The Mystery of Grigori Perelman

Russian Grigori “Grisha” Perelman

Fischer never returned to the U.S., living for a time in Japan, where he was arrested for using a revoked passport. Fischer eventually received an Icelandic passport, and lived in that country until his death in 2008.

Someone else did something even more amazing than Fischer, then — disappeared, Russian Grigori “Grisha” Perelman. Perelman solved the Poincaré conjecture, the only one of the seven Millennium Prize Problems that has been solved. The award for a solution to any of the problems is US$1 million.

The seven problems were stated by the Clay Mathematics Institute on May 24, 2000, and they are:

• The Birch and Swinnerton-Dyer conjecture
• The Hodge conjecture
• The Navier-Stokes existence and smoothness problem
• The P versus NP problem
• The Poincaré conjecture
• The Riemann hypothesis
• The Yang-Mills existence and mass gap problem

The Poincaré Conjecture

The Poincaré conjecture is a problem in the mathematical field of topology, which focuses on the intrinsic properties of spaces. To a topologist, a bagel and a coffee cup with a handle are the same since each has a single hole, and each can be manipulated to resemble the other without being torn or cut.

Henri Poincaré (1854 – 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

Poincaré used the term “manifold” to describe these abstract topological spaces. The simplest possible two-dimensional manifold is a sphere. The proof that an object is a so-called two-sphere is that it is “simply connected”, meaning that no holes puncture it. If you tie a slipknot around a sphere, you can pull the slipknot closed by sliding it along the surface of the sphere. By contrast, if you tie a slipknot around a bagel through its hole, you cannot pull the slipknot closed without tearing the bagel.

Since the 1960s, Poincare’s conjecture had been been proven for all dimensions except the third. The four-dimensional case was solved in 1982 by Michael Freedman. Beginning in 2002, and over the course of eight months, Perelman published three papers on the public website arXiv. Building on the work of the mathematician Richard S. Hamilton, the papers used what’s called the Ricci flow to attempt to solve the Poincaré conjecture.

Hamilton had introduced a modification called Ricci flow with surgery to remove problem regions as they arose, but he had been unable to complete the proof.

By 2006, several teams of mathematicians verified that Perelman’s proof was correct, and in August 2006, Perelman won the coveted Fields Medal, which is equivalent to the Nobel Prize, but in mathematics.

The Fields Medal

The Fields Medal is awarded only once every four years, during the International Congress of the International Mathematical Union (IMU). The medal is awarded to up to four mathematicians, but they must all be under 40 years of age.

Perelman shockingly declined the award stating, “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” He additionally said of the medal, “It was completely irrelevant for me … everybody understood that if the proof is correct then no other recognition is needed.”

On December 22, 2006, the scientific journal Science recognized Perelman’s proof of the Poincaré conjecture as the scientific “Breakthrough of the Year”. It was the magazine’s first such recognition in the field of mathematics.

The Millennium Prize

On March 18, 2010, The Millennium Prize committee announced that Perelman had met its criteria to receive the first Clay Millennium Prize for his resolution of the Poincaré conjecture. On July 1, 2010, Perelman rejected the prize, saying that his contribution was no greater than that of Richard Hamilton.

“Grisha”

Grigori “Grisha” Perelman was born on June 13, 1966 in Leningrad, Soviet Union, now called Saint Petersburg, Russia, to an electrical engineer father and a mathematician mother. Perelman’s mother gave up her graduate studies to raise Grigori and his younger sister.

Perelman’s mathematical talent was apparent early, and he attended the Leningrad Secondary School #239, which was a specialized school with advanced mathematics and physics programs. In 1982, Perelman was named a member of the Soviet team competing in the International Mathematical Olympiad, an international competition for high school students. Perelman won a gold medal, achieving a perfect score.

At 16, Perelman entered the School of Mathematics and Mechanics at the Leningrad State University, completing his Ph.D. in 1990. After working at the Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences, Perelman accepted a position at the Courant Institute in New York University, and at the State University of New York at Stony Brook.

In New York, Perelman was miserable, subsisting on traditional Russian brown bread and cheese. In 1993, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley, and it was there that he proved the soul conjecture in 1994.

