Video Lectures

Minimax optimization, especially in its general nonconvex formulation, has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs) and adversarial training. It brings a series of unique...

The unique games conjecture gives a very strong PCP theorem, which, if true, leads to a clean understanding of a broad family of approximation problems. We will describe recent progress on the conjecture and how certain type of expansion and...

A large number of close-in super-Earths have now been discovered. These planets are most likely formed while the proto-planetary disks are still alive, hence the title of this talk. I will report some recent observational results, regarding their...

High dimensional expansion generalizes edge and spectral expansion in graphs to higher dimensional hypergraphs or simplicial complexes. Unlike for graphs, it is exceptionally rare for a high dimensional complex to be both sparse and expanding. The...

The tasks of finding and randomly sampling solutions of constraint satisfaction problems over discrete variable sets arise naturally in a wide variety of areas, among them artificial intelligence, bioinformatics and combinatorics, and further have...

I will describe my work, all joint with D. Gaiotto and some also joint with E. Witten, to understand the homotopy type of the space of (1+1)d N=(0,1) SQFTs --- what a condensed matter theorist would call "phases" of SQFTs. Our motivating hypothesis...

I will introduce an isoperimetric inequality for the Hamming cube and some of its applications. The applications include a “stability” version of Harper’s edge-isoperimetric inequality, which was first proved by Friedgut, Kalai and Naor for half...

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1...

The PCP theorem says that any mathematical proof can be written in a special "PCP" format such that it can be verified, with arbitrarily high probability, by sampling only a few symbols in the proof. Hence the name, Probabilistically Checkable...