The Poincare conjecture is one the seven Clay millenium problems, each worth a million-dollar prize. It says something about all 3-dimensional “simply connected” manifolds being “equivalent” to the sphere in R4. A Fields medal was awarded for the proof for k-dimensional manifolds, k>4, and another Fields medal for k=4. (In those cases the “equivalence” is with the sphere in Rk+1.)
A few years ago, the Russian mathematician Grigori Perelman posted a series of papers on the arxiv. The papers outlined a proof of the “Thurston geometrization conjecture,” a statement that implies the Poincare conjecture.
Ever since, the status of the Poincare conjecture has remained uncertain. Perelman’s papers do not contain a full proof, but various experts have been in agreement that most likely Perelman’s ideas can be successfully formalized. Meanwhile, Perelman’s himself seemed to be oblivious to the matters of writing a full paper and claiming the prize.
Now comes the news (via Not Even Wrong) that two Chinese mathematicians have written a 300-page paper that formalizes Perelman’s argument, and the paper is about to be published in the Asian Journal of Mathematics.
Update 6/6/06: The Guardian technology blog picks up the story.
Update 6/6/06: The Notices of the AMS have an article on the genesis of the Clay millenium prizes. The stories are very interesting. (For example Wiles’s insights into mathematical politics.) I noticed one paragraph in particular. Perhaps with Perelman’s case in mind, Arthur Jaffe writes
The rules for the prize resulted from a fair amount of thought. […] One major safeguard involved the importance of publication of the solution. […] Of course there can also be unforeseen circumstances. For example, an author of a solution may not write it down completely […]
Update 6/7/06: An April 10 press releas from the organizers of ICM’06 suggests that there were plans to announce in Madrid that Perelman’s proof has been “checked.”
Update 6/8/06: see also the May 15 ICM bulletin.