After that, Perelman had his pick of jobs at the U.S.’s top universities, including Stanford and Princeton, but he rejected them all, and in 1995, he returned to the Steklov Institute in Saint Petersburg. There, he had a research position that paid less than a hundred dollars a month.

Perelman told a colleague at the Steklov, “I realize that in Russia I work better.”

A Disappearance

In 2006, Perelman abruptly quit his job at the Steklov Institute and withdrew from view, living on the carefully-saved money from his time in America. Perelman had been the victim of an attack.

In an extraordinary piece of detective work, Sylvia Nasar and David Gruber described in an August 28, 2006 article in The New Yorker magazine what had occurred. Nasar is the author of “A Beautiful Mind” about Nobel Prize winning mathematician John Forbes Nash, which was turned into a 2001 movie starring Russell Crowe.

Nasar and Gruber described how in a June 20, 2006 lecture, Harvard mathematician Shing-Tung Yau had implied that the Poincaré conjecture had actually been solved by two of his graduate students – Xi-Ping Zhu and Huai-Dong Cao. Criticizing Perelman’s proof, Yau had said, “We would like to get Perelman to make comments. But Perelman resides in St. Petersburg and refuses to communicate with other people.”

According to Nasar and Gruber, Yau had a history of trying to negate other mathematicians’ proofs. In 1997, a former student of Yau’s, Kefeng Liu, presented a paper he co-authored with Yau on mirror symmetry. It was strikingly similar to a paper presented by a young geometer at Berkeley named Alexander Givental.

To add insult to injury, at the same time Givental had received an e-mail from Yau and his collaborators, saying that they had found his arguments impossible to follow and his notation baffling, and that they had come up with a proof of their own. They praised Givental for his “brilliant idea” and wrote, “In the final version of our paper your important contribution will be acknowledged.”

A few weeks later, the paper, “Mirror Principle” had appeared in the Asian Journal of Mathematics, which is co-edited by Yau. In it, Yau and his coauthors described their result as “the first complete proof” of the mirror conjecture. Of Givental’s proof, they wrote, “Unfortunately, [his proof] which has been read by many prominent experts, is incomplete.” However, Yau and his coauthors failed to identify what in Givental’s proof was incomplete.

In June 2006, the Asian Journal of Mathematics published Zhu and Cao’s paper which was entitled, “A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the Hamilton-Perelman Theory of the Ricci Flow.” The abstract stated, “This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.”

Crowning Achievement Indeed

Zhu and Cao wrote that they had to “substitute several key arguments of Perelman by new approaches based on our study because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program.” As quoted in The New Yorker article, mathematician John Morgan said. “I don’t see that they [Zhu and Cao] did anything different.”

On May 25, 2006, Bruce Kleiner and John Lott, of the University of Michigan, had posted a paper on arXiv that filled in the details of Perelman’s proof of the Geometrization conjecture, and thus the Poincaré conjecture. In November 2006, Cao and Zhu admitted in the A.J.M. that they had failed to cite properly the work of Kleiner and Lott, and in that same issue, the A.J.M. editorial board issued an apology for what it called “incautions” in the Cao–Zhu paper.

On December 3, 2006, Cao and Zhu retracted their original paper, which had been entitled “A Complete Proof of the Poincaré and Geometrization Conjectures — Application of the Hamilton–Perelman Theory of the Ricci Flow” and posted a renamed version that was more modestly titled, “Hamilton–Perelman’s Proof of the Poincaré Conjecture and the Geometrization Conjecture”.

Nasar and Gruber Find Perelman

In 2006, Nasar and Gruber traveled to St. Petersburg and tracked down Perelman in his apartment. When asked, Perelman repeatedly said that he had retired from the mathematics community and no longer considered himself a professional mathematician. He said that he was dismayed by the field’s lax ethics. Of Yau, Perelman said, “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.”

Discussing with Nasar and Gruber Perelman’s refusal of the Fields Medal and Millennium Prize, a colleague of his, Mikhail Gromov, said, “The ideal scientist does science and cares about nothing else.” “[Perelman]” wants to live this ideal.”