Video Lectures

Associated to a star-shaped domain in ℝ2nR2n are two increasing sequences of capacities: the Ekeland-Hofer capacities and the so-called Gutt-Hutchings capacities. I shall recall both constructions and then present the main theorem that they are the...

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we...

Multi-variate residues on Grassmannians G(k,n) and moduli spaces M0,n are ubiquitous in the study of scattering amplitudes; they provide a powerful and essential tool. Amenable theories include the biadjoint scalar, NLSM, Yang-Mills, gravity and N=4...

A classical construction associates a Poincare duality algebra to a homogeneous polynomial on a vector space. This construction was used to give a presentation for cohomology rings of complete smooth toric varieties by Khovanskii and Pukhlikov and...

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of...

In this talk I will try to motivate the interest and some of the mystery in the Ramsey numbers R(k), which are fundamental quantities in combinatorics. I will go on to discuss some recent progress on our understanding of these numbers and make some...

The classical Serrin’s overdetermined theorem states that a C^2 bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of...

Relative symplectic cohomology, an invariant of subsets in a symplectic manifold, was recently introduced by Varolgunes. In this talk, I will present a generalization of this invariant to pairs of subsets, which shares similar properties with the...

The well-known Zig-Zag product and related graph operators, like derandomized squaring, are fundamentally combinatorial in nature. Classical bounds on their behavior often rely on a mix of combinatorics and linear algebra. However, these traditional...

Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the...