Mathematics Colloquium

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Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications

Speaker: Richard Bamler, UC Berkeley

Location: Warren Weaver Hall 1302

Date: Monday, December 12, 2016, 3:45 p.m.

Synopsis:

In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed 3-dimensional 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 there must be a canonical, weak flow that automatically "flows through its surgeries”, at an infinitesimal scale.

A few years ago, Kleiner and Lott showed the existence of weak Ricci flows, which exhibit this desired behavior. In this talk, I will first review their result. I will then present recent work of Bruce Kleiner and myself, in which we show that these weak flows are in fact unique and fully determined by their initial data. Therefore, these flows can be viewed as “canonical”, hence confirming Perelman’s Conjecture. Finally, I will discuss topological applications of this uniqueness statement.