Mathematical Physics Seminar

Geometric heat flows like Ricci flow and mean curvature flow
have proven to be remarkably successful tools to investigate the
geometry and topology of manifolds. They are nonlinear PDE with a
diffusion-reaction structure that makes them prone to finite-time
singularities. Perhaps counterintuitively, these singularities are
aids, not obstacles, to these flows' applications -- because regions
of high curvature tend to be very special. In this talk, we will
survey these phenomena and present evidence in favor of the
conjectures that (1) solutions asymptotically acquire extra symmetries
as they become singular, and (2) generic solutions may be constrained
to a small catalog of universal asymptotic profiles.