Geometry

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…

 I will discuss joint work with Gunther Uhlmann regarding the anisotropic fractional Calderon problem for Dirac operators on closed manifolds; these give fractional analogues of Maxwell systems. Namely we show that knowledge of the source-to-solution map of the fractional Dirac…

In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can…

In dimensions four and higher, the Ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature.  It may be possible to resolve such singularities and continue the flow using expanding Ricci solitons asymptotic to these cones, if they exist.  I will…

Embeddability is a natural geometric criteria for CR structures. In dimensions higher than three, there is local integrability condition that implies embeddability. In this dimension, this condition is vacuous and is replaced by a global condition on the sign of the Paneitz operator…