Lagrangian mean curvature flow out of conical singularities

Lagrangian mean curvature flow (LMCF) is a way to deform Lagrangian submanifolds inside a Calabi-Yau manifold according to the negative gradient of the area functional. There are influential conjectures about LMCF due to Thomas-Yau and Joyce, describing the long-time behaviour of the flow, singularity formation, and how one may flow past singularities. In this talk, we will show how to flow past a conically singular Lagrangian by gluing in expanders asymptotic to the cone, generalizing an earlier result by Begley-Moore. We solve the problem by a direct P.D.E.-based approach, along the lines of recent work by Lira-Mazzeo-Pluda-Saez on the network flow. The main technical ingredient we use is the notion of manifolds with corners and a-corners, as introduced by Joyce following earlier work of Melrose.