Video Lectures

Expanders are highly connected sparse graphs. Simplicial complexes are a natural generalization of graphs to higher dimension, and the notions of connectedness and expansion turn out to have interesting analogues, which relate to the homology and...

Consider a rational homogeneous variety \(X\). (For example, take \(X\) to be the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\)-planes in complex \(n\)-space.) The Schubert classes of \(X\) form a free additive basis of the integral homology of \(X\)...

In this talk, I'll describe connections between the unique games conjecture (or more precisely, the closely relatedly problem of small-set expansion) and the quantum separability problem. Remarkably, not only are the problems related, but the...

We will describe formulas for the asymmetric simple exclusion process (ASEP) starting from half-flat and flat initial data. The formulas are for the exponential moments of the height function associated with ASEP. They lead to explicit formulas for...

We will discuss space and parallel complexity, ranging from some classical results which motivated the study, to some recent results concerning combinatorial lower bounds in restricted settings. We will highlight some of their connections to boolean...

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even \(n\), there exists an explicit bijection \(f\) from the \(n\)-dimensional Boolean cube to the Hamming ball of...

Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series convergent in a neighborhood of zero in the complex plane. Every scheme \(\cal X\) of finite type over \(R\) defines a complex analytic space \({\cal X}^h\) over an open disc \(D\) of small...

The stochastic Burgers equation (equivalent to the one-dimensional KPZ equation) is a hyperbolic conservation law with random currents. In applications, one often has to deal with several conservation laws, a little explored case. We discuss...

In several naturally occurring (infinite) point processes, we show that the number (and other statistical properties) of the points inside a finite domain are determined, almost surely, by the point configuration outside the domain. This curious...