In this talk, we shall describe some topological rigidity and its relationship with positive scalar curvature. Precisely, we will present a proof that a complete contractible 3-manifold with

The study of positive scalar curvature on noncompact manifolds has seen significant progress in the last few years. A major role has been played by Gromov's results and conjectures, and in

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second

This is a talk about my works with Damin Wu concerning those manifolds with negative holomorphic sectional curvature. I shall describe our theorem that such manifold must have negative first Chern

Given a complete Riemannian manifold that looks enough like Euclidean space at infinity, physicists have defined a quantity called the “mass” that measures the asymptotic deviation of the geometry

In this talk, I will discuss some recent developments on the topology of closed manifolds admitting Riemannian metrics of positive scalar curvature. In particular, we will prove if a closed PSC

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

Given a closed and oriented manifold M and Riemannian tensors g0, g1, ... on M that satisfy g0 < gj, vol(M, gj)→vol (M, g0) and diam(M, gj)≤D we will see that (M, gj) converges to (M, g0) in the