Eccentric genius

Russian mathematician Grigori Perelman refuses the top-most honour in mathematics for solving the Poincare Conjecture.

R. RAMACHANDRAN in New Delhi

The biggest buzz today in the mathematics world is that the Poincare Conjecture has been proved. To the lay reader this perhaps does not mean much. But to mathematicians it is indeed a big achievement. For, a seemingly simple poser by the French polymath, Henri Poincare (1854-1912), in 1904 concerning the nature of three-dimensional space of finite extent had defied the best of the brains in the business all these years.

At the International Congress of Mathematicians (ICM), held in Madrid from August 22-30, the Conjecture was elevated to the status of a theorem after top mathematicians found no errors in the tersely written proofs by the Russian mathematician Grigori Yakovlevich (`Grisha') Perelman, which had been posted on the Internet over three years ago.

It would, therefore, seem extraordinarily strange that, having cracked such a monumental problem, Perelman should refuse the topmost honour in mathematics and shut himself out from the world of mathematics. At the end of May, a committee of nine prominent mathematicians had selected Perelman for the award of a Fields Medal, regarded as the equivalent of the Nobel Prize in mathematics , for his work on the Poincare Conjecture. In June, Sir John M. Ball, president of the International Mathematics Union (IMU), went to St. Petersburg to persuade Perelman to accept the prize to be awarded at the inaugural ceremony of ICM 2006.

Ball spent nearly ten hours over two days to convince him to accept the prize. He had even offered him the choice of accepting but not coming to receive it. But Perelman, it seems, plainly, but firmly, refused. To the outside world, mathematicians included, the attitude would smack of an arrogant genius. "He was very polite and cordial, and open and direct," Ball has stated in an interview. "The reasons centre around his feeling of isolation from the mathematics community and, in consequence, his not wanting to be a figurehead for it or wanting to represent it."

A recent article titled `Manifold Destiny: A legendary problem and the battle over who solved it' by Sylvia Nasar (the author of A Beautiful Mind) and David Gruber in the online edition of New Yorker dated August 21, provides an insight into the man's thoughts and views, his intellect and psyche. He had absolutely no interest in the coveted Fields Medal, he told the magazine in a rare press interview that he gave in June at his home in St. Petersburg. "It was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed," he said.

These words almost echo the famous 19th century mathematician Franz Ernst Neumann who said, "The discovery of truth is the greatest joy; recognition can add almost nothing to it." Indeed, this is not the first time that Perelman has refused an award. In 1996, when he was 30, he did not accept the award given by the European Mathematical Society to outstanding young mathematicians every four years.

"To do great work, you have to have a pure mind," Mikhail Gromov, the Russian Geometer, who saw logic in Perelman's action, told New Yorker. "The ideal scientist does science and cares about nothing else. He wants to live this ideal. Now, I don't think he really lives on this ideal plane. But he wants to."

Perelman was born on June 13, 1966, in a Jewish family in Leningrad (now St. Petersburg). His early mathematical education occurred at the world-famous Leningrad Secondary School, a specialised school with advanced mathematics and physics programmes. In 1982, as a member of the Union of Soviet Socialist Republics (USSR) team competing in the International Mathematical Olympiad, he won a gold medal, achieving a perfect score. In the late 1980s, Perelman got his doctoral degree at the Mathematics & Mechanics Faculty of the Leningrad State University, one of the leading universities in the former Soviet Union.

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences in St. Petersburg. In the late 1980s and early 1990s, he worked at several universities in the United States during which period he wrote several strikingly original papers. He was invited to give a lecture at the 1994 ICM in Zurich. He got job offers from Stanford, Princeton, the Institute for Advanced Study, and the University of Tel Aviv, all of which he spurned.

He returned to the Steklov Institute in 1996. Since then, he has worked in complete isolation and the rest of the mathematics community had little idea of his engagement with the Poincare Conjecture. While in the U.S. he had realised that his earlier results in an apparently unconnected field (known as Alexandrov spaces) could be applied to get over the serious obstacles encountered by other workers.

