# Video Lectures

### Surface Law and Charge Rigidity for the Coulomb Gas on Zd

Christophe Garban

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...

### Counting Low Degree Number Fields with Almost Prescribed Successive Minima

Sameera Vemulapalli

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...

### The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together II

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...

### The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together

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...

### The Erdős-Szekeres problem in three (and higher) dimensions

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...

### Locally Decodable Codes

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...

### Non-measurability of the inverse theorem for the Gowers norms

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)...

### Langevin Dynamics of Yang-Mills in 2D and 3D

Hao Shen

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...

### Arnold Conjecture Over Integers

Shaoyun Bai

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...

### Regularity of Vortex and SQG Patches

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...