summation formulae for cubic fields."

Bump: "Crystals, Gauss Sums and Multiple Dirichlet Series." Notes

Weyl Group multiple Dirichlet series are Dirichlet series in several complex variables whose coefficients have a twisted multiplicativity that reduces their description to those of their p-parts (though they are not Euler products). The p-parts themselves are extremely interesting. I will describe joint work with Brubaker, Chinta, Friedberg, Gunnells and Hoffstein with the general aim of showing that one may describe such a p-part as a sum over a crystal. Crystals are combinatorial objects introduced by Kashiwara in the context of quantum groups. They are colored directed graphs whose vertices may be parametrized by Young Tableaux or Gelfand-Tsetlin patterns. As a biproduct we will discuss deformations of the Weyl character formula and interesting phenomena involving metaplectic Whittaker functions.

Farmer: "Computing GL(3) L-functions."

I will describe current methods for computing Maass forms on GL(3) and the associated L-functions. Included will be a list of the first several eigenvalues of the Laplacian for SL(3,Z).

Gan : "Non-vanishing of L-values and theta lifts."

Goldfeld: "The Voronoi summation formula for GL(n)." This talk will be about the Voronoi formula, beginning with the classical one and its derivation using Mellin inversion. It will then be shown how to generalize the proof to obtain a Voronoi formula on GL(n). This is joint work with Xiaoqing Li.

Lapid: " Arthur's non-invariant trace formula in higher rank and applications." ( Notes)

One of the simplest applications of the trace formula (and the original one due to Selberg) is to Weyl's law. I'll explain what is needed for the higher rank case, and in particular explicate the spectral side of Arthur's trace formula. Joint work with Tobias Finis and Werner Muller.

Li : "Triple L-functions in the splitting case."

Lindenstrauss: " Effective equidistribution and density on the torus."
Given an irrational point x in R/Z, Furstenberg has shown that {2^n 3^k x mod 1 : n,k>=1} is dense in R/Z. How fast does this happen? One can also consider the related problem of how an orbit of a rational point is distributed in R/Z. Now consider the n torus T^n = R^n / Z^n and let A,B in SL(n,Z). Suppose e.g. that A and B generate a Zariski dense subgroup of SL(n). Perform a random walk on T^n by randomly applying A and B to x (and iterating this procedure). If x is irrational this random walk becomes equidistributed (in a quantitative way). This answers a question of Guivarch as well as a question of Furstenberg about stiffness of such actions (a notion I will define in my talk), and is in contrast to the behavior of 2^n 3^k x on R/Z. I will eplain the two problems and how they relate, as well as their relation to recent results in arithmetic combinatorics, specifically the sum product phenomenon. Based on joint works with Bourgain, Michel, Venkatesh and Bourgain Furman and Mozes.

Michel (lecture series): "Distribution of CM points, subconvexity, and generalizations."

Michel (research lecture): Uniform subconvexity for L-functions on GL(2) and global periods.

Miller (lecture series) (Notes available from Steve's webpage) "Automorphic distributions and analytic properties of L-functions."

This mini-course consists of three lectures on the analytic properties of L-functions, using the technique of automorphic distributions. The course will begin with an overview of GL(2) L-functions and some prominent conjectures and theorems about them. We will then discuss automorphic forms on GL(n) and Langlands' conjectures. Finally, we will describe the technique of changing vectors in an automorphic representation, and some analytic facts that can be obtained from it.

Problem session: Steven J. Miller's notes, Rob Rhoades' notes.

Reception: To be eaten and drunk.

Reznikov: "On Rankin-Selberg identities."

Sarnak: " "The Spectral Gap for Gamma\G" (notes to be posted)

Skinner: `` Some analytic problems from an algebraic number theorist.'' Unedited notes taken by A. Venkatesh; not verified by the speaker.

In keeping with the request of the conference organizers I will discuss some questions regarding automorphic forms and their L-values that have an analytic flavor (or at least their solutions are likely to) and that are of interest to those who - like the speaker - are motivated by algebraic questions.

Venkatesh: "Some local computations related to bounds for L-functions."