Towards a theory of Ricci flow in dimension 4 (and higher)
Towards a theory of Ricci flow in dimension 4 (and higher)
Online Talk
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The Ricci flow (with surgery) has proven to be a powerful tool in the study of 3-dimensional topology — its most prominent application being the verification of theÌýPoincaré and Geometrization Conjectures by Perelman about 20 years ago. SinceÌýthen further research has led to a satisfactory understanding of the flow and surgeryÌýprocess in dimension 3.Ìý
In dimensions 4 and higher, on the other hand, Ricci flows have been understoodÌýrelatively poorly and a surgery construction seemed distant. Recently, however, thereÌýhas been some progress in the form of a new compactness and partial regularityÌýtheory for higher dimensional Ricci flows. This theory relies on a new geometricÌýperspective on Ricci flows and provides a better understanding of the singularityÌýformation and long-time behavior of the flow. In dimension 4, in particular, it mayÌýeventually open up the possibility of a surgery construction or a construction of aÌý"flow through singularities".Ìý
The goal of this talk will be to describe this new compactness and partial regularityÌýtheory and the new geometric intuition that lies behind it. Next, I will focus on 4-dimensional flows. I will present applications towards the study of singularities of suchÌýflows and discuss several conjectures that provide a possible picture of a surgeryÌýconstruction in dimension 4. Lastly, I will discuss potential topological applications.