Video Lectures

It is well-known that any Boolean function f:{-1,+1}^n \to {-1,+1} can be written uniquely as a polynomial f(x) = \sum_{S subset [n]} f_s \prod_{i in S} x_i. The collection of coefficients (f_S's) this expression are referred to (with good reason)...

I'll talk on work in progress on algebraic and analytic geometry over the field of one element F_1. This work originates in non-Archimedean analytic geometry as a result of a search for appropriate framework for so called skeletons of analytic...

We attach Galois representations to automorphic representations on unitary groups whose weight (=component at infinity) is a holomorphic limit of discrete series. The main innovation is a new construction of congruences, using the Hasse Invariant...

Non-relativization of complexity issues can be interpreted as giving evidence that these issues cannot be resolved by “black-box” techniques. We show that the assumption $DistNP \subseteq AvgP$ does not imply that $NP\subseteq BPP$ by...

I will introduce two basic problems in random geometry. A self-avoiding walk is a sequence of steps in a d-dimensional lattice with no self-intersections. If branching is allowed, it is called a branched polymer. Using supersymmetry, one can map...

We show a (3/2-\epsilon)-approximation algorithm for the graphical traveling salesman problem where the goal is to find a shortest tour in an unweighted graph G. This is a special case of the metric traveling salesman problem when the underlying...

In the early 80's, Shimura made a precise conjecture relating Petersson inner products of arithmetic automorphic forms on quaternion algebras over totally real fields, up to algebraic factors. This conjecture (which is a consequence of the Tate...

Let G be a connected reductive group over Q such that G(R) has discrete series representations. I will report on some statistical results on the Satake parameters (w.r.t. Sato-Tate distributions) and low-lying zeros of L-functions for families of...