Student Analysis Seminar (Spring 2026): Recent developments in Mean Curvature Flow

Abstract

This seminar will survey recent developments in the analysis of singularities in mean curvature flow, with an emphasis on flows that are locally modeled on cylinders ("asymptotically cylindrical"). We will begin by reviewing the modern “weak-solution” viewpoint (Brakke/level-set formulations), blow-up and tangent-flow analysis, and the role of cylindrical models in the structure theory near generic singularities. The main focus of the seminar will be the recent Bamler-Lai proof of the Mean Convex Neighborhood Conjecture: roughly, whenever a singular point has a multiplicity-one cylindrical tangent flow, the ambient flow is forced to become mean-convex in a neighborhood of that point and admits a canonical-neighborhood description by explicit local models. 

 We will then see how these ideas feed into a classification of ancient asymptotically cylindrical flows—showing that such solutions fall into a small number of canonical families (ancient ovals, bowl-type translators, and flying-wing translators)—and how this classification underpins the proof of the mean-convex neighborhood and canonical-neighborhood theorems. 

Time permitting, we will discuss related progress on the Multiplicity One Conjecture (notably for the flow of surfaces in ℝ3) and consequences for well-posedness and fattening/ nonfattening phenomena. We will also cover recent advances in the classification program for ancient noncollapsed flows and translators in ℝ4, including bubble-sheet ovals and analytic foundations for the linearized translator equation.

Primary readings will be drawn from: arXiv:2512.24524, 2512.25050, 2312.02106, 2509.07629, 2509.06667, 2412.10581, 2305.19137, 2209.04931.