Title: Circle bundles with PSC over large manifolds Abstract: In his seminal work on metric inequalities for scalar curvature, Gromov asked whether total spaces of circles bundles over enlargeable manifolds can admit metrics with positive scalar curvature. We answer this question in all dimensions and construct infinitely many such examples over manifolds dimension 4 and above, […]
Title : Scalar curvature comparison and rigidity of 3-dimensional weakly convex domains Abstract: I will discuss a comparison and rigidity result of scalar curvature and scaled mean curvature on the boundary for weakly convex domain in Euclidean space, which is a joint work with Xuan Yao. This result is a smooth analog of Gromov's dihedral […]
Title: Non-uniqueness of mean curvature flow Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes […]
Title: Finite-Time Singularities of the Ricci Flow on Kähler SurfacesAbstract: By work of Song-Weinkove, it is understood that the Ricci flow on any Kähler surface can canonically be continued through singularities in a continuous way until its volume collapses. This talk will discuss recent progress in understanding a more detailed picture of the singularity formation […]
Abstract - Lecture 3 (lecture 2 will take place on March 24th at the Analysis & PDE seminar, see https://sites.google.com/view/hopkins-pde-seminar for details)In this series of lectures, we will explore viscosity solutions to fully nonlinear elliptic equations, following the foundational book of L. Caffarelli and X. Cabr´e in Fully Nonlinear Elliptic Equations (AMS Colloquium Publications, 1995). Our primary goal […]
Title - Stability Theorems for the Width Abstract - In this talk, I'll discuss some recent and ongoing work about the stability of min-max widths of spheres under various lower curvature bounds. Some of this is joint with Davi Maximo and Paul Sweeney Jr.
Title - Variations of the Yang-Mills Lagrangian in high dimensionAbstract - In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^2-control of their curvature. We will start from the classical theory in critical […]
Title: Rigidity of ancient ovals in mean curvature flow. Abstract: We will discuss the classification of ancient noncollapsed flows as singularity models of mean curvature flow. In particular, I will discuss my recent joint work with Beomjun Choi and Jingze Zhu about spectral rigidity, stability and symmetry of the so-called k-ovals in general dimensions. These […]