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. 

The homotopy groups of spheres classify continuous functions from an n-dimensional sphere to an m-dimensional sphere up to continuous deformation. They are of great interest as they control many classification problems in geometric topology, but are rich and complicated. I will explain some of what we know about them, and describe the telescopic approach to understanding them systematically. I will then explain how algebraic K-theory has refined our understanding of the telescopic approach, and can be used to obtain asymptotic lower bounds on the complexity of these homotopy groups that were previously out of reach.