Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

I will present 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.At the heart of our proof is a new uniqueness and stability theorem for singular Ricci flows. Singular Ricci flows can be seen as an improvement of Ricci flows with surgery, which were used in Perelman’s proof of the Poincaré and Geometrization Conjectures. The latter flows had the drawback that they were not uniquely determined by their initial data, as their construction depended on various auxiliary surgery parameters. Perelman 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.The lectures will roughly be structured as follows:(1) Preliminaries of Ricci flow, Blow-up analysis of singularities, Statement of the main results(2) Local stability Analysis(3) Comparing singular Ricci flows, Proof of the uniqueness and stability result(4) Continuous families of singular Ricci flows, Proof of the topological theorems.