Ricci flows are a powerful geometric-analytical tool, as they have been used to prove important results in low-dimensional topology.

In the first part of this talk I will focus on Ricci flows in dimension 3. I will briefly review Perelman’s construction of Ricci flow with surgery, which led to the resolution of the Poincare and Geometrization Conjectures. Then I will discuss recent work of Lott, Kleiner and myself on an improved version of this flow, called “singular Ricci flow”. This work allowed us to resolve the Generalized Smale Conjecture, concerning the structure of diffeomorphism groups, and a conjecture concerning the contractibility of the space of positive scalar curvature metrics on 3-manifolds.

In the second part of the talk, I will focus on Ricci flows in higher dimensions. I will present a new compactness, which can be used to study the singularity formation of the flow, as well as its long-time asymptotics. I will discuss these and some further consequences. I will also convey some intuition of the new terminology that had to be introduced in connection with this compactness theory.