Seminars & Colloquia Calendar

• April 15, 2022, Friday, 2:00 - 3:00pm (EST), 

Abstract: 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 the Poincaré and Geometrization Conjectures by Perelman about 20 years ago. Since then further research has led to a satisfactory understanding of the flow and surgery process in dimension 3. In dimensions 4 and higher, on the other hand, Ricci flows have been understood relatively poorly and a surgery construction seemed distant. Recently, however, there has been some progress in the form of a new compactness and partial regularity theory for higher dimensional Ricci flows. This theory relies on a new geometric perspective on Ricci flows and provides a better understanding of the singularity formation and long-time behavior of the flow. In dimension 4, in particular, it may eventually open up the possibility of a surgery construction or a construction of a "flow through singularities". The goal of this talk will be to describe this new compactness and partial regularity theory 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 such flows and discuss several conjectures that provide a possible picture of a surgery construction in dimension 4. Lastly, I will discuss potential topological applications.