  Once one knows that singularities occur, one naturally wonders what the
singularities are like. For minimal varieties the first answer, already known
to Federer-Fleming in 1959, is that they weakly resemble cones. For mean
curvature flow, by the combined work of Huisken, Ilmanen, and White,
singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave
open the possibility that a minimal variety or a mean curvature flow looked at
under a microscope will resemble one blowup, but under higher magnification, it
might (as far as anyone knows) resemble a completely different blowup. Whether
this ever happens is perhaps the most fundamental question about singularities.
It is this long standing open question that we settle here for mean curvature
flow at all generic singularities and for mean convex mean curvature flow at
all singularities.
