Wednesday, August 23, 2006

Perelmania

The New Yorker has a nice article on Perelman, his proof of the geometrization and Poincare conjectures, the politics surrounding the proof, and S.T. Yau's antics. If the article is correct (see end of excerpts below), Yau's maneuvering is directly related to Perelman's declining the Fields medal. Note he has not said whether he would decline the $1 million Clay Foundation prize ;-)

One of the authors is Sylvia Nasar, who wrote A Beautiful Mind. The writers actually went to St. Petersberg to track down the elusive genius, and have reconstructed the history behind his decade of work to produce the proof.

...In 1982, the year that Shing-Tung Yau won a Fields Medal, Perelman earned a perfect score and the gold medal at the International Mathematical Olympiad, in Budapest.

...In 1993, he began a two-year fellowship at Berkeley. While he was there, Hamilton gave several talks on campus, and in one he mentioned that he was working on the PoincarĂ©. Hamilton’s Ricci-flow strategy was extremely technical and tricky to execute. After one of his talks at Berkeley, he told Perelman about his biggest obstacle. As a space is smoothed under the Ricci flow, some regions deform into what mathematicians refer to as “singularities.” Some regions, called “necks,” become attenuated areas of infinite density. More troubling to Hamilton was a kind of singularity he called the “cigar.” If cigars formed, Hamilton worried, it might be impossible to achieve uniform geometry. Perelman realized that a paper he had written on Alexandrov spaces might help Hamilton prove Thurston’s conjecture—and the PoincarĂ©—once Hamilton solved the cigar problem. “At some point, I asked Hamilton if he knew a certain collapsing result that I had proved but not published—which turned out to be very useful,” Perelman said. “Later, I realized that he didn’t understand what I was talking about.” Dan Stroock, of M.I.T., said, “Perelman may have learned stuff from Yau and Hamilton, but, at the time, they were not learning from him.”

By the end of his first year at Berkeley, Perelman had written several strikingly original papers. He was asked to give a lecture at the 1994 I.M.U. congress, in Zurich, and invited to apply for jobs at Stanford, Princeton, the Institute for Advanced Study, and the University of Tel Aviv. Like Yau, Perelman was a formidable problem solver. Instead of spending years constructing an intricate theoretical framework, or defining new areas of research, he focussed on obtaining particular results. According to Mikhail Gromov, a renowned Russian geometer who has collaborated with Perelman, he had been trying to overcome a technical difficulty relating to Alexandrov spaces and had apparently been stumped. “He couldn’t do it,” Gromov said. “It was hopeless.”

Perelman told us that he liked to work on several problems at once. At Berkeley, however, he found himself returning again and again to Hamilton’s Ricci-flow equation and the problem that Hamilton thought he could solve with it. Some of Perelman’s friends noticed that he was becoming more and more ascetic. Visitors from St. Petersburg who stayed in his apartment were struck by how sparsely furnished it was. Others worried that he seemed to want to reduce life to a set of rigid axioms. When a member of a hiring committee at Stanford asked him for a C.V. to include with requests for letters of recommendation, Perelman balked. “If they know my work, they don’t need my C.V.,” he said. “If they need my C.V., they don’t know my work.”

...Perelman had posted a thirty-nine-page paper entitled “The Entropy Formula for the Ricci Flow and Its Geometric Applications,” on arXiv.org, a Web site used by mathematicians to post preprints—articles awaiting publication in refereed journals. He then e-mailed an abstract of his paper to a dozen mathematicians in the United States—including Hamilton, Tian, and Yau—none of whom had heard from him for years. In the abstract, he explained that he had written “a sketch of an eclectic proof” of the geometrization conjecture.

Perelman had not mentioned the proof or shown it to anyone. “I didn’t have any friends with whom I could discuss this,” he said in St. Petersburg. “I didn’t want to discuss my work with someone I didn’t trust.” Andrew Wiles had also kept the fact that he was working on Fermat’s last theorem a secret, but he had had a colleague vet the proof before making it public. Perelman, by casually posting a proof on the Internet of one of the most famous problems in mathematics, was not just flouting academic convention but taking a considerable risk. If the proof was flawed, he would be publicly humiliated, and there would be no way to prevent another mathematician from fixing any errors and claiming victory. But Perelman said he was not particularly concerned. “My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased,” he said. “I never set out to be the sole solver of the PoincarĂ©.”

Gang Tian was in his office at M.I.T. when he received Perelman’s e-mail. He and Perelman had been friendly in 1992, when they were both at N.Y.U. and had attended the same weekly math seminar in Princeton. “I immediately realized its importance,” Tian said of Perelman’s paper. Tian began to read the paper and discuss it with colleagues, who were equally enthusiastic.

On November 19th, Vitali Kapovitch, a geometer, sent Perelman an e-mail:

Hi Grisha, Sorry to bother you but a lot of people are asking me about your preprint “The entropy formula for the Ricci . . .” Do I understand it correctly that while you cannot yet do all the steps in the Hamilton program you can do enough so that using some collapsing results you can prove geometrization? Vitali.

Perelman’s response, the next day, was terse: “That’s correct. Grisha.”

