Ricci flow, positive curvature, and symmetry

Abstract: The Ricci flow is a geometric PDE for evolving with time a Riemannian metric on a given manifold.  The Ricci flow is heuristically like a heat equation for the metric, and its regularizing properties have been used with great success in solving problems in geometry and topology. In this talk I will discuss joint work with R. Bettiol showing that positive curvature is not preserved under the Ricci flow on closed manifolds in dimension 4. This is in contrast to dimension 3, where the preservation of positive curvature under the flow is a crucial step in Hamilton’s theorem that a simply-connected closed 3-manifold with positive Ricci curvature is diffeomorphic to the 3-sphere. Our examples come from metrics on $S^4$ and $\mathbb{C}P^2$ that have a large group of isometries. In my talk I will discuss background on positive curvature and the Ricci flow, and explain how symmetry enters the proof. If time permits I will also discuss work in progress about how ancient homogeneous Ricci flows have “more symmetries” than expected.