Elusive Proof, Elusive Prover: A New Mathematical Mystery

By DENNIS OVERBYE (August 15, 2006)

Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.

After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.

Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.

As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.

“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”

In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.

Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”

But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.

Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.

“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.

... ... ... ...

Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.

Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.

In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.

“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.

Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.

Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.

Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.

But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.

“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”

Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.

In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures.

Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”

Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).

Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.

James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.

“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.

In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.

Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.

Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”

为了节省篇幅，我略去了中间介绍Poincaré猜想及早期工作的部分。为了方便国内读者，下面对其中几点内容作一下转述及评论：这篇文章引用了Yau的话，但只是Yau阐述Poincaré猜想重要性的话，不包括Yau对曹朱两人工作的推荐。另外，这篇文章引述了Clay研究所主席的话，明确表示Perelman完全有可能得奖（即他的得奖不违反章程），并且为Perelman可能拒绝领奖准备了下台阶（即可以把奖金转而用来支持俄罗斯数学）。文章最后引述的纽约州立大学石溪分校（杨振宁所在学校）的Anderson的话则把包括曹朱等人的文章在内的一系列文章都视为是对Perelman的工作的同行评定。这实际上是对Clay研究所评奖要求中的同行评定部分作了较宽的解读，为Perelman的可能获奖进一步扫清障碍。

看了这些报道感觉前一阵对曹、朱两人的报道还有一个误导之处，那就是给人一个印象，即Poincaré猜想的“封顶”是由曹、朱两人完成的。那段时间哪怕是不同意见，也大都只是对该“封顶”是否具有某些报道所说的30%这样的重要性提出异议，但对“封顶”工作由曹、朱两人完成并无异议。现在看来，这后一点也大可商榷。曹、朱两人虽然的确是封顶者（这里我们假定有关工作在技术上成立），但国际数学界似乎没有将之视为仅有的封顶者，而是把曹、朱的工作与其他几组数学家的工作并列对待。从这个意义上讲，即使封顶工作的重要性真的如前一阵子的报道所说，这一重要性也会由若干组数学家共享。

：：Anderson的话则把包括曹朱等人的文章在内的一系列文章都视为是对Perelman的工作的同行评定
=============================================
有三个小组在为他人做嫁衣裳。

丘大师如是说：
“但是事实上只能说佩雷尔曼解决了庞加莱猜想其中一个重要的难点，但是不能说完全证明了，这一点也是数学界公认的”
老仙生这次又走眼了。遗憾啊。

昨日又耳闻八卦，关于Hamilton的女朋友、Siu在哈佛的见证、Milnor为什么出头。很多事情又打了个折扣。希望自己以后不再在这个版议论这件事了。还是王元说得好，中国数学界是

池小王八多
庙小妖风大

还是王元说得好，中国数学界是

池小王八多
庙小妖风大
＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝

小时候脑子里数学家的伟大形象，毁于一旦！！！

偶觉得，抱着平常心对待就好了。
数学家也是人，只不过是比较特殊的一群人，这群人都喜欢数学，就这样。
其他的，他们与平常人没什么两样，在平常人身上有的问题，在他们身上同样会有，并不会因为他们是数学家，他们就变成了神。

个人愚见