# School of Mathematics

## On some consequences of exponential mixing

In this talk I will discuss some definitions of exponential mixing and other rates of mixing and discuss some of its consequences.

## Canonical integral models of Shimura varieties

We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.

## Unique and 2:2 Games, Grassmannians, and Expansion

The unique games conjecture gives a very strong PCP theorem, which, if true, leads to a clean understanding of a broad family of approximation problems. We will describe recent progress on the conjecture and how certain type of expansion and hypercontractivity of the Grassmannian complex plays a key role.

## Nonconvex Minimax Optimization

Minimax optimization, especially in its general nonconvex formulation, has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs) and adversarial training. It brings a series of unique challenges in addition to those that already persist in nonconvex minimization problems. This talk will cover a set of new phenomena, open problems, and recent results in this emerging field.

## High Dimensional Expansion and Error Correcting Codes

High dimensional expansion generalizes edge and spectral expansion in graphs to higher dimensional hypergraphs or simplicial complexes. Unlike for graphs, it is exceptionally rare for a high dimensional complex to be both sparse and expanding. The only known such expanders are number-theoretic or group-theoretic.

## Constraint Satisfaction Problems and Probabilistic Combinatorics I

The tasks of finding and randomly sampling solutions of constraint satisfaction problems over discrete variable sets arise naturally in a wide variety of areas, among them artificial intelligence, bioinformatics and combinatorics, and further have deep connections to statistical physics.

## The PCP theorem

## The singular set in the fully nonlinear obstacle problem

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\eps}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

## An isoperimetric inequality for the Hamming cube and some consequences

I will introduce an isoperimetric inequality for the Hamming cube and some of its applications. The applications include a “stability” version of Harper’s edge-isoperimetric inequality, which was first proved by Friedgut, Kalai and Naor for half cubes, and later by Ellis for subsets of any size. Our inequality also plays a key role in a recent result on the asymptotic number of maximal independent sets in the cube.

This is joint work with Jeff Kahn.