## The Graph Removal Lemma

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^{h}) copies of H can be made H-free by removing o(n^{2}) 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.

## Edward T. Cone Concert Talk

Listen to members of the Borromeo String Quartet discuss music with Institute Artist-in-Residence Derek Bermel.

## Ground States of the 2D Edwards-Anderson Spin Glass

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).

## Lecture 4

## A Satake Isomorphism $mod.p$

## Local-Global Compatibility in the p-Adic Langlands Progra for GL(2) over Q

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.

## Fourier Spectrum of Polynomials Over Finite Fields

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.

## 3/2 Firefighters Are Not Enough

## The Topology of Restricted Partition Posets

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