Thursday, August 17, 2006

The curiouser and curiouser world of higher math, or Poincare vindicated

And now for a scheduled break from Lebabnon, Leiberman, and US foreign policy.

The Poincare Conjecture has been proved! Although I only learned about this yesterday, via a long overdue Times editorial, the proof has been circulating since 2003. Except that “proof” may be too strong a term; “sketch of proof by enigmatic genius leading to feverish gap-filling by eminent mathematicians” is a better description. And therein lies a tale.

The precise formulation and significance of the Poincare conjecture is inaccessible to those without a considerable mathematical background. Suffice to say that it is the major outstanding problem in topology, and one of the most storied challenges in mathematics, having resisted attempts at proof (or a counterexample) for over a century. It is also one of the Millennium Problems: a set of seven mathematical problems each carrying a million dollar prize for its resolution, courtesy of the Clay Institute.

The Conjecture states, roughly, that if all loops in a bounded three-dimensional space can be shrunk to points through deformations of the space, then the space must be topologically equivalent to a three-sphere (note that a three-sphere is not what we normally think of as a sphere; topology deals with surfaces, and the surface of an everyday sphere has only two dimensions, not three). The analogue conjecture for all higher dimensions was proved over twenty years ago, but the three-dimensional problem has been a much harder nut to crack.

Enter a Russian, an unworldly genius who is the spitting image of Rasputin. Until recently Grisha Perelman was a forgotten man, a mathematician who hadn’t published in eight years, a ghost whose colleagues were unaware even of what field he was working in. He first showed promise as a teen, winning the 1982 International Mathematical Olympiad with a perfect score. After earning his PhD at the St. Petersburg State University, he was a post-doctoral student at the Courant Institute, SUNY, and at UC Berkeley. Colleagues from that era recall his brilliance, his inability or unwillingness to talk about anything other than math, and his many-inch long fingernails, which he claimed to grow so that he could open a book at the exact page he wanted. In 1996 he returned to Russia—spurning positions at various American Universities—to join the Steklov Institute of Mathematics in St. Petersburg. Once there, he effectively fell off the academic map.

In early 2003 Perelman posted two papers on arXiv, a website popularly used by scientists and mathematicians to disseminate their research (usually as a precursor to submitting the work to a peer-reviewed journal). Neither paper mentioned the Poincare conjecture by name, but they claimed to provide “a sketch of a proof” of a more general conjecture—the Thurston Conjecture—of which the Poincare Conjecture is a special case. Both papers were concise in the extreme, often outlining arguments while omitting the details, full of statements such as “the proof of [this lemma] can be extracted from the proof of [an earlier lemma]”, which are fine in a Calculus 100 textbook, but a bit difficult to swallow in an extraordinarily original and difficult exegesis at the frontier of mathematics. Nonetheless, the cognoscenti soon sensed that something big was afoot. In May Perelman accepted an invitation to tour America, lecturing for weeks on end to packed audiences at Princeton and the Courant Institute, and fielding questions on his approach from the best mathematicians in the world. By all accounts he was clear and convincing in every detail. At the end of the tour, he had satisfied everybody who was anybody that he had at the very least achieved a seminal advance in mathematics, even if the jury was still out on Poincare’s Conjecture.

And then he returned to St. Petersburg, apparently smarting from the unwelcome attention of the media, and eager to return to his hermetic existence. He showed zero interest in pursuing publication or moving up the academic totem pole. It seemed that from his point of view he had solved the problem, and that was all that mattered. Others could fill in the gaps if they wished—and indeed, since 2003 there has been a flurry of papers devoted to explicating his proof in detail—but he was done. His role would be confined to answering e-mails from a select group of mathematical colleagues requesting help at tricky junctures of the proof.

Gradually a consensus has gathered that Perelman’s proof is indeed valid, and that Poincare’s Conjecture can be laid to rest, to rise again phoenix-like as a theorem. None of the researchers in the field have detected any significant flaw in his approach, and slowly and painfully all the gaps in his papers have been filled. This summer saw the arrival of two book-length manuscripts (one of them is available here) that purport to formally prove the Poincare Conjecture on the basis of Perelman’s approach. It is expected that the forthcoming International Congress of Mathematics will impress its official seal of approval on the proof.

Three twists or variations in the plot deserve comment.

First, it would appear that Perelman is a shoo-in for this year’s Fields Medal. This is mathematics’ equivalent of a Nobel Prize, except that it is much more difficult to earn, being awarded only every four years, and only to candidates under the age of forty. But, as is always the case with Perelman, there are problems. In order of intractability these are: Perelman is unlikely to accept the medal, and, indeed, is not even attending the Congress at which the medal will be announced and where he will play the role of invisible superstar; and nobody seems to know if he is under the age of forty and thus eligible for the medal. The blogosphere is rife with conflicting reports of his age. The most promising theory is that he was probably 16 years old when he won the Math Olympiad, which would put his current age at exactly 40!

Second, there is the little matter of the Clay Institute’s million dollar prize for solving one of the Millennium Problems. If Perelman were to be awarded the money, it would probably be shared with Richard Hamilton, who developed the central technique refined and employed by Perelman. But the Institute requires that a proof be published in a peer-reviewed journal and subjected to public scrutiny for at least two years before it receives consideration for the prize. Perelman has not, and will not, publish his proof. As noted, others have published extensively on the subject, and Perelman’s arXiv papers have already been subjected to scrutiny that can only be described with considerable understatement as intense. This may suffice to qualify Perelman for the prize, although it would be disctinctly odd for a mathematician to receive an award on the basis of his colleagues’ publications. In fact, officials from the Clay Institute have stated that they may be willing to change the prize criteria to accommodate Perelman. The real stumbling block, as you might have guessed, is that Perelman does not seem to be remotely interested in either the money or the glamour of being the first recipient of the prize.

Third, there is a mystery bordering on tragedy at this story’s end. On returning from his American tour in 2003, Perelman sporadically answered e-mails from mathematical colleagues, but over time, his responses grew more and more infrequent, and then stopped altogether. According to Russian colleagues, he has resigned his position at the Steklov Institute. His current whereabouts are unknown to the mathematical world.


shadhu said...

This is indeed quite a story. The NY Times Science section had a long piece on this earlier this week.

Last Tuesday's NY Times Podcast has a discussion on this too.

Curmudgeon said...

A point of clarification. Perelmen did not win the 1982 International Mathemetics Olympiad. He received a perfect score and a gold medal, one of three contestants to do so that year. I'm not sure how many gold medals were given out in 1982, but generally contestants with perfect or near perfect scores receive gold medals to reflect their acheivement, no their placing. Typically, 10-20 gold medals are given out per competition.

kitab said...

Thanks for the clarification, curmudgeon. So does the Olympiad have no "winner"? Or is one chosen from among the gold medalists (and if so, how)?

Curmudgeon said...

The Math Olympiad is a team contest. Each team consists of six participants. The country team with the most points is deemed the winner. For example, China won this year with 214 points out of a possible 252, with all its team receiving gold medals. (The cutoff for gold medals this year was 28 out of 42 points for each contestant, with 43 of 498 contestants receiving a gold medal)

Also, it was announced today that Perelman, along with three others, received the Fields Medal.