Video Lectures

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the “best” for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its...

Over 40 years ago, Karp, Upfal, and Wigderson posed a central open question in parallel computation: how many adaptive rounds are needed to find a basis of a matroid using only independence queries? Their pioneering work gave an upper bound of O(n‾√...

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...

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...

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...