Johns Hopkins

Title: Positive scalar curvature and exotic structures on simply connected four manifolds.

Abstract: We address Gromov’s band width inequality and Rosenberg’s S1-stability conjecture for smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless, we show that both of these results hold upon considering simply connected smooth four manifolds up to homeomorphism. We also obtain a weaker related result in the general case. Based on joint work with B. Sen.

Title: Arnold-Thom conjecture for the arrival time of surfaces

Abstract: Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. In this talk, I will explain the proof of Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions of mean convex mean curvature flows of surfaces. This is joint work with Tang-Kai Lee.

Title: Some recent developments on the fully nonlinear Yamabe problems 

Abstract: In recent joint work with YanYan Li and Zongyuan Li, we broaden the scope of fully nonlinear Yamabe problems by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor. Our results accommodate conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities of the associated equations. In addition to above results, I will briefly describe our developments on the fully nonlinear Yamabe problems on manifolds with boundary, discussing both boundary mean curvature and boundary curvatures arising from the Chern–Gauss–Bonnet formula.

Title: Finite time explosion for SPDEs

Abstract: The classical existence and uniqueness theorems for SPDEs prove that, under appropriate assumptions, SPDEs with globally Lipschitz continuous forcing terms have unique global solutions. This talk outlines recent results about SPDEs exposed to superlinearly growing deterministic and stochastic forcing terms. I describe sufficient conditions that guarantee that, despite the superlinear growth, the SPDEs have unique global solutions.