Between November 2002 and July 2003, Perelman posted three papers on the subject on the Internet. Given the history of the large number of failed attempts at proving the conjecture, mathematicians were naturally sceptical of Perelman's claims. However, given his significant contributions in geometric analysis, his papers were taken seriously. Writing in the Notices of the American Mathematical Society, Allyn Jackson points out that Perelman's deep new ideas in geometric analysis were able to overcome the earlier difficulties with spectacular effect.

In the spring of 2003, after his first two papers had appeared on the Web, Perelman was invited to lecture on his work at several U.S. universities, including Princeton, Massachusetts Institute of Technology (MIT), Columbia and Stony Brook. Thereafter he returned to St. Petersburg and has given very few lectures since then. Initially he had responded to mathematical questions by e-mail but later even this stopped. He did not seem interested in publishing them either.

His papers were tough going even to the experts but they found them to be extremely carefully written. It became clear that they had to fill in lot of details that Perelman had skipped in his condensed style to understand them fully. And several major projects were undertaken to verify and understand Perelman's claims. Notable among these are the works of Bruce Kleiner and John Lott, and John Morgan and Gang Tian, which have been sponsored by the Clay Mathematical Institute (CMI). Perelman is now a serious contender for CMI's million-dollar Milennium Prize announced in 2000 for solving the Poincare Conjecture. The Kleiner-Lott paper and the 400-page work of Morgan and Tian, which fill in all the details in Perelman's proof of the Poincare Conjecture, were posted on the Internet on May 25 and July 25 respectively. The latter is due to appear as a book in early 2007.

However, to be eligible for the CMI award the work should have been published in a peer-reviewed journal. While Perelman's papers have not been refereed in the traditional sense, they have been subjected to extraordinary scrutiny in the three and a half years since their posting on the Internet. "The fact that Perelman pursued an unorthodox route... is not an obstacle," Jim Carlson, CMI president, is said to have stated. "At the appropriate time," Carlson has said, "the Clay Institute will consider all the available materials and make a judgment if the proof is really correct."

One question that CMI will have to face is whether the award should go to Perelman alone or others too, notably Richard Hamilton, who formulated the geometric framework for the approach to the solution. It is not clear if Perelman's rejection of awards extended to the Millennium Prize. "I'm not going to decide whether to accept the prize until it is offered," he told New Yorker.

There is a visible twist in the offing, however. In June 2006, the Asian Journal of Mathematics published a paper by Xi-Ping Zhu of Sun Yat-sen University in China and Huai-Dong Cao of Lehigh University in Pennsylvania, U.S., claiming to give a complete proof. The true extent of the contribution of Zhu and Cao as well as the ethics of Yau's involvement remain a matter of contention. Yau is both editor-in-chief of the journal as well as Cao's doctoral adviser. It has been suggested that Yau was intent on being associated, directly or indirectly, with the proof of the conjecture and pressured the journal's editors to accept Zhu and Cao's paper on unusually short notice.

In fact, Indian members of the journal's editorial board have written to Yau taking serious objections to the manner in which the paper was hurriedly published without the laid-down process of peer review.

In a lecture on the Poincare Conjecture that Yau presented at the Beijing International String Theory conference in June, he has taken special care to mention dates of various works that have followed Perelman's papers with a clear attempt at claiming priority for the Cao-Zhu paper over others for the Millennium Prize. "In Perelman's work," Yau said in a lecture, "many key ideas of the proof are sketched or outlined, but complete details of the proofs are often missing. The recent paper of Cao-Zhu, submitted to the Asian Journal of Mathematics in 2005, gives the first complete and detailed account of the proof of the Poincare Conjecture and the Geometrisation conjectures." MIT mathematician Dan Stroock has been quoted as saying, "I find it a little mean of [Yau] to seem to be trying to get a share of this as well."

"It is not clear to me what new contribution they did make," Perelman told New Yorker. "Apparently, Zhu did not quite understand the argument and reworked it."

Speaking of Yau, Perelman is supposed to have 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."

Nasar and Gruber say in their New Yorker article that the prospect of being awarded a Fields Medal had forced Perelman to make a complete break with his profession. "As long as I was not conspicuous, I had a choice... Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit."