Video Lectures

(This lecture will be self-contained.) In high dimensions, what does it look like when we take the intersection of a set of random half-spaces with either the sphere or the Hamming cube? This is one phrasing of the so-called perceptron problem...

Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for...

The talk introduces the Kazhdan-Lusztig categories of representations of affine Lie algebras

(This lecture is related to the preceding lecture, but I will try to make it self-contained as much as possible.) In this lecture I will elaborate on some of the existing mathematical approaches to the study of random CSPs, particularly involving...

In the mid 80's, Bourgain asked the following question: Is there a universal constant c>0c>0 such that every compact convex set KK in RnRn, of unit volume, must contain a slice (namely, a set of the form K∩HK∩H for some affine hyperplane HH) whose...

Consider a point particle flying freely on the torus and elastically bouncing back from the boundary of fixed smooth convex obstacles. This is the celebrated Sinai billiard, a rare example of a deterministic dynamical system where rigorous results...

I will describe recent progress in determination of asymptotic behavior in random constraint satisfaction problems, including the independent set problem on random graphs, random regular NAE-SAT, and random SAT. The results include sharp phase...

What is the "right" topological invariant of a large point cloud XX? Prior research has focused on estimating the full persistence diagram of XX, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic...

Histories of ancient cultures often present the image of clearly recognizable peoples. Those who are centered in historical canons—Egyptians, Greeks, Romans—are often subject to the most longstanding and unyielding historical expectations. In this...

Conformal Field Theories (CFTs) have rich dynamics in heavy states. We describe the constraints due to spontaneously broken boost and dilatation symmetries in such states. The spontaneously broken boost symmetries require the existence of new low...