Gauge Theory and Topology Seminar: Recent progress in 4-dimensional Ricci flow and potential applications to topology

The Ricci flow (with surgery or through singularities) has proven to be a powerful tool in the study of 3-dimensional topology, its most prominent application being the verification of the Poincaré and Geometrization Conjectures by Perelman. Recent developments suggest that a theory of Ricci flow through singularities in dimension 4 is now within reach. The approach draws on a new compactness and partial regularity theory for Ricci flow, a degree theory for expanding solitons developed jointly with Eric Chen, and the resolution of the mean convex neighborhood conjecture for mean curvature flow in joint work with Yi Lai. Together, these results provide tools for understanding and resolving conical and cylindrical singularities and point toward a construction of a 4-dimensional flow through singularities. I will begin with a brief overview of Ricci flow in dimension 3 and explain the geometric intuition behind the expected 4-dimensional picture. I will describe the tools that are already available, how they fit into the proposed construction, and the remaining problems and possible approaches to them. Finally, I will discuss potential topological applications and further analysis of the flow, including ways of obtaining stronger geometric control, that may reveal new connections between 4-dimensional topology and geometric analysis.