Seminars & Colloquia Calendar

Abstract: Based on Hamilton's program, Perelman solved the Poincar'{e} conjecture and Thurston's geometrization conjecture using Ricci flow with surgeries. It is crucial to understand the singularity formation in order to perform surgeries. To capture the geometry in the large of the singularity models, Perelman developed a space-time comparison geometry, the L-geometry, to prove the existence of the asymptotic shrinkers, which are smooth blow-down limits as time goes \(-infty\). Despite the miraculous success in dimension 3, Perelman's machinery is not suitable for higher dimensions partly due to the complexity of curvatures. Recently, Richard Bamler developed a compactness and partial regularity regularity theory for noncollapsed Ricci flows, which greatly advanced the study of Ricci flows in dimension greater or equal to 4. Among other important results, Bamler introduced a notion of tangent flow at infinity, which is a blow-down limit with respect to Bamler's new \(mathbb{F}\)-distance, and its existence is canonical thanks to Bamler's compactness theory. In a recent work with Chan and Zhang, roughly speaking, we proved that the two notions of blow-downs coincide. We also mention the study of tangent flows at infinity of 4-dimensional steady Ricci solitons and how the geometry at infinity determines the geometry in the large, which is a recent work joint with Bamler, Chow, Deng and Zhang. Moreover, we will talk about the uniqueness of closed and smooth tangent flows at infinity and the \(mathbb{F}\)-convergence rate, which are recent works joint with Chan and Zhang.