Louis Yudowitz: Ricci Flow, the Poincaré Conjecture, and Bubbles

Abstract

 So far, the only millennium problem that has been solved is the Poincaré conjecture. This was done by Perelman in the early 2000s using Ricci flow, which was introduced by Richard Hamilton in 1982. Since then, Ricci flow has been used to solve a variety of other problems in geometry and topology. In this talk, I will give a general introduction to Ricci flow and a broad overview of Perelman's proof. After this, I will discuss the difficulties faced in extending Perelman's arguments to higher dimensions and some recent work which aims to address some of these issues. The talk is aimed at a general audience and only some basic familiarity with partial derivatives will be assumed.