Video Lectures

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...

I will describe a double scaled matrix and tensor integral whose Feynman diagrams can be organized in a 3d topological expansion which agrees term by term with partition functions of 3d gravity. The integral is taken over CFT2 data, and the limiting...

This will be an informal presentation / discussion of the basics of holographic theories that live on Cauchy slices (arXiv:2204.00591), which can be constructed by the TTbar deformation.  Please feel free to bring in questions and I will try to...

 "We know that we live in a global age, but did medieval people? Historians typically reserve the concept of a 'globalized world' for the fifteenth century and beyond, following the rise of worldwide commercial shipping. But in recent years...

Recent horizon-scale images of Messier 87* and Sagittarius A* have been used to demonstrate that their spacetimes are well-described by the Kerr metric. The latter is a solution to the vacuum Einstein equations of general relativity, and is used to...

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...