Video Lectures

The loop equation for circulation PDF as functional of the loop shape is derived from the Hopf equation. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity playing the role of Planck's constant. This...

We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions...

We discuss some recent progress on the model-theoretic problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include Castle’s recent solution of the Restricted Trichotomy...

A seminal result in learning theory characterizes the PAC learnability of binary classes through the VC dimension. Extending this characterization to the general multiclass setting has been open since the late 1980s.

We resolve this problem by...

The Markoff equation x2+y2+z2=3xyz, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss (briefly) asymptotics of integer points, and (in some detail) recent...

Our goal is to develop a generic framework for converting modern gradient-descent based optimization algorithms into mechanisms where inputs come from self-interested agents.

We focus on aggregating preferences from n players in a context...

We review results about random walks on groups, discussing results and conjectures relating critical constant for recurrence/transience, growth and Poisson boundary.

Much less is known about behavior of random walks on Schreier graphs. Since any...

For each central charge c∈(0,1], we construct a conformally invariant field which is a measurable function of the local time field L of the Brownian loop soup with intensity c and i.i.d. signs given to each cluster. This field is canonically...

In this talk, we start by reviewing recent results on the dynamics of Reeb vector fields defined by contact forms on three-dimensional manifolds, and then introduce Reeb fields defined by stable Hamiltonian structures. These are more general and...

The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves.  Here we describe a proof of an asymptotic for the second moment of...