University of Miami
Department of Mathematics
College of Arts and Sciences

Professor John Morgan
Columbia University

will present

The Poincare Conjecture and Classification of 3-manifolds

Thursday, March 22, 2007, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: For over 100 years the Poincare Conjecture has been the central problem in topology. It conjectures a characterization of the 3-sphere as the only simply connected, closed 3-manifold. It has been generalized to a conjectured characterization of the sphere in all dimensions, and by 1986 had been solved in all dimensions except dimension 3. In the early 1980's Richard Hamilton proposed an attack on this conjecture, and a more general conjecture, due to Thurston, about the classification of all compact 3-manifolds. The method of attack proposed by Hamilton was analytic and differential geometric in nature. His idea was to use an evolution equation for a Riemannian metric on a manifold, deforming the metric by it curvature, which is an analogue of the heat equation. The intuition is that just as the heat equation distributes the temperate evenly around the manifold, the analogous tensor equation should distribute the curvature equally around the manifold. Hamilton showed that under certain geometric assumptions this program works perfectly, and the evolution equation converges to a round metric, from which it is easy to reach topological conclusions. In spite of this positive evidence, there was a serious obstacle in Hamilton's program. The equation is non-linear and, as is usual in non-linear evolution equations, singularities develop in finite time. In order to establish topological conclusions, one must continue the evolution process through these singularities and define an evolution for all positive time. Perelman provided the insights that allowed the extension of the equation through the singularities and showed how to complete Hamilton's program and prove the Poincare Conjecture and the more general classification of all closed 3-manifolds.

In this talk we will describe some of the history surrounding the Poincare Conjecture, indicating its central importance in much of 20th century topology. Then we will describe Hamilton's program and briefly indicate how Perelman overcame the difficulties that Hamilton had encountered in carrying out the program.

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BREAKTHROUGH OF THE YEAR: The Poincare Conjecture-Proved

On August 22, 2006, the ICM awarded Perelman the Fields Medal for his work on the conjecture, but Perelman refused the medal. John Morgan spoke at the ICM on the Poincare conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincare Conjecture."