Marston Morse 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...

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...

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...

The question of stability of approximate group homomorphisms was first formulated by Ulam in the 1940s. One of the most famous results in this area is Kazhdan's 1982 result on stability of approximate unitary representations of an amenable group...

In this lecture I will present basic elements of the theory of nonlocal games from quantum information theory and give some examples. I will then introduce the idea of "compressing" the complexity of nonlocal games, and show how the right form of...

The three problems referred to in the title originate in the theory of von Neumann algebras, C* algebras, and quantum information theory respectively. Each of them has been a deep long-standing open problem in its respective area. Surprisingly, the...

(This lecture will be self-contained.) In high dimensions, what does it look like when we take the intersection of a set of random half-spaces with either the sphere or the Hamming cube? This is one phrasing of the so-called perceptron problem...