Video Lectures

I will start by introducing and motivating the (two-component) Coulomb gas on the d-dimensional lattice Zd. I will then present some puzzling properties of the fluctuations of this Coulomb gas. The connection of this model with integer-valued fields...

The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed...

Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave...

Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave...

Finding the smallest integer N=ES_d(n) such that in every configuration of N points in R^d in general position there exist n points in convex position is one of the most classical problems in extremal combinatorics, known as the Erdős-Szekeres...

A Locally Decodable Codes (LDC) is an error correcting code which allows the decoding of a single message symbol from a few queries to a possibly corrupted encoding.  LDCs can be constructed, for example, from low degree polynomials over a finite...

Over a high-dimensional vector space Fnpover a fixed finite field Fp, the inverse theorem for the Gowers norm asserts that a bounded function f on this space that has a large Gowers Uk+1 norm, must necessarily correlate with a phase polynomial e(P)...

We will discuss stochastic quantization i.e. Langevin dynamic of the Yang-Mills model on two and three dimensional torus. This is described by a Lie algebra valued stochastic PDE driven by space-time white noise. We construct (local) solution to...

We show that for any closed symplectic manifold, the number of 1-periodic orbits of any non-degenerate Hamiltonian is bounded from below by a version of total Betti number over Z, which takes account of torsions of all characteristics. The proof...

I will review some recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to (C2) class at all...