Friday, August 11, 2006

The Poincare Conjecture

Huai-Dong Cao and Xi-Ping Zhu published A Complete Proof of the Poincare Conjecture and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow (PDF) in the June issue of the Asian Journal of Mathematics. The paper is 328 pages of dense mathematics understandable only by experts. However, the title of the paper is electrifying to those with some familiarity with The Poincare Conjecture, for it has been one of the most famous unresolved problems of mathematics since first proposed by Henri Poincare at the beginning of the Twentieth Century. The Clay Mathematics Institute will award a million dollars to any person who proves it.

The Poincare Conjecture is stated thus:
Every simply connected closed 3-manifold is homeomorphic to a 3-sphere.

3-manifolds and 3-spheres are objects in four dimensional space so it requires some tricks to imagine them. The analogues in three dimensions are the 2-sphere, the surface of a ball, and 2-manifolds, which also exist in three dimensional space. Two objects are homeomorphic if they can be stretched, without tearing, to look like each other. So, crudely, the Poincare Conjecture asks whether all simply connected closed 3-manifolds can be stretched to look like the 3-sphere, which is the surface of a four dimensional ball.

To arrive at a proof of the Poincare Conjecture one can prove the Geometrization Conjecture proposed by William Thurston. This conjecture says that there only certain distinct geometries that 3-manifolds can have. The Geometrization Conjecture implies the truth of the Poincare Conjecture as a special case. Richard Hamilton pioneered a technique called Ricci Flows to attack the Geometrization Conjecture and made significant progress towards its proof. In 2003, Russian mathematician Grisha Perelman published three papers on the Internet that seemed to have finished the proof of the Geometrization Conjecture, and thusly the Poincare Conjecture. Perelman extended the use of Ricci Flows to arrive at his results. His work has withstood the scrutiny of experts in the field and the paper cited at the beginning puts Q. E. D. to the proof.

In fact, on page 320, the authors state:
Thurston’s geometrization conjecture is true.

Now, we see (kind of) the meaning of the paper cited at the beginning of this post.

A headline in a recent issue of Nature claims that Perelman may get the Nobel Prize for his work. (I don’t have a subscription so I have not read the article.) The Nobel does not have a category for mathematics, so I assume he will get it for physics. Why physics?

To the extent that the space in which we live is some kind of 3-manifold in four dimensional space, the Geometrization Conjecture says that there are only certain shapes it can have. This means that scientists measuring the shape of the universe have a limited set of choices as to its geometry.

As for the Clay Mathematics Prize, it requires that a proof must be published in a refereed mathematics journal and withstand two years of scrutiny by the mathematics community. Perelman is reported to be reclusive, even though he gave a set of lectures in the US in 2004, and has not published his papers other than on the Internet. One wonders if the Clay Math Institute might not bend the rules in his case if he should receive a Nobel for his work.

Jeffrey R. Weeks has written an outstanding book called The Shape of Space covering a lot of ideas about 3-manifolds. He claims in the preface that the interested high school mathematics student should be able to understand the book. He delivers on that promise. The challenge of the book is to use one’s imagination to understand what 3-manifolds look like. He gently guides the reader, via pictures, through some very interesting mathematics without all the formalism of a mathematics text. He connects the topology and geometry to the physics at the end of the book.

I am sure we will see many good books published within the next couple of years about the history and mathematics of the Poincare Conjecture. It is that exciting and interesting. Under the hand of the able writer, its importance will be made accessible to the interested lay reader.

P. S. I found Ricci Flow and the Poincare Conjecture (PDF) by John Morgan and Gang Tian via Ars Mathematica. This is a complete proof of the Poincare Conjecture.

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