Some recent developments in Ricci flow

Abstract

In this article we survey some of the recent developments in Ricci flow. We present a new theory of weak, 3-dimensional Ricci flows “through singularities,” which can be viewed as an improvement of Perelman’s Ricci flow with surgery. We point out two topological applications: the resolution of the Generalized Smale Conjecture regarding the diffeomorphism groups of 3-manifolds and the resolution of a conjecture regarding the space of positive scalar curvature metrics on 3-manifolds. We also describe ongoing research on the formation of singularities in higher dimensions, which may yield further interesting applications in the future.