Video Lectures

A compact hyperkahler manifold is a higher-dimensional analog of a K3 surface; Lagrangian fibrations of hyperkahler manifolds are higher-dimensional versions of elliptic fibrations of K3 surfaces. A result of Voisin shows that these fibrations yield...

On a projective variety, Simpson showed that there is a homeomorphism between the moduli space of semisimple flat bundles and that of polystable Higgs bundles with vanishing Chern classes. Recently, Bakker, Brunebarbe and Tsimerman proved a version...

The symplectic area of a Lagrangian submanifold L in a symplectic manifold is defined as the minimal positive symplectic area of a smooth 2-disk with boundary on L. A Lagrangian torus is called extremal if it maximizes the symplectic area among all...

In this talk, I will discuss Cosmohedra. These are geometric objects which encode the vacuum wavefunction for theories with colored scalars with cubic interactions. I will start by introducing the vacuum wavefunction, both in Minkowski and de Sitter...

We give a lower bound on the codimension of a component of the non-abelian Hodge locus within a leaf of the isomonodromy foliation on the relative de Rham moduli space of flat vector bundles on an algebraic curve. The bound follows from a more...

A Hodge structure is a certain linear algebraic datum.  Importantly, the cohomology groups of any smooth projective algebraic variety come equipped with Hodge structures which encode the integrals of algebraic differential forms over topological...

Can we encode data in a way that it is recoverable even when 1) most data becomes corrupted, and 2) we can only read a sub-constant fraction of the database? This is the central question of local list decoding, a powerful tool from coding theory...

At the center of nearly every galaxy in the Universe resides a supermassive black hole. When galaxies collide, their supermassive black holes sink to the center of the newly forming galaxy. There in this nascent galactic nucleus a supermassive black...

A conjecture of Komlós states that the discrepancy of any collection of unit vectors is O(1), i.e., for any matrix A with unit columns, there is a vector x with -1,1 entries such that |Ax|∞=O(1). The related Beck-Fiala conjecture states that any set...

It is often said that a geometer has only two basic tools to study a space: to slice it or to project it. Cartography provides a vivid example: contour lines are simply the intersections of a landscape with horizontal planes. In the nineteenth...