The mean curvature flow is an evolution of hypersurfaces satisfying a geometric heat equation. The flow naturally develops singularities and changes the topology of the hypersurfaces at singularities, Therefore, one can study topological problems via singularity analysis for parabolic partial differential equations. Indeed, linearly stable singularities are just round cylinders, and thus we can expect how to change the topology of generic solutions to the flow.
In this talk, we first introduce the mean curvature flow, the blow-up analysis, and the stability of singularities. If time permits, we also discuss the well-posedness problem through linearly stable singularities.