School of Mathematics

The Graph Removal Lemma

Jacob Fox
Massachusetts Institute of Technology
November 8, 2010

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(nh) copies of H can be made H-free by removing o(n2) edges. We give a new proof which avoids Szemeredi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.

Ground States of the 2D Edwards-Anderson Spin Glass

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).

Fourier Spectrum of Polynomials Over Finite Fields

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.

Shimura Varieties, Local Models and Geometric Realizations of Langlands Correspondences

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.

Semiclassical Eigenfunction Estimates

Melissa Tacy
Institute for Advanced Study
October 29, 2010

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

Concentration phenomena for Laplacian eigenfunctions can be studied by obtaining estimates for their $L^{p}$ growth. By considering eigenfunctions as quasimodes (approximate eigenfunctions) within the semiclassical framework we can extend such estimates to a more general class of semiclassical operators. This talk will focus on $L^{p}$ estimates for quasimodes restricted to hypersurfaces and the links between such estimates and properties of classical flow.

The Mathematical Truth

Enrico Bombieri
Institute for Advanced Study
October 29, 2010

In this lecture, Enrico Bombieri, IBM von Neumann Professor in the School of

Mathematics, attempts to give an idea of the numerous different notions of truth in mathematics. Using accessible examples, he explains the difference between truth, proof, and verification. Bombieri, one of the world’s leading authorities on number theory and analysis, was awarded the Fields Medal in 1974 for his work on the large sieve and its application to the distribution of prime numbers. Some of his work has potential practical applications to cryptography and security of data transmission and identification.