Video Lectures

About 20 years ago, Gurvits developed the notion of polynomial capacity to give a simple proof of the famous van der Waerden lower bound on the permanent of a doubly stochastic matrix. Since then, similar techniques have led to various other...

This talk examines the historical origin of the attempts to understand, control, and use noise in modern times. Today, the concept of noise is employed to characterize random fluctuations in general. Before the twentieth century, however, noise only...

We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the...

Let G be a compact Lie group acting on a closed manifold M. Partially motivated by work of Uhlenbeck (1976), we explore the generic properties of Laplace eigenfunctions associated to G-invariant metrics on M. We find that, in the case where 𝕋 is a...

Despite the apparent complexity of galaxies, some of their most basic properties (mass M, binding energy E, angular momentum J, etc) obey simple power-law scaling relations. Such relations provide some of the clearest insights into galaxy formation...

I will introduce Apollonian circle packings, and describe the local-global conjecture, which predicts the set of curvatures of circles occurring in a packing. I will then describe reciprocity obstructions, a phenomenon rooted in reciprocity laws...

How did dogs become scientific beings? Although it is now widely understood that a Border Collie or a Golden Retriever is a cognitively complex creature meriting advanced nutrition and monthly pharmaceuticals, this way of thinking about dogs was the...

I will discuss escape of mass estimates for SL(d,ℝ)-horospheres embedded in the space of affine lattices, which depend on the Diophantine properties of the shortest affine lattice vector. These estimates can be used, in conjunction with Ratner's...

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed...

Agreement testing (aka direct product testing), checks if consistent local information reveals global structure. Beyond its theoretical connections to probabilistic checkable proofs (PCPs), constructing agreement testers is a fundamental...