Video Lectures

The method of hypergraph containers is a widely-applicable technique in probabilistic combinatorics. The method enables one to gain control over the independent sets of many `interesting' hypergraphs by exploiting the fact that these sets exhibit a...

Structures which minimise area appear in numerous geometric contexts often related to degeneration phenomena. In turn, in many situations these structures also reflect the ambient geometry in some way (they are ‘calibrated’) and so they may provide...

Since the beginning of the subject, it has been speculated that Gromov-Witten invariants should admit refinements in complex cobordism. I will propose a resolution of this question based on joint work-in-progress with Abouzaid, building on recent...

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...