Mean curvature flow (or MCF) is a nonlinear heat equation for hyper-surfaces, where the surface evolves by moving in the direction where volume locally decreases the fastest.  The simplest non-static examples are round concentric spheres, where the radius shrinks until it becomes zero at "extinction" (a singularity of the flow).  Singularities are unavoidable as the flow contracts any closed surface and thus one of the most important problems in MCF is understanding the singularities.  Matt Grayson, Mike Gage and Richard Hamilton proved that this is the only singularity for simple closed curves in the plane.  However, many examples were discovered in higher dimensions.I will describe recent work with Toby Colding, MIT, where we:
Classify the generic singularities of MCF of closed embedded hyper-surfaces.
Prove compactness of all (even non-generic) singularities.
I will also discuss an application, where we construct a "generic mean curvature flow."