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 there must be a canonical flow that automatically "flows through its surgeries”, at an infinitesimal scale.

Recently, Kleiner and Lott constructed so-called Ricci flow space-times, which exhibit this desired behavior. In this lecture series, I will first review their construction. I will then present recent work of Bruce Kleiner and myself, in which we show that these Ricci flow space-times are in fact unique and fully determined by their initial data. Therefore, these flows can be viewed as “canonical”, hence confirming Perelman’s Conjecture. I will also discuss further applications of this uniqueness statement.

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