  Mean curvature flow is the most natural evolution equation in extrinsic
geometry, and shares many features with Hamilton's Ricci flow from intrinsic
geometry. In this lecture series, I will provide an introduction to the mean
curvature flow of surfaces, with a focus on the analysis of singularities. We
will see that the surfaces evolve uniquely through neck singularities and
nonuniquely through conical singularities. Studying these questions, we will
also learn many general concepts and methods, such as monotonicity formulas,
epsilon-regularity, weak solutions, and blowup analysis that are of great
importance in the analysis of a wide range of partial differential equations.
These lecture notes are from summer schools at UT Austin and CRM Montreal, and
also contain a detailed discussion of open problems and conjectures.
