Thursday, June 08, 2006

Poincaré Conjecture

The American Mathematical Society has posted a news item on two key papers on the Poincaré Conjecture. The posting reads:

Within the past two weeks two papers have been issued on proofs of the Poincaré Conjecture: "A Complete Proof of the Poincaré and Geometrization Conjectures--Application of the Hamilton-Perelman Theory of the Ricci Flow," by Huai-Dong Cao and Xi-Ping Zhu (Asian Journal of Mathematics, June 2006) and "Notes on Perelman's Papers," by Bruce Kleiner and John Lott (on ArXiv, May 25, 2006).

The first of these two papers, which is roughly 300 pages long, is published in an archival journal, but not one that a mathematician would consider to be top-notch. This is odd, considering the importance and the financial worth of the result. If that paper is correct, who will receive the 1 million dollars from the Clay Mathematical Institute? To whom should the credit for the result go? Perelman or the Chinese mathematicians? And will Perelman receive the Fields medal at ICM 2006? I believe that there will be interesting developments in this story.

For the record, here is a quote from the recent 192 page preprint of Kleiner and Lott: "Regarding the proofs, [Perelman's papers] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader... We did not find any serious problems, meaning problems that cannot be corrected by using the methods introduced by Perelman."

No comments: