Natasa Sesum
Introduction to Ricci flow
Ricci flow has been studied extensively recently, as an analytic tool that can be used to prove important topological results about a manifold, whose metric we involve in time by Ricci flow. It is a parabolic equation, so it is expected that the properties of the initial metric improve in time, and in the best case that the flow converges to a metric of constant curvature.
We will learn how various geometric quantities evolve by the flow, discuss the short time existence of the flow (since it is only a weakly parabolic equation), discuss the regularity properties of the equation, we will characterize singular time of the equation. In geometric equations, like Ricci flow, it is very important to understand singularity formation, since that is the first step in possibly finding ways how to continue the flow past singularities. We will introduce Ricci solitons, that often model singularities of the flow, and talk about their properties. We will also introduce Perelman's monotonicity formula, and its important consequences, such as noncollapsing property of the flow. We will also talk about ancient solutions and canonical neighborhood theorem.
Introduction to Ricci flow by Peter Topping and various papers on Ricci flow
Differential Geometry and PDE knowledge is recommended