Video Lectures

In 2014, Fargues realized that one can formulate the local Langlands correspondence over p-adic fields as a geometric Langlands correspondence on the Fargues-Fontaine curve. This raises the question of a similar realization of the local Langlands...

In joint work with Bulent Tosun, it was shown that Heegaard Floer theory provides an obstruction for a contact 3-manifold to embed as a contact type hypersurface in standard symplectic 4-space. As one consequence, no Brieskorn homology sphere admits...

Where are the hard problems? In the absence of a proof of P ≠ NP, researchers have spent years proving unconditional lower bounds for constrained models of computation. In time, a distinct theme arose: random problems (in particular random...

Interaction models discussed here are the (asymmetric) quantum Rabi model (QRM), which describes the interaction between a photon and two-level atoms, and the non-commutative harmonic oscillator (NCHO). The latter can be considered as a covering...

A central theme in differential geometry involves studying Riemannian metrics satisfying various curvature positivity conditions. The weakest condition one can impose is the positivity of the scalar curvature. Inspired by Toponogov's triangle...

The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. From the point of view of PDE, the Ricci flow is a system of linear parabolic equations, which can be viewed as the heat equation analogue of the Einstein...

Given two univariate polynomials, how does one compute their greatest common divisor (GCD)? This problem can be solved in polynomial time using the Euclidean algorithm, and even in quasi-linear time using a fast variant of the Euclidean algorithm...

The isoperimetric inequality has a long history in the geometry. In this lecture, we will discuss how the isoperimetric inequality can be generalized to submanifolds in Euclidean space. As a special case, we obtain a sharp isoperimetric inequality...

In their 2001 paper, Hofer, Wysocki and Zehnder conjectured that every autonomous Hamiltonian flow has either two or infinitely many simple periodic orbits on any compact star-shaped energy level; in the same paper, the authors prove this assuming...

Over the last two decades we have developed good understanding how to quantify the impact of strategic user behavior on outcomes in many games (including traffic routing and online auctions) and showed that the resulting bounds extend to repeated...