# Video Lectures

et K be a convex body in Rn. In some cases (say when K is a cube), we can tile Rn using translates of K. However, in general (say when K is a ball) this is impossible. Nevertheless, we show that one can always cover space "smoothly" using translates...

Following Birkhoff's proof of the Pointwise Ergodic Theorem, it has been studied whether convergence still holds along various subsequences. In 2020, Bergelson and Richter showed that under the additional assumption of unique ergodicity, pointwise...

Can you hear the shape of LQG? We obtain a Weyl law for the eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows linearly with n, with the proportionality constant given by the Liouville area of the domain (times a certain...

How long does it take for a random walk to cover all the vertices of a graph?

And what is the structure of the uncovered set (the set of points not yet visited by the walk) close to the cover time?

We completely characterize the...

Graph Crossing Number is a fundamental and extensively studied problem with wide ranging applications. In this problem, the goal is to draw an input graph G in the plane so as to minimize the number of crossings between the images of its edges. The...

Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel--Voevodsky cannot be employed in studying the...

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While the first results in this direction were obtained in the 1960-ies, various facets of this question remain largely open. I will...

Determining the complexity of matrix multiplication is a fundamental problem of theoretical computer science. It is popularly conjectured that matrices can be multiplied in nearly-linear time. If true, this conjecture would yield similarly-fast...

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...

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...