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).
Here is the link to the scores for that 1982 Olympiad
http://www.srcf.ucam.org/~jsm28/imo-scores/1982/scores-order.html
As a non-mathematician, I’d sure like to see this thread grow before it gets replaced by something else. These blogs (the good ones) are like gardens where the new blossoms crowd out the old ones, so you end up with only the topmost flowers. What a waste.
This place needs a forum.
Ooops.
I just accidentally deleted two interesting comments. They pointed out that Perelman was the winner of the Math Olympiad competition in 1982, that if he was under 17 then he would be under 41 in 2006, and that Noam Elkies came in 4th behind Perelman that year.
There is a nice, short article in Notices by John Milnor on the Poincare conjecture and Hamilton’s work and a few words on Perelman’s results, which I found helpful.
http://www.ams.org/notices/200310/fea-milnor.pdf
The nice property of Ricci flow equation is that is it like a heat equation (clear in the weak-field approximation, see Bakas). Just as heat flows from hot to cold so object gets uniform in temperature, the Ricci flow behaves similarly so “curvature tries to become more unoform”, though there are several complications. My understanding is that Perelman showed how to take care of all that.
The article also seems to suggest that the choice of the Ricci flow equation chosen by Hamilton was analogous to Einstein’s derivation of his field equation: essentially R_{ij} is the unique 2-index
tensor arising naturally from the first and second order derivatives of the metric. Persumably all other terms that can be written (terms of higher order in the Hilbert action in an effective field theory approach) will contain no additional geometric information.
I am merely scratching the surface, I am sure, but very interesting stuff…
Peter,
Many thanks for your patient explanations; you are an invaluable resource! I will try to get a hold of the original references and try to understand it better. At least, now I have a pretty good chance of understanding it!
Just talked to Richard about this since he was upstairs at tea. He started work on this in the late seventies and didn’t hear about the connection to the renormalization group and non-linear sigma models until many years later. He thinks the first time these equations occurred in physics were in Dan Friedan’s thesis (1980), which he only heard about years later.
As far as I know, the way Hamilton and Perelman are using these equations has nothing to do with the fact that they are approximate renormalization group equations for a non-linear sigma model. (If I see Hamilton maybe I’ll ask him if he knew about renormalization group eqs. when he started studying this equation). For them, the important point is just that the topology of the manifold doesn’t change as you evolve the metric according to the Ricci flow (at least until you hit a singularity in the solution of the PDE). The idea of the proof of geometrization (drastically over-simplified…) is to show that, no matter what metric you start with, you end up at one of the finite number of possibilities on the list that Thurston conjectured were all the possible topologies of 3d manifolds. Solutions to the Ricci flow eq. certainly do develop singularities, which is one of the things that makes this very hard.
I read with great interest the talk of Bakas and tried to understand Perlman’s papers.
I have a basic question that perhaps somebody can answer.
I understand that the Ricci flow equation is nothing but a 1-loop RG equation of the 2-dim (world-sheet) sigma-models with target space metric G_{\mu\nu}(X), i.e., \beta(G_{\mu\nu})= -(Ricci Tensor)_{\mu\nu}. The higher order terms are nonzero, but may be neglected for weak curvature. This Ricci flow equation is the basis for Perelman (and Hamilton’s) analysis.
How does this 1-loop approximate result become the foundation of studying geometry, especially global toopolgical questions like the Poincare conjecture.
Issues like manifold surgery, Thurston geometrization conjecture are definitely way over my head.
Thanks
Re age limits:
I think Nobel’s will stipulates that the Prize should go to the person who, within the appropriate field, has made the most important contribution to the benefit of mankind during the past year. One may argue whether academic research really benefits mankind the most – the Swedish inventors organization challenge that, arguing that society benefit more from inventions (like those of AN himself) than from basic research. Be that as it may, it is still hard to argue that your average Nobel laureate did his best work during the last year.
Thanks for pointing this out, the link to the conference with the Bakas talk should now work correctly.
The link to Bakas’ talk is broken.
I thank you for your field-theoretic explanation of the general idea of the method of Perelman’s proof (and the work of Richard Hamilton).
dolt
It looks like most of the recent media articles were generated by a program at the British Association Science Festival in Exeter a couple days ago called “Million Dollar Maths” where Perelman’s proof was discussed by Keith Devlin.
I’ve been hearing more about Perelman recently from my colleague John Morgan, who was involved in the recent workshop in Princeton. He was the one who told me that Perelman might be young enough for the Fields medal, something I hadn’t realized.
It has suprised many people during the last few months to see that maybe Perelman won’t bother to write a real paper about this and really doesn’t seem to care about the money.
I don’t know if there will ever be any sort of definite point at which people working on Perelman’s papers announce that, yes, it is a proof. As far as I can tell, everyone now believes Perelman’s Poincare result (if not the full Geometrization), so how the story of the million dollars plays out will be interesting to watch.
btw why is there a sudden burst of interest in the poincare conjecture again ? has there been a recent verification of the proof ? I have been reading about Perelman’s proofs for at least a year now.
It is kind of dumb, and most recently this was made clear when Wiles proved Fermat’s last theorem, but was too old for a Fields. The positive argument for this age cut-off is that all too often what happens with these prizes is that a bunch of old guys sit around and decide to give prizes to each other. The age cut-off is a way of ensuring that recognition goes to people who have recently done something exciting and are not already well-established.
Age discrimination is a terrible problem in IT work. I can’t believe the Fields Medal is tied to an arbitrary number! What’s the point of that?
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