Video Lectures

The goal of these lectures is to present the fundamentals of Simpson’s correspondence, generalizing classical Hodge theory, between complex local systems and semistable Higgs bundles with vanishing Chern classes on smooth projective varieties.

Let q,Q be two integral quadratic forms in m less than n

variables. One can ask when q can be represented by Q - that is,

whether there exists an n×m-integer matrix T such that Q∘T=q.  Naturally, a necessary condition is that such a representation...

In the Calculus of Variations, convexity plays a seemingly irreplaceable role.

For vectorial problems, however, for good reasons a whole zoo of its generalizations was introduced. It ranges  from notions based on very classical

observation over...

In this talk we will construct (approximate) locally list decodable codes from high dimensional expanders (HDXs).

Last week we saw that locally list decoding is a powerful tool from coding theory. We also got a subspace-based construction of...

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...