Geometry

We discuss some regularity results for mean curvature flows from smooth hypersurfaces with conical singularities. We then discuss how to use these results to tackle two conjectures of Ilmanen. 

Recently there have been significant developments in how we can think about singularities in minimal submanifolds. I will discuss this circle of ideas, in particular how the new planar frequency function of B. Krummel & N. Wickramasekera allows for a more efficient and refined study of…

A fundamental result about the dynamics and geometry of hyperbolic manifolds is Besson-Courtois-Gallot's entropy inequality. The volume entropy of a Riemannian metric measures the growth rate of geodesic balls in the universal cover. The result says that given a closed hyperbolic manifold (M,g_0…

In their seminal work on the minimal surface system, Lawsonand Osserman conjectured that Lipschitz graphs that are critical pointsof the area functional with respect to outer variations are alsocritical with respect to domain variations. We will discuss the proof ofthis conjecture for two-…

The interplay between curvature and topology is always one of the central topics in Riemannian geometry. For open (noncompact and complete) manifolds with nonnegative Ricci curvature, it is known that the fundamental groups can be torsion-free nilpotent. This is distinct from open manifolds with…

In this talk, I will show that for certain one-parameter families of initial conditions in R^3, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts…

Atiyah and Bott generalized the Lefschetz fixed point theorem to elliptic complexes on smooth manifolds, and its various incarnations now appear in many areas of mathematics and physics. 

I will describe a generalization of this theorem for Hilbert complexes associated to Dirac type…

We discuss on a new systolic inequality for 3-manifolds with positive scalar curvature. It was proved by Bray, Brendle and Neves that if a closed 3-manifold has scalar curvature at least 1 and has nonzero second homotopy group, then its spherical 2-systole is bounded from above by 8π. Moreover,…

In this talk I will discuss a collection of related compactness problems in gauge theory. After some basic set-up and motivation for studying these problems, I will explain how non-compactness phenomena naturally lead to degenerate elliptic equations which govern the behavior at the boundary of…

Abstract: In this talk, I will present a construction of regular initial data for the Einstein-Maxwell-charged scalar field system collapsing to exactly extremal Reissner--Nordström black holes within finite advanced time. In particular, our result can be viewed as a definitive disproof of “the…