The mean curvature flow is an evolution of hypersurfaces satisfying a ?geometric heat equation. The flow naturally develops singularities so ?that it changes the topology of the hypersurfaces. There are ?infinitely many types of singularities, but it has been conjectured ?that the flow generically develops spherical or cylindrical ?singularities. Hence, it is important to show the well-posedness of ?the flow through cylindrical singularities. In this talk, we introduce ?the mean-convex neighborhood hood conjecture and its application to the well-posedness problem. If time permits, we discuss how to prove ?the conjecture by classifying ancient low entropy flows.