Title: Closed Mean Curvature Flows with Prescribed Tangent Flows

Abstract: We show that every nonflat smooth embedded shrinker that is closed, smoothly asymptotically conical, or a product of such a shrinker with a Euclidean factor can arise as a multiplicity-one tangent flow at the first singularity of a mean curvature flow starting from a closed embedded hypersurface. We also discuss how certain first-order asymptotics near the singularity can be prescribed. This is joint work with Jingwen Chen, Tang-Kai Lee, and Ao Sun.

Title: Isoperimetry by stretching 

Abstract: Isoperimetric boundaries minimise area for fixed enclosed volumes, with sharp regularity theory ensuring they are smooth away from a closed singular set of codimension seven. I will discuss recent work, with G. Niu, which constructs isoperimetric regions from hypersurfaces in closed manifolds. As a direct application, we show that a wide variety of topological types and singular sets arise in isoperimetric boundaries.

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