### May 26, 2021

TITLE: Ancient solutions to the Ricci flow in dimension 3

ABSTRACT:

The Ricci flow is a natural evolution equation for Riemannian metrics on a manifold. The central problem is to understand singularity formation. In other words, what does the geometry look like at points where the curvature is large? In his spectacular 2002 breakthrough, Perelman showed that, for a solution to the Ricci flow in dimension 3, the high curvature regions are modeled on so-called ancient -solutions: By definition, these are solutions to the Ricci flow which are defined for and satisfy certain extra conditions (most importantly, a noncollapsing condition). Moreover, Perelman achieved a qualitative understanding of ancient -solutions in dimension 3; this is sufficient for topological conclusions.

In this lecture, I will discuss recent work which gives a complete classification of all noncompact ancient -solutions in dimension 3, thereby confirming a conjecture of Perelman. Time permitting, I will mention joint work with Panagiota Daskalopoulos and Natasa Sesum which gives a complete classification of all compact ancient -solutions in dimension 3.

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