Welcome, this is the course webpage for Math 760, "Topics in differential geometry" for the Spring 2010 semester. The topic this semester will be introduction to Ricci flow.
Text: "Lecture notes on Ricci flow" by Peter Topping, Cambridge University Press (available electronically from Topping's webpage.)
Overview: To start we will follow the text closely. It covers many of the interesting aspects of Ricci flow theory, including many of the key ideas in Perelman's famous proofs of the Poincare conjecture and geometrization of 3-manifolds. The text culminates with a proof of Hamilton's theorem (which began the subject) that uses the more recently developed tools. Time permitting we will also try to go further into Perelman and Hamilton's work. Along the way we will encounter many interesting and important topics in geometric analysis such as collapsing, entropies, Sobolev and Harnack inequalities, the maximum principle, diffeomorphism invariance, and convergence of manifolds.
Other References: There are many, many good textbooks and monographs about the Ricci flow and geometric analysis which I will also use as references. I will compile a (very incomplete, I am sure) list of the references I use here.
Course Calendar: I will keep a course calendar where I will list the topics covered each day.
pre-requisites: I would like to emphasize that I will try to keep pre-requisites to a minimum. A introductory manifold course like Math 600-601 is probably necessary, but I will go over the basics from Riemannian geometry and parabolic PDEs as we go.
One of the things I find very beautiful about the Ricci flow is the way it combines elements of geometry, topology, and analysis. Thus, I hope this course will at least serve as motivation for geometers, topologists, and analysts to learn a little more about each other's subjects. That has certainly been my experience with Ricci flow!
Text: "Lecture notes on Ricci flow" by Peter Topping, Cambridge University Press (available electronically from Topping's webpage.)
Overview: To start we will follow the text closely. It covers many of the interesting aspects of Ricci flow theory, including many of the key ideas in Perelman's famous proofs of the Poincare conjecture and geometrization of 3-manifolds. The text culminates with a proof of Hamilton's theorem (which began the subject) that uses the more recently developed tools. Time permitting we will also try to go further into Perelman and Hamilton's work. Along the way we will encounter many interesting and important topics in geometric analysis such as collapsing, entropies, Sobolev and Harnack inequalities, the maximum principle, diffeomorphism invariance, and convergence of manifolds.
Other References: There are many, many good textbooks and monographs about the Ricci flow and geometric analysis which I will also use as references. I will compile a (very incomplete, I am sure) list of the references I use here.
Course Calendar: I will keep a course calendar where I will list the topics covered each day.
pre-requisites: I would like to emphasize that I will try to keep pre-requisites to a minimum. A introductory manifold course like Math 600-601 is probably necessary, but I will go over the basics from Riemannian geometry and parabolic PDEs as we go.
One of the things I find very beautiful about the Ricci flow is the way it combines elements of geometry, topology, and analysis. Thus, I hope this course will at least serve as motivation for geometers, topologists, and analysts to learn a little more about each other's subjects. That has certainly been my experience with Ricci flow!