## Columbia Mathematics Department Colloquium

Uniqueness of blowups and Lojasiewicz inequalities

by

# Tobias Colding

# (MIT)

Abstract:

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 the proof of this long standing open question that we discuss for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities. This is joint work with Bill Minicozzi.

It is the proof of this long standing open question that we discuss for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities. This is joint work with Bill Minicozzi.