Nondegenerate neck pinches along the mean curvature flow

It was shown recently by Chodosh-Choi-Mantoulidis-Schulze that for a generic initial surface in R^3, the mean curvature flow only has spherical and cylindrical singularities, confirming a well known conjecture of Ilmanen. I will discuss recent work showing that in fact for generic initial surfaces the cylindrical singularities are nondegenerate, at least at the first singular time. By slightly perturbing the flow after such nondegenerate singularities, and using work on Sun-Wang-Xue on the structure of nondegenerate cylindrical singularities, this gives a new approach to mean curvature flow with surgery, where the flow itself performs the topological changes.