Monday

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. This is joint work with Richard Bamler.

We will begin by discussing rather broadly the relationship between nilpotent groups and curvature. There turn out to be surprising and deep connections, beginning in some sense with the work of Gromov on almost flat manifolds and the polynomial growth groups, but also with the works of Milnor on the fundamental groups of Ricci curvature. These works were extended by Fukaya/Yamaguchi and later Kapovitch/Wilking in order to understand more completely the fundamental groups of spaces with lower sectional and lower Ricci curvature bounds. The first part of this talk will give a broad introduction to these ideas and concepts, hopefully with relatively minimal background requirements, and hopefully with a focus on the leftover open questions and conjectures from the topic. The last part of the talk will focus on some recent work with Brue and Semola, where we answer some of these questions. In particular, we show the existence of compact manifolds with nonnegative Ricci curvature which are not (uniformly) almost abelian.

We study Riemannian prisms (D^2\times [0,1], g) whose boundary are minimal surfaces meeting at constant angles (called capillary prisms). For well-chosen angles, these are platonic building blocks for various closed 3-manifolds. Under a positive scalar curvature condition, we obtain estimates on the area of the minimizing disks in capillary prisms. An interesting feature is that, the optimal shape is not obtained by foliations of totally geodesic capillary disks. Our proofs rely on new estimates on capillary isoperimetric profiles. As an application, we construct metrics achieving large pi_2-systole on 3-manifolds with positive scalar curvature, whose prime factors are S^2 \times S^1 and lens spaces. Joint work with Shihang He.

A classic subject in geometry has been the count of closed geodesics on negatively curved spaces. I will explain why minimal Lagrangians of genus g with area bounded by L in a product of Riemann surfaces grows on the order of L^{6(k−1)}, with an explicit leading constant given in terms of the Mirzakhani function. This is joint work with Lowe and Marques.

Expanders are a special class of solutions to mean curvature flow in which each time slice is a rescaled (expanded) copy of an earlier one. They play a fundamental role in understanding both the long-time behavior of the flow and the asymptotic structure of solutions arising from cone-like singularities. In this talk, I will survey recent developments in the theory of expanders and discuss their applications to related problems in geometric analysis.