Davies Professor of Mathematics

Columbia University

**Lecture I (Public Lecture): The Ricci Flow and the **Poincaré **Conjecture **

Monday, April 15

Lewis Lab 270

7:30 pm; Lobby reception at 6:45pm

On the compact surfaces, the sphere is the only one where every curve bounds a disk; on the others, a curve going around a hole does not. Such a manifold is called simply connected. In 1904, Poincaré conjectured the only compact simply connected three-manifold is the sphere. Famously, a proof of this conjecture was given by Dr Perelman in 2002 - 2003, using the Ricci flow.

A manifold can carry a metric, measuring the distances between points. In differential geometry, the metric is given as a quadratic function on tangent vectors, measuring the square of their lengths. The length of a path is given by an integral of the lengths of tangent vectors.

The heat equation on a surface (or manifold) is a partial differential evolution equation for a function whose solution describes how the initial given heat will spread by diffusion around the surface (or manifold) to approach a constant equal distribution of heat. The Ricci flow is a partial differential evolution equation for a metric which tries to spread the curvature to approach a constant equal distribution. The idea for the Poincaré conjecture is that if the metric converges to constant positive curvature, the manifold must be the sphere.

(This lecture will be suitable for a general audience.)

**Lecture II: Estimates for the Ricci Flow**

Wednesday, April 17

Neville 2

4:10 pm

**Lecture III: Surgery on the Ricci Flow**

Thursday, April 18

Neville 3

4:10 pm

Richard S. Hamilton was born in Cincinnati, Ohio, in 1943. He received his B.A in 1963 from Yale University, and Ph.D. in 1966 from Princeton University at age 23. He taught at Cornell University, UC San Diego, and UC Irvine before joining Columbia University where he is currently Davies Professor of Mathematics.