Richard Bamler, UC Berkeley

Title: Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications

Abstract: In this talk I will survey recent work with Kleiner in
which we verify two topological conjectures using Ricci flow. First,
we classify the homotopy type of every 3-dimensional spherical space
form. This proves the Generalized Smale Conjecture and gives an
alternative proof of the Smale Conjecture, which was originally due to
Hatcher. Second, we show that the space of metrics with positive
scalar curvature on every 3-manifold is either contractible or
empty. This completes work initiated by Marques.

Our proof is based on a new uniqueness theorem for singular Ricci
flows, which I have previously obtained with Kleiner. Singular Ricci
flows were inspired by Perelman's proof of the Poincaré and
Geometrization Conjectures, which relied on a flow in which
singularities were removed by a certain surgery construction. Since
this surgery construction depended on various auxiliary parameters,
the resulting flow was not uniquely determined by its initial
data. Perelman therefore conjectured that there must be a canonical,
weak Ricci flow that automatically "flows through its singularities"
at an infinitesimal scale. Our work on the uniqueness of singular
Ricci flows gives an affirmative answer to Perelman's conjecture and
allows the study of continuous families of singular Ricci flows
leading to the topological applications mentioned above.