The Econophysics Blog

This blog is dedicated to exploring the application of quantiative tools from mathematics, physics, and other natural sciences to issues in finance, economics, and the social sciences. The focus of this blog will be on tools, methodology, and logic. This blog will also occasionally delve into philosophical issues surrounding quantitative finance and quantitative social science.

Friday, August 25, 2006

Waiting for Grisha: What REALLY Happened to The Man Who Proved the Poincaré Conjecture

There was a minor scandal in the mathematics community this week when -- a Russian mathematician that is widely credited with having proved the (one of the $1 million offered by the ) -- refused to accept the prestigious (in the world, the Fields Medal is considered to be on par with the Nobel Prize).

The has been an important unsolved problem in . Topology is the branch of mathematics that deals with what can be loosely termed 'fundamental shapes' -- spatial properties that are perserved even after deformation. For example, to a topologist, a coffee mug is topologically equivalent to a doughnut -- they both have one hole. Thus, the 'fundamental shape' of both a coffee mug and a doughnut is a torus.

Dr. Perelman used a technique called a (which smooths out the 'bumps' of geometric space to reveal it's 'fundamental shape') in order to prove the conjecture. Other mathematicians had tried to use Ricci flow in order to tackle the conjecture, but they stumbled over singularities whose properties were difficult to predict and manage. Grisha Perelman overcame this obstacle by proving that these singularities could be 'managed' (so to speak) and the Ricci flow allowed to continue to its logical conclusion, thus solving Poincaré's age old conundrum. (For those interested in the gory technical details, you can find Dr. Perelman's papers -- as well as related material -- at Notes and commentary on Perelman's Ricci flow papers.)

Most media reports of Dr. Perelman's refusal to accept the Fields Medal has chalked it up to his alleged 'eccentricities' (some have even gone as far as using the word "crazy"). Some examples in this vein include stories in the New York Times, The Guardian (UK), and The New Scientist.

From a journalistic/editorial point of view, this is all suppose to be reminiscent of the brilliant but troubled mathematician (of A Beautiful Mind fame (BTW, the real John Nash doesn't look like Russell Crowe) ) -- the inventor of in . I suppose that type of analogy has some superficial appeal for media types, the public at large, and, apparently, even professional mathematicians.

Despite the general tenor of how the majority of the press (and some mathematicians) are portraying Dr. Perelman, I'm a little skeptical of this 'kooky math wizzard hiding out in the Russian woods' tag people have tried to place on Dr. Perelman. My suspicions are confirmed by an excellent article in the New Yorker magazine, Manifold Destiny: A legendary problem and the battle over who solved it (Aug. 28, 2006) by Sylvia Nasar (the author of A Beautiful Mind) and David Gruber.

According to the New Yorker article, several prominent mathematicians have unfairly tried to minimize Grisha Perelman's contributions in proving the Poincaré Conjecture (to be fair, others have made important contributions leading to the proof, but, as I believe Sir Issac Newton said, scientists and mathematicians always "stand on the shoulders of giants" and Dr. Perelman -- although standing on others' shoulders (which he generously acknowledges) -- saw what others did or could not). I strongly believe that this type of situation (not specifically this incident since Dr. Perelman seems to be a genuinely humble and shy man who would rather defer to others) -- the perception of unfairness and, even, injustice in what is suppose to be a pure and noble intellectual pursuit -- is the real reason behind Dr. Perelman's refusal to accept the Fields Medal.

The following passage from the New Yorker article seems to back up my conjecture:

Perelman repeatedly said that he had retired from the mathematics community and no longer considered himself a professional mathematician. He mentioned a dispute that he had had years earlier with a collaborator over how to credit the author of a particular proof, and said that he was dismayed by the discipline’s lax ethics. “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.”

(Emphasis added.)

That's my two cents on the matter. Frankly -- although I wish Dr. Perelman will eventually accept the accolades (including the Millennium Prize and the Fields Medal) that he so richly deserves -- I have to say that I admire his integrity and sincerity in trying to stand up for and protect the purity of mathematics. The Mandarin Chinese word for 'sincere' is 'zhenxin' -- it literally means 'pure heart' or 'genuine heart.' Dr. Perelman, from the bottom of my heart, I want to let you know that you have both a pure mind and zhenxin, a pure heart.


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