Video Lectures

Let C be a closed surface and Σ⊂T*C a real exact Lagrangian surface associated to a spectral curve. In this talk we will first try to explain the context of this work (e.g., Higgs bundles and spectral curves). We then construct a homomorphism from...

Jacobi's theta function Θ(q):=1+2q+2q4+2q9+…, and more generally the theta functions associated to positive-definite quadratic forms, have the property that they are modular forms of half-integral weight. The usual proof of this fact is completely...

The Pila–Zannier strategy has proved to be a particularly fruitful approach: first applied to reprove the Manin–Mumford conjecture, it was later exploited decisively in proofs of the André–Oort conjecture. Crucially relying on a recent counting...

Plasmas exhibit a wide range of behaviors across fluid and kinetic regimes, governing key processes in diverse laboratory, space, and astrophysical environments. Understanding these systems requires numerical models with varying levels of physical...

Rank 1 transformations are a long studied, flexible class of measure preserving transformations. King proved a variety of deep results about rank 1 rank transformations including: A) A rank 1 transformation’s centralizer is the weak closure of its...

In the '80s, Yau conjectured that every closed Riemannian 3-manifold contains infinitely many smooth minimal hypersurfaces. In this talk, I will present a Yau-type existence result for nonlocal minimal surfaces and discuss how one can recover the...

A closed hypersurface in a contact manifold is convex if it admits a transverse contact vector field. A closed hypersurface is robustly non-convex if it cannot be smoothly approximated by a convex hypersurface. The existence of such hypersurfaces...

I plan to survey the many ways we have today of constructing expanding Cayley graphs of finite groups, and the ideas behind their analysis (some useful beyond that purpose). I want to highlight that despite a comprehensive understanding of achieving...

Given a compact hyperbolic surface of fixed topology, we consider its Laplace eigenvalues together with the structure constants for multiplication with respect to a suitable orthonormal basis of Laplace eigenforms. These numbers obey algebraic...