Mean curvature flow can be viewed as the ‘heat equation’ for an embedding function. Indeed, on short time scales MCF will improve the regularity of the embedding. Moreover, smooth MCF is an isotopy, making it an attractive tool for answering questions concerning the geometry and topology of embeddings. Unfortunately, MCF from a compact initial condition will form singularities in finite time, posing an obstruction to the construction of isotopies. MCF with surgery was introduced by Huisken—Sinestrari for 2-convex hypersurfaces in order to avoid the formation of singularities. I will show how recent results of Choi—Haslhofer—Hershkovits and Choi—Haslhofer—Hershkovits—White regarding mean-convex neighbourhoods can be combined with the 2-convex surgery procedure of Haslhofer—Kleiner to construct a MCF with surgery that approximates a mean curvature flow in which 2-convexity is only assumed on the singularity models, rather than on the entire flow. I will also present some topological applications.

No knowledge of MCF will be assumed, though familiarity with the heat equation and hypersurface geometry will be helpful.