MOSCOW - In his office overlooking the faded pastel mansions along a St. Petersburg canal, a young Russian mathematician spent eight solitary years grappling with the Poincare Conjecture, one of the most famous and frustrating conundrums in math.
Now, colleagues say, Grigori Y. Perelman may not only have solved the century-old riddle. He may have helped advance many areas of math and physics, and made it possible to better understand the shape of the universe. "It seems like a very beautiful idea," one American colleague said.
If "Grisha" Perelman's proof of the Poincare is correct - and many mathematicians suspect it is - it will seal his transformation from an obscure researcher into one of the world's leading scientists.
And he will become the first person eligible to claim a $1 million prize offered by the Clay Mathematics Institute of Cambridge, Mass., for solving what it calls one of the seven central problems in math.
But the 37-year-old native of Leningrad, now St. Petersburg, doesn't seem interested in money or acclaim. While he could probably get a far more lucrative job in the West, he earns only about $200 a month at the Steklov Institute of Mathematics in St. Petersburg.
And he has rejected at least one mathematical prize. In 1996, he refused to accept an award in Budapest, Hungary, from the European Mathematical Society.
Perelman himself won't discuss any of this. "In my opinion, any public discussion of my work at the moment is premature and counterproductive," he wrote in an e-mail to The Sun several months ago. He didn't respond to another request for an interview this week.
Colleagues describe Perelman as jealous of his privacy, and fearful that public attention would distract him from his work. He may also be concerned that the talk of the prize money could make him a target of the Russian underworld.
"I think he wants to be a private person," said John W. Milnor, director of the Institute for Mathematical Science at the State University of New York at Stony Brook. "He doesn't want to be a media hero, where he can't walk out without being recognized, where he has a fear for his life."
Partly, Perelman is probably leery of prematurely claiming victory. Dozens of researchers have tackled the Poincare Conjecture. It goes to the heart of topology, or the mathematical study of surfaces, which holds that the world consists of two basic shapes, the sphere and the doughnut. Poincare speculated, in effect, that certain rules governing these three-dimensional shapes also apply to the same shapes projected into four and more dimensions.
Two years ago, the mathematician Martin Dunwoody of Southampton University in Britain caused a stir when he published a proposed proof. But Dunwoody, like all his predecessors, was later proved wrong.
Milnor cautioned that Perelman's proof could have hidden flaws. "It's been a puzzle and a challenge for 100 years," he said. "There have been many positive first steps and many false proofs. It's the kind of a subject where it's very easy to make a mistake if you're not careful."
Experts say it could take another six months to a year to verify Perelman's work, which is being scrutinized by teams around the world. But the work appears to have avoided the pitfalls of past efforts. Colleagues say even if his proof has hidden flaws, it represents a major advance in math.
"Although at the moment it is still too soon to declare a definitive solution to the problem, Perelman's ideas are highly original and of deep insight," wrote Michael T. Anderson of Stony Brook, in next month's issue of Notices of the American Mathematical Society.
Perelman has so far refused to publish his arguments, a mixture of esoteric math jargon and formulas, in a recognized journal, the traditional method for announcing scientific discoveries. Instead, he has posted his proof as three separate "preprints" -- or draft scientific papers - on an obscure Web site (www.arvix.org/abs/math.DG). A fourth preprint, containing the final elements of his proof, is expected to be posted there soon.
Excitement in the tiny international mathematics community has been building since November 2002, when the first preprint appeared.
Last spring, Perelman gave a series of lectures on his proof at the Massachusetts Institute of Technology, Stony Brook, the University of California at Berkeley and Princeton University. At each appearance, those in attendance say, he parried probing questions with rock-solid answers.
These lectures triggered news coverage that rivaled that given the Princeton mathematician Andrew Wiles a decade ago. Wiles' proof of Fermat's Last Theorem, a problem that had tormented number theorists for 350 years, was confirmed in 1995.
Although Perelman made little stir in math circles before his recent discovery, he made a big impression on his professors. "Grigori Perelman is one of the brilliant successors of earlier Petersburg mathematicians," said a former teacher, Gennadi A. Leonov, dean of the faculty of mathematics and mechanics at St. Petersburg State University.
After his appointment, Perelman published a few works in his areas of expertise, geometry and topology. Then for eight years, he toiled doggedly on the Poincare Conjecture, telling his Russian colleagues little about his work and publishing nothing.
Unlike most of their counterparts in the United States, researchers at the numerous institutes run by the Russian Academy of Sciences are not required to teach or publish. But even by Russian standards, Perelman's long academic silence was remarkable.
Over the past two years he has given numerous lectures to fellow mathematicians. But he has resisted becoming a scientific celebrity. He has refused interviews and asked colleagues not to disclose details of his personal life.
"He doesn't want any intrusions," said Eldar Ibradimov, director of the Steklov, the home to about 85 scholars and researchers. He referred to Perelman respectfully, using his first name and patronymic: "Grigori Yakovlevitch."
The mathematician Jean Pierre Serre recently called Poincare's Conjecture "central to our understanding of the mathematical world." It was proposed in 1904 by the Frenchman Henri Poincare, a nearsighted mining engineer and university professor who was described by one of his early teachers as "a monster of mathematics."
Poincare has been called the last of the great "universalists" in his field, meaning he was a master of all areas of math. But he is probably most famous as a founder of modern topology, the study of the geometric properties of elastic surfaces that don't change when they are stretched or bent.
In topology, a sphere is the only way to bend a two-dimensional plane into a shape without holes. In 1904, at the age of 31, Poincare speculated, in effect, that there was likewise only one way to bend three-dimensional space into a shape without holes. But he couldn't prove this was so.
This talk of bending space may sound like it has nothing to do with the real world. But Albert Einstein said that gravity does indeed bend space, meaning that our three-dimensional universe is curved. If you travel long enough in a straight line through the cosmos, physicists say, you will eventually wind up back where you started.
According to the Clay Institute rules, its so-called Millennium prizes can only be awarded two years after the publication of a proof. But the president of the institute has said that, if the proof holds up, Perelman may still be eligible to win .