Previous Special Year Seminar

The (Simon-Smith) min-max width of a Riemannian three dimensional sphere is a geometric invariant that measures the tightest way, in terms of area, of sweeping out the three-sphere by two-spheres. In this talk, we will explore the properties of this...

In the 60's Almgren initiated a program for developing Morse theory on the space of flat cycles. I will discuss some simplifications, generalizations and quantitative versions of Almgren's results about the topology of the space of flat cycles and...

We will present the project of using the Willmore elastic energy as a quasi-Morse function to explore the topology of immersions of the 2-sphere into Euclidean spaces and explain how this relates to the classical theory of complete minimal surfaces...

A translator for mean curvature flow is a hypersurface $M$ with the property that translation is a mean curvature flow. That is, if the translation is $t\rightarrow M+t\vec{v}$, then the normal component of the velocity vector $\vec{v}$ is equal to...

In this talk we will survey recent progress on the Beresticky-Caffarelli-Nirenberg Conjecture in Space Forms; that is, let $\Omega$ be an open connected domain of a complete connected Riemannian manifold ($M,g$) and consider the OEP given by
\begin...

Preliminary I will expose a technique developed with T. Rivi\`{e}re to prove energy identities (weak compactness) for sequences of solutions of any conformally invariant problem of second order in dimension 2, see [1]. Then after introducing some...

In the early 80’s, Yau conjectured that in any closed 3-manifold there should be infinitely many minimal surfaces. I will review previous contributions to the question and present a proof of the conjecture, which builds on min-max methods developed...

We'll describe a joint project with X. Zhou in which we use min-max techniques to prove existence of closed hypersurfaces with prescribed mean curvature in closed Riemannian manifolds. Our min-max theory handles the case of nonzero constant mean...

Following a program proposed by Gromov, we study metric singularities of positive scalar curvature of codimension two and three. In addition, we describe a comparison theorem for positive scalar curvature that is captured by polyhedra. Part of this...