...Perelman repeatedly said that he had retired from the mathematics community and no longer considered himself a professional mathematician. ... “It is not people who break ethical standards who are regarded as aliens,” he said. “It is people like me who are isolated.” We asked him whether he had read Cao and Zhu’s paper. “It is not clear to me what new contribution did they make,” he said. “Apparently, Zhu did not quite understand the argument and reworked it.” As for Yau, Perelman said, “I can’t say I’m outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.”

The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession. “As long as I was not conspicuous, I had a choice,” Perelman explained. “Either to make some ugly thing”—a fuss about the math community’s lack of integrity—“or, if I didn’t do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit.” We asked Perelman whether, by refusing the Fields and withdrawing from his profession, he was eliminating any possibility of influencing the discipline. “I am not a politician!” he replied, angrily. Perelman would not say whether his objection to awards extended to the Clay Institute’s million-dollar prize. “I’m not going to decide whether to accept the prize until it is offered,” he said.

Mikhail Gromov, the Russian geometer, said that he understood Perelman’s logic: “To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness.” Others might view Perelman’s refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. “The ideal scientist does science and cares about nothing else,” he said. “He wants to live this ideal. Now, I don’t think he really lives on this ideal plane. But he wants to.”

8 comments:

PS said...

“Hamilton contributed over fifty per cent; the Russian, Perelman, about twenty-five per cent; and the Chinese, Yau, Zhu, and Cao et al., about thirty per cent.”

Vanity is definitely my favorite sin!

Anonymous said...

It is difficult to accurately evaluate Perelman's motivations from the article. Only the last part of the article supports this argument, that Perelman declined because of his disgust in the politics. Earlier parts of the article suggest that his motivation was partly also simply disinterest in claiming credit. He's happy with his proof.

The article seems fairly clear, but I'm skeptical of a story like this, by an author who doesn't really know Perelman (a brief interview doesn't count). I think he might be pushing the story to fit his compelling storyline.

Anonymous said...

Very nice article.

I think Perelman will decline his share of the Clay Prize. We shall see...

Antics of Yau nothwistanding, the devil is in the details in pure mathematics. So I can see why Zhu and Cao deserve some credit. In any case, I found Cao and Zhu's paper easier reading than Morgan and Tian, or, of course Perelman; it offers hope to me about learning about surgeries on manifolds!! I note in some mathematical circles, this direct "computational approach" is derided. Well, I say being obscure is not the same as being profound. I much prefer physicists way of being transparent in their intuition. The calculations and discussion in Cao and Zhu seem easier to follow to me.

Note Morgan and Tian's paper/book, which is longer (with smaller font!), only proved Poincare; Cao and Zhu did the full Thurston geometrization conjecture based on Perelman.

MFA

Anonymous said...

This is an email that grad students at Stony Brooks got from Michael Anderson who is quoted in the article. It is interesting to note that such celebrated journalists from reputed magazines can misquote people to fit their stories. Or maybe Dr. Anderson had second thoughts...



" The New Yorker article badly distorted my comments and
the quote given is very inaccurate and misleading. I've already
discussed it with Yau and expressed to him my apologies and disgust
at using my name in this respect. I tried to have the quote removed,
but was unsuccessful, partly because I was travelling in Europe while
all this happened very quickly and I had no time respond.
I spent a good deal of time talking with Sylvia Nasar
trying to convince her to avoid discussion of the Tian-Yau fight
since it is irrelevant to Perelman, Poincare, etc. But obviously I
was not successful. In this particular respect, I feel the New Yorker
has done a disservice to mathematicians.
Sincerely, Michael Anderson"

Anonymous said...

Newton and Leibniz developed calculus. However, their works are obviously not as readable and lucid as most of the calculus textbooks we have nowaday. You can pick basically anyone of the popular textbooks on Calculus--it's better than Newtonian and Leibnizian books on calculus. Can we say that the better authors get the credit of developing calculus? Of course NOT!

Anonymous said...

en.wikipedia.com says:"Perelman's outline can indeed be expanded into a complete proof ..."

An outline, or in Perelman's own word "sketch", is not a proof.

Moreover, in his second famous preprint, Perelman said that another paper with the proof of Theorem 7.4 would be given, but until now nobody have seen it. Is it the reason that this "honest" person refused to come to meet other experts? Note that in April 2003, nobody can fully digest his first two articles.

There are two versions of Perelman's so called Theorem 7.4. One is the strong version with only two conditions, which no proof is available except for Perelman's "sketch". The other is a weak version with an additional condition, which might be too weak to use according to Kleiner & Lott.
It is also quite strange that this "honest" person cannot see the difference.

Anonymous said...

Idiots !

Perelman did all the work without any financial support, living on his personal savings for many years...
Other guys took money from NSF and other sources to extend and refine Perelman's proof and now they also want to take a part of Perelman's credit for solving the puzzle..
How is that for fairness ?

Anonymous said...

Unfortunately, your anger cannot change the facts.

I believe that nobody can "take a part of Perelman's credit" and vice versa.

No matter how well somebody plays with politics and media, mathematics as we know it would not be changed.

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