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...

The interstellar medium (ISM) is a turbulent, multi-phase, magnetic environment. It is home to vastly different gas phases, from cold, dense clouds to hot, tenuous plasma. Magnetic fields thread this interstellar environment, helping to sculpt...

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‾√...

Analog Superpowers explores the early history of computing, intellectual property law, government secrecy, and Anglo-American relations to draw out surprising connections with today’s world of digital devices and great-power competition. It follows...

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...