Introduction to Ricci flow

In preparation for the colloquium talk scheduled on Feb 4, we will discuss some of the basics of Ricci flow. Ricci flow is a parabolic evolution equation for Riemannian metrics introduced by Richard Hamilton in 1982 to study the three-dimensional Poincare conjecture and geometrization of 3-manifolds. It has since been used to solve several difficult geometric and topological problems, including the aforementioned Poincare and geometrization conjectures and the differentiable 1/4-pinched sphere theorem. We will provide an introduction to Ricci flow, introducing the method, some of the major results, and describe some of the key techniques that have been influential in the field. We may discuss traditional methods like parabolic maximum principle for systems, singularity blow-up, and Li-Yau type Harnack inequalities as well as newer techniques such as spacetime energies and reduced volume of Perelman.