p-Adic Analytic Continuation of Genus 2 Overconvergent Hilbert Eigenforms in the Inert Case

Yichao Tian
Princeton University; Member, School of Mathematics
March 17, 2011

A well known result of Coleman says that p-adic overconvergent (ellitpic) eigenforms of small slope are actually classical modular forms. Now consider an overconvergent p-adic Hilbert eigenform F for a totally real field L. When p is totally split in L, Sasaki has proved a similar result on the classicality of F. In this talk, I will explain how to treat the case when L is a quadratic real field and p is inert in L.

Periods Over Spherical Subgroups: An Extension of Some of the Langlands Conjectures

Yiannis Sakellaridis
Member, School of Mathematics
March 16, 2011

Periods of automorphic forms over spherical subgroups tend to: (1) distinguish images of functorial lifts and (2) give information about L-functions.

This raises the following questions, given a spherical variety X=H\G: Locally, which irreducible representations admit a non-zero H-invariant functional or, equivalently, appear in the space of functions on X? Globally, can the period over H of an automorphic form on G be related to some L-value?

A PRG for Gaussian Polynomial Threshold Functions

Daniel Kane
Harvard University
March 15, 2011

We define a polynomial threshold function to be a function of the form f(x) = sgn(p(x)) for p a polynomial. We discuss some recent techniques for dealing with polynomial threshold functions, particular when evaluated on random Gaussians. We show how to use these ideas to produce a pseudo random generator for degree-d polynomial threshold functions of Gaussians with seed length poly(2^d,log(n),epsilon^{-1}) .

On the Fourier Spectrum of Symmetric Boolean Functions

Amir Shpilka
Technion; on leave at Microsoft Research New England
March 14, 2011

It is well-known that any Boolean function f:{-1,+1}^n \to {-1,+1} can be written uniquely as a polynomial f(x) = \sum_{S subset [n]} f_s \prod_{i in S} x_i. The collection of coefficients (f_S's) this expression are referred to (with good reason) as the Fourier spectrum of f. The Fourier spectrum has played a central role in modern computer science by converting combinatorial and algorithmic questions about f into algebraic or analytic questions about the spectrum.