# Video Lectures

The Stefan problem, dating back to the XIX century, aims to describe the evolution of a solid-liquid interface, typically a block of ice melting in water. A celebrated work of Luis Caffarelli from the 1970's established that the ice-water interface...

Physics inspired mathematics helps us understand the random evolution of Markov processes. For example, the Kolmogorov forward and backward differential equations that govern the dynamics of Markov transition probabilities are analogous to the...

We study the vortex sheet solutions of the Euler equation, which correspond to the tangent discontinuity of the velocity field.

We observe that stationary flows correspond to the Hamiltonian's minimization by the tangent discontinuity density ?...

I will explain that incompressible Navier-Stokes is the wrong equation to describe turbulence in low Mach number molecular fluids because it neglects the effects of thermal noise. There should, in fact, be strong effects of thermal noise throughout...

We present an elementary summary of known results, and open questions, on scaling problems in hydrodynamic turbulence in three dimensions. The goal is to provide some background for the two talks to follow, and summarize Victor Yakhot's theory...

I will explain the construction of a functor from the exact symplectic cobordism category to a totally ordered set, which measures the complexity of the contact structure. Those invariants are derived from a bi-Lie infinity formalism of the rational...

Let ?? be the Liouville function and P(x)P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence ?(P(n))?(P(n)) changes sign infinitely often. We present a solution to this problem for...

Motivated by a formal similarity between the Hard Lefschetz theorem and the geometric Satake equivalence we study vector spaces that are graded by a weight lattice and are endowed with linear operators in simple root directions. We allow field...

In his seminal paper from 1973, Garland introduced a machinery for proving vanishing of group cohomology for groups acting on Bruhat-Tits buildings. This machinery, known today as “Garland’s method”, had several applications as a tool for proving...

Expander graphs in general, and Ramanujan graphs in particular, have played an important role in computer science and pure mathematics in the last four decades. In recent years the area of high dimensional expanders (i.e. simplical complexes with...