One of the great stories of mathematics in recent years has been the proof of the Poincare conjecture by Grisha Perelman. This has been one of the most famous open problems in mathematics and has been around for about one hundred years. In technical terms the conjecture is that if a space is homotopically equivalent to a three-dimensional sphere it is homeomorphic to the three-sphere. In less technical terms it says that if you have a bounded three-dimensional space in which all loops can be shrunk down to points, it has to be the three-sphere. In dimensions other than three the analog conjecture has been proved, but the case of three dimensions has resisted all attempts to solve it.
Perelman spent time as a visiting mathematician at Stony Brook, Berkeley and NYU, then went back to St. Petersburg where for eight years he seemed to disappear from mathematics research. In November 2002 he posted a preprint on the arXiv, which quickly drew a lot of attention. He seemed to be claiming to have a proof of an even more general conjecture than Poincare, known as the Thurston Geometrization conjecture, but the way his preprint was written, it wasn’t clear whether he was claiming to really have a proof. The method he was using was one pioneered by my Columbia colleague Richard Hamilton, called the “Ricci flow method”. This involves something like a renormalization group flow to a fixed point (for more about this, see the talks by Ioannis Bakas at a recent conference in Crete). If you start with an arbitrary metric on a space you think might be a three-sphere, the hope was that Hamilton’s Ricci flow would take you to the standard metric for the three-sphere. Hamilton had made a lot of progress using his techniques, but as far as pushing them through to give a proof of Poincare, he was stuck.
In the spring of 2003, Perelman traveled to the US and gave talks at several places, including a long series at Stony Brook. By then he was explicitly claiming to have a proof, but few of the details were written down, although he did post two more preprints to the arXiv. His talks were major events in the math community, and at them he was able to answer anyone who asked for details on specific points of his argument. He gave a somewhat informal talk at Columbia one Saturday, a talk that I attended sitting next to Hamilton, who was hearing Perelman speak for the first time. Hamilton was clearly very impressed, and soon thereafter he and most other experts began to become convinced that Perelman really did have a way of proving the conjecture.
By now the situation seems to be that the experts are pretty convinced of the details of Perelman’s proof for the Poincare conjecture. The full Geometrization conjecture requires some more argument and I gather that Perelman is supposed to at some point produce another preprint with more about this. A workshop was held a couple weeks ago about Perelman’s work at Princeton and several people have been carefully working through the details needed to be completely sure the proof works. For this material, see a web-site maintained at Michigan by Bruce Kleiner and John Lott.
One interesting part of this story is that the Poincare conjecture is one that the Clay Mathematics Foundation has put a one-million dollar price tag on. There’s an elaborate set of rules that Perelman should follow to collect his million dollars. This is supposed to begin with the submission of a detailed proof to a well-known refereed journal, something Perelman hasn’t done and shows no signs of doing. As far as anyone can tell, his attitude is that he’s not interested in the million dollars. If you look closely at the rules, it doesn’t necessarily have to be Perelman who writes up the proof. Someone else may do it, with Perelman still getting the money. Ultimately the question of the million dollars is to be decided by the Scientific Advisory Board of the Clay Mathematics Institute, and one question they will have to face is whether to split the award between Perelman and Hamilton.
Another interesting question concerns the Fields medal, the most prestigious award in mathematics. These are awarded every four years at the International Congress of Mathematicians, the next one of which will take place in Madrid in the summer of 2006. One stipulation for the award of the Fields medal is that a recipient must be under the age of 40. Seeing Perelman speak, I had assumed he was already at least forty, but this is not so clear. No one seems to be sure exactly what his age is and whether he will be under 40 in 2006. Some news reports from spring 2003 referred to him as being in his late 30s or even 40, some recent ones claim that he is now 37. His first scientific paper was published in 1985, so he would have had to have been 19 or younger at the time to be under 40 in 2006. If Perelman really is under 38 now, he’s a sure thing for a 2006 Fields medal.
For a really dumb news article about this, go here (no, proving the Riemann hypothesis won’t bring down the internet, and Perelman’s Poincare proof won’t explain the nature of the universe).
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