## The James Construction and $pi_4(S^3)$

## Norm Convergence of Nonconventional Ergodic Averages

Consider a group of measure preserving transformations acting on a probability space. The limiting behavior of the nonconventional ergodic averages associated with this action has been the subject of much attention since the work of Furstenberg on Szemerédi’s theorem. We will discuss this problem, and how to establish the convergence of these averages whenever the group is nilpotent.

## Special Lectures in Analysis/Number Theory

## A Zero-Density Approach to Smooth Numbers

A number is said to be $y$-smooth if all of its prime factors are less than $y$. Such numbers appear in many places throughout analytic and combinatorial number theory, and much work has been done to investigate their distribution.

## Mean Values of L-Functions for the Hyperelliptic Ensemble

## Statistics of the Zeros of the Zeta Function: Mesoscopic and Macroscopic Phenomena

We review the well known microscopic correspondence between random zeros of the Riemann zeta-function and the eigenvalues of random matrices, and discuss evidence that this correspondence

extends to larger mesoscopic collections of zeros or eigenvalues. In addition, we discuss interesting phenomena that appears in the statistics of even larger macroscopic collections of zeros. The terms microscopic, mesoscopic, and macroscopic are from random matrix theory and will be defined in the talk.

## Natural Models of Type Theory

## Dimers and Integrability

This is joint work with A. B. Goncharov. To any convex integer polygon we associate a Poisson variety, which is essentially the moduli space of connections on line bundles on (certain) bipartite graphs on a torus. There is an underlying integrable Hamiltonian system whose Hamiltonians are weighted sums of dimer covers.