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.

Title: Minimal Surfaces in Negative Curvature

Abstract: It follows from work by Kahn-Markovic that every closed negatively curved 3-manifold contains essential minimal surfaces in great abundance.  Since then the goal of better understanding these minimal surfaces has been a focus of activity, both in analogy to the geodesic flow one dimension lower and the more positive-curvature-centric min-max theory of minimal surfaces.   This talk will survey recent developments in this area, which brings together techniques from dynamical systems, geometric analysis, and hyperbolic geometry.

Title: Restriction and local smoothing estimates using decoupling and two-ends inequalities.

Abstract: I will discuss restriction estimates and local smoothing estimates derived from decoupling theorems and two-ends inequalities, with a focus on the differences between these two problems. Specifically, I will introduce the wave packet density, which is the key to the induction on scales argument for the local smoothing problem. Also, I will show how an additional careful analysis of L^2 orthogonality leads to the sharp local smoothing estimates for wave equations on compact Riemannian manifolds.

Title: Triple junction solution for the Allen-Cahn system

Abstract: In this talk, I will discuss recent results on the vector-valued Allen-Cahn system with a triple-well potential. We establish the existence of an entire minimizing solution that asymptotically converges to a triple junction, corresponding to a planar minimal cone, at infinity.  Furthermore, we prove the uniqueness of the blow-down limit at infinity by deriving precise estimates for the location and size of the diffuse interface. We also show that the solution is almost invariant along the diffuse interface and, at infinity, closely resembles one dimensional heteroclinic connections between two energy wells. Our proof is primarily variational and does not assume any symmetry of the solution. These results are based on joint works with Nicholas Alikakos.

Title: Generic Regularity of Minimal Submanifolds with Isolated Singularities

Abstract: Singularities commonly arise in geometric variational problems such as minimal submanifolds, where they are locally modeled by minimal cones. Although a wide variety of singularity models have been constructed in the literature, it is conjectured that under generic setting, the they are substantially simpler.

In this talk, we will present a characterization of regular minimal cones that can appear as singularity models for minimal submanifolds in a generic Riemannian manifold, without any restrictions on the dimension or codimension. As an application, we will discuss a generic finiteness result for low-area minimal hypersurfaces in nearly round 4-spheres. This is based on joint work with Alessandro Carlotto and Yangyang Li.