Understanding ancient solutions is a central problem in the study of geometric flows, which in the case of curve shortening flow (CSF) has seen significant recent progress. This includes the classification of compact, convex ancient solutions to CSF by P. Daskalopoulos, R. Hamilton, and N. Sesum. In this talk, we discuss the construction of a non-convex ancient solution to CSF which is at all times compact and embedded—which shows that the convexity assumption cannot be dropped in the D-H-S classification. This talk is based on joint work with S. Angenent, C. Olson, and Y. Zhang.
