To characterize scalar curvature, Gromov proposed the dihedral rigidity conjecture which states that a positively curved polyhedron having dihedral angles less than those of a corresponding flat polyhedron should be isometric to a flat one. In this talk, we will discuss some recent progress on this conjecture and its connection with general relativity (ADM mass and quasilocal mass). 

Schoen-Yau proved the spacetime positive energy theorem by reducing
it to the time-symmetric (Riemannian) case using the Jang equation. To
acquire solutions to the Jang equation, they introduced a family of
regularized equations and took the limit of regularized solutions, whereas a
sequence of regularized solutions could blow up in some bounded regions
enclosed by apparent horizons. They analyzed the blowup behavior near but
outside of apparent horizons, but what happens inside remains unknown. In
this talk, we will discuss the blowup behavior inside apparent horizons
through two common geometric treatments: dilation and translation. We will
also talk about the relation between the limits of blowup regularized
solutions and constant expansion surfaces.

There are many interesting examples of complete non-compact Ricci-flat metrics in dimension 4, which are referred to as ALE, ALF, ALG, ALH gravitational instantons. In this talk, I will describe some examples of these geometries, and other types called ALG^* and ALH^*. All of the above types of gravitational instantons arise as bubbles for sequences of Ricci-flat metrics on K3 surfaces, and are therefore important for understanding the behavior of Calabi-Yau metrics near the boundary of the moduli space.  I will describe some general aspects of this type of degeneration, and some recent work on degenerations of Ricci-flat metrics on elliptic K3 surfaces in which case ALG and ALG^* bubbles can arise. This is based on joint work with Hein-Song-Zhang (to appear in JAMS) and with Chen-Zhang (to appear in CAG).

The mean curvature flow is an evolution of hypersurfaces satisfying a geometric heat equation. The mean curvature flow in general develops singularities, and the topology of the hypersurface is changed through singularities. To study the topological change, we consider ancient flows which can be obtained by the blow-up of the flow at singularities. In this talk, we will discuss how to use the classification result of ancient flows for singularities analysis and topology.

Joint with Analysis seminar.

Pluripotential solutions to the complex Monge-Ampere equations are very useful  in modern complex geometry. It is thus very natural to seek analogues for other Hessian type complex equations that appear in geometric analysis. In the talk I will discuss several approaches towards such a theory and their limitations. Then I will focus on a newly coined notion of a $L^p$-viscosity solutions and discuss uniform estimates for a large class of Hessian equations. This is a joint work with S. Abja and G. Olive.