  We prove Ilmanen's resolution of point singularities conjecture by
establishing short-time smoothness of the level set flow of a smooth
hypersurface with isolated conical singularities. This shows how the mean
curvature flow evolves through asymptotically conical singularities and
resolves a particular case of Ilmanen's strict genus reduction conjecture.
  Precisely, we prove that the level set flow of a smooth hypersurface
$M^n\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, with an isolated conical
singularity is modeled on the level set flow of the cone. In particular, the
flow fattens (instantaneously) if and only if the level set flow of the cone
fattens.
