## Erdos Distinct Distance Problem in the Plane

Erdos conjectured that N points in the plane determine at least c N (log N)^{-1/2} different distances. Building on work of Elekes-Sharir, Nets Katz and I showed that the number of distances is at least c N (log N)^{-1} . (Previous estimates had lower bounds like N^{.86}.)

## Colouring Tournaments

A ``tournament'' is a digraph obtained from a complete graph by directing its edges, and ``colouring'' a tournament means partitioning its vertex set into acyclic subsets (``acyclic'' means the subdigraph induced on the subset has no directed cycles). This concept is quite like that for graph-colouring, but different. For instance, there are some tournaments H such that every tournament not containing H as a subdigraph has bounded chromatic number. We call them ``heroes''; for example, all tournaments with at most four vertices are heroes.

## Uniform Well-Posedness and Inviscid Limit for the Benjamin-Ono-Burgers Equation

We prove that the Cauchy problem for the Benjamin-Ono-Burgers equation is uniformly globally well-posed in H^1 for all "\epsilon\in [0,1]". Moreover, we show that for any T>0 the solution converges in C([0,T]:H^1) to that of Benjamin-Ono equation as "\epsilon --> 0". Our results give a new proof for the global well-posedness of the BO equation in H^1(R) without using gauge transform, which was first obtained by Tao using gauge transform, and also solve the problem about the inviscid limit behavior in H1.

## Edward T. Cone Concert Talk

As a part of the Mallet Madness concerts, percussionists Lisa Pegher, Joe Locke, and Bernard Woma join in conversation with Institute Artist-in-Residence Derek Bermel and composer Paul Lansky

## Univalent Foundations of Mathematics

The correspondence between homotopy types and higher categorical analogs of groupoids which was first conjectured by Alexander Grothendieck naturally leads to a view of mathematics where sets are used to parametrize collections of objects without "internal structure" while collections of objects with "internal structure" are parametrized by more general homotopy types. Univalent Foundations are based on the combination of this view with the discovery that it is possible to directly formalize reasoning about homotopy types using Martin-Lof type theories.

## A Classical Approximation Point of View on Some Results in the Spectral Theory of Jacobi Matrices

Deift--Simon and Poltoratskii--Remling proved upper bounds on the measure of the absolutely continuous spectrum of Jacobi matrices. Using methods of classical approximation theory, we give a new proof of their results, and generalize them in several ways. First, we prove a sharper inequality taking the distribution of the values of the potential into account. Second, we prove a generalization of a "local" inequality of Deift--Simon to the non-ergodic setting. Based on joint work with Sasha Sodin

## Understanding Area Preserving Disk Maps Through Holomorphic Curves

## Parahoric Subgroups and Supercuspidal Representations of p-Adic groups

This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be.