The Ricci flow is a geometric evolution equation which has received enormous attention in the last few decades due to its overwhelming tendency to “improve” extremely rough initial geometries. One important prerequisite to understanding the forwards behaviour of Ricci flows in full generality is the thorough understanding of those Ricci flows which are ”ancient” i.e., those Ricci flows which have already existed for an infinite amount of time. Geometric analysts currently have a fairly comprehensive theory of ancient Ricci flows that are non-collapsed, but the collapsed ones have received comparatively little attention. In this talk, I will discuss the recent construction of an ancient Ricci flow on \(\mathbb {S}^n\) (for any integer \(n\geq 3\)) which is both collapsed and positively-curved.
This is joint work with Theodora Bourni, Mat Langford and Ramiro Lafuente.