## On the Structure of Cubic and Quartic Polynomials

In our work we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results.

Elad Haramaty

Technion

November 1, 2010

In our work we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results.

Elena Mantovan

California Institute of Technology; Member, School of Mathematics

November 1, 2010

I will introduce Shimura varieties and discuss the role they play in the conjectural relashionship between Galois representations and automorphic forms. I will explain what is meant by a geometric realization of Langlands correspondences, and how the geometry of Shimura varieties and their local models conjecturally explains many aspects of these correspondences. This talk is intended as an introduction for non-number theorists to an approach to Langlands conjectures via arithmetic algebraic geometry.

Shachar Lovett

Institute for Advanced Study

November 2, 2010

Let $f(x_1,...,x_n)$ be a low degree polynomial over $F_p$. I will prove that there always exists a small set $S$ of variables, such that `most` Fourier coefficients of $f$ contain some variable from the set $S$. As an application, we will get a derandomized sampling of elements in $F_p^n$ which `look uniform` to $f$.

The talk will be self contained, even though in spirit it is a continuation of my previous talk on pseudorandom generators for $CC0[p]$. Based on joint work with Amir Shpilka and Partha Mukhopadhyay.

Rani Hod

Tel Aviv University

November 2, 2010

Richard Ehrenborg

University of Kentucky; Member, School of Mathematics

November 2, 2010

The d-divisible partition lattice is the collection of all partitions of an n-element set where each block size is divisible by d. Stanley showed that the Mobius

Matthew Emerton

Northwestern University

November 2, 2010

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine—Mazur conjecture, and to a conjecture of Kisin.

Matthew Emerton

Northwestern University

November 3, 2010

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine--Mazur conjecture, and to a conjecture of Kisin.

Pierre Colmez

National Center for Scientific Research; Member, School of Mathematics

November 4, 2010

Guy Henniart

University of Paris-Sud

November 4, 2010

Michael Damron

Princeton University

November 5, 2010

I will discuss the problem of determining the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass model on $Z^D$ for $D \geq 2$. There are no complete results for this problem even in $D=2$. I will focus on this case and explain recent results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP).