In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed Ricci flows in which singularities are removed by a surgery process. His construction depended on various auxiliary parameters, such as the scale at which surgeries are performed. At the same time, Perelman conjectured that his Ricci flow with surgery converges to a canonical Ricci flow through singularities when the surgery parameters are sent to zero.

A few years ago, John Lott and I introduced singular Ricci flows, which are a kind of generalized solution to Ricci flow in dimension three. We proved the existence of a singular Ricci starting from a prescribed compact Riemannian 3-manifold M. Recently Richard Bamler and I have proven that 3d Ricci flow has a strong stability property that implies Perelman’s convergence conjecture, and that the singular Ricci flow with intiial condition M is unique and depends continuously on M.