The Schedule

Title: Gradient estimates for scalar curvature

Abstract: I will talk about some work with Toby Colding on monotonicity formulas for harmonic functions and give an application to a sharp integral gradient estimate for three manifolds with nonnegative scalar curvature. 

Title: Harmonic Maps into Euclidean Buildings and Their Applications

Abstract: In this talk, we discuss the construction of equivariant pluriharmonic maps, building on the foundational work of Gromov and Schoen. We also explore applications to the study of fundamental groups of smooth projective and quasiprojective varieties. 

Title: Minimization of ADM Mass in General Relativity

Abstract: The positive mass theorem states that an asymptotically flat initial data set satisfying an energy condition has a future timelike or null ADM energy-momentum vector, with the null case known as the equality case. This is closely linked to the problem of mass minimization in the context of the Bartnik quasi-local mass, known as the stationary conjecture. I will describe progress on both problems, as well as counterexamples in higher dimensions. 

Title: Controlling Geometry and Topology in Dimensions 3 and 4 via Scalar Curvature

Abstract: In this talk, we will discuss recent applications of variational and parabolic methods to control the geometry and topology of manifolds with positive scalar curvature in dimensions three and four.

Title: On the topology of stable minimal hypersurfaces in a homeomorphic S^4

Abstract: Given an n-dimensional manifold (with n at least 4), it is generally impossible to control the topology of a homologically minimizing hypersurface M. In this talk, we construct stable (or locally minimizing) hypersurfaces with optimal restrictions on its topology in a 4-manifold X with natural curvature conditions (e.g. positive scalar curvature), provided that X admits certain embeddings into a homeomorphic S^4. As an application, we obtain black hole topology theorems in such 4-dimensional asymptotically flat manifolds with nonnegative scalar curvature. This is based on joint work with Boyu Zhang.