Mathematics Colloquia and Seminars

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Ricci flow is a powerful tool for probing the geometry and topology of Riemannian manifolds. From the analytic point of view, it is a quasilinear weakly parabolic diffusion-reaction system for an evolving Riemannian metric. Because of the reaction terms present, finite-time singularity formation is inevitable for large classes of initial data. It is therefore natural to want to know as much as possible about what singular behaviors are possible. We will review some of what is known about the classification of singularity types, and we will discuss recent progress (by the speaker and collaborators) in determining the rates at which singularities can form.