The Perelman Enigma

Photos of the reclusive genius are rare

Without doubt a mathematical genius, Grigori Yakovlevich Perelman aka Grisha Perelman is also the ultimate unsung anti-hero.

In 1981, Dr Richard Hamilton of Columbia devised a method, known as the Ricci flow, to investigate the shapes of spaces. Dr Shing-Tung Yau of Harvard was enthusiastic that this method might finally crack the Poincaré conjecture. He began working with Dr Hamilton and urging others to work on it, with little success.

Then, in 2003, a Russian mathematician, Grigori Perelman, sketched a way to jump a roadblock that had stymied Dr Hamilton and to prove the hallowed theorem as well as a more general one proposed by the Cornell mathematician William Thurston. Dr Perelman promptly disappeared, leaving his colleagues to connect the dots.

Having solved the famous Poincaré conjecture, posed in 1904 and regarded as one of the most important and difficult open problems in mathematics, Perelman has refused to accept one of the discipline's top awards, the prestigious Fields Medal, which was awarded to him on August 22, 2006. A week later a drawing in The New Yorker showed Dr Shing-Tung Yau trying to grab the Fields Medal from the neck of Dr Perelman.

Apparently, Perelman is currently jobless, living with his mother in St Petersburg, and subsisting on her modest pension. His friends are said to have stated that he currently finds mathematics a painful topic to discuss; some even say that he has abandoned mathematics entirely.

He has stated that he is disappointed with mathematics' ethical standards, in particular of Fields medalist Shing-Tung Yau's effort to downplay his role in the proof and up-play the work of Huai-Dong Cao of Lehigh University in Pennsylvania and Xi-Ping Zhu of Sun Yat-sen University in China.

In a speech in June 2006 during the string theory conference, Dr Yau said, ?In Perelman?s work, many key ideas of the proofs are sketched or outlined, but complete details of the proofs are often missing,? adding that the Cao-Zhu paper had filled some of these in with new arguments.

This annoyed many mathematicians, who felt that Dr Yau had slighted Dr Perelman. Other teams who were finishing their own connect-the-dots proofs said they had found no gaps in Dr Perelman?s work. ?There was no mystery they suddenly resolved,? said John Morgan of Columbia, who was working with Gang Tian of Princeton on a proof.

On his Web site, doctoryau.com, Dr Yau has posted a 12-page letter showing what he and his lawyer say are errors in the New Yorker article. The New Yorker has said it stands by its reporting.

Perelman has said that "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. It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated."

This led him to quit professional mathematics. He has said that "As long as I was not conspicuous, I had a choice. Either to make some ugly thing" (a fuss about the mathematics 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."

Professor Marcus du Sautoy of Oxford University has said that "He has sort of alienated himself from the maths community. He has become disillusioned with mathematics, which is quite sad. He's not interested in money. The big prize for him is proving his theorem."

In a new twist, a flaw has been discovered in the Cao-Zhu paper. One of the arguments that the authors used to fill in Dr Perelman?s proof is identical to one posted on the Internet in June 2003 by Bruce Kleiner, of Yale, and John Lott, of the University of Michigan, who had been trying to explicate Dr Perelman?s work.

In an erratum to run in The Asian Journal of Mathematics, Dr Cao and Dr Zhu acknowledge the mistake, saying they had forgotten that they studied and incorporated that material into their notes three years ago.

In an e-mail message, Dr Yau said the incident was ?unfortunate? but reaffirmed his decision to expedite the paper?s publication. ?Even after the correction, the paper provides many important new details and clarifications of Hamilton and Perelman?s proof of the Poincaré and Thurston conjectures.?

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