The mean curvature flow is an evolution of hypersurfaces under a geometric heat equation. As a solution to a parabolic PDE, the flow converges to a self-similar solution at its singularity after rescaling. Hence, we can find a scalar-valued function defined over the self-similar solution whose graph is the rescaled flow. Therefore, the function is a solution to a parabolic PDE, and thus we can study the fine asymptotic behavior by using the spectrum of the linearized operator. Indeed, we can also apply this theory for the classification of ancient flows.

In this talk, we discuss it applications to the optimal regularity of arrival time, namely solutions to the 1-Laplace equation. Also, we talk about a potential application to the generic mean curvature flow and the knot theory.