Mean curvature flow
We will introduce the mean curvature flow and some of its basic properties, such as characterization of the first singular time, Huisken's monotonicity formula, special solutions to the flow that only move by homotheties and are called the self-similar solutions, the notion of weak solutions to the flow. We will also discuss singularity analysis of the flow, give a possible description of singularities in some special situations, such as in the mean convex curvature setting. We will also talk about the importance of ancient solutions in singularity analysis of the flow and mention how the classification of those can help us prove some nice theorem in the field, such as the Mean convex neighborhood conjecture for neck singularities or uniqueness of weak solutions through spherical or neck singularities.
Regularity of mean curvature flow by Klaus Ecker
Basic PDE course