Million Dollar Math Problem

Grigory Perelman and the mathematical breakthrough of the century—the Poincaré Conjecture.

Grigory Perelman
Grigory Perelman.

In 2000, the Cambridge, Massachusetts-based Clay Mathematics Institute — a nonprofit organization devoted to popularizing mathematical ideas and encouraging their professional exploration — identified seven exceptionally difficult math problems, and offered a million dollars for the solution of each. One was the Poincaré Conjecture, a classic of topology that was formulated by Henri Poincaré in 1904. No one expected that this particular problem—or any of the six others—would be solved anytime soon, which explains why the mathematics community was thrown for a loop when Russian mathematician Grigory Perelman, 43, posted his proof of the Poincaré Conjecture on the Internet in November 2002. Even more stunning—except to those familiar with his work—was that his proof turned out to be correct.

In the wake of this feat, Perelman did not behave as one might expect. As Russian journalist Masha Gessen notes in her new biography, “Perfect Rigor: A Genius + the Mathematical Breakthrough of the Century” (Houghton Mifflin Harcourt), [he] “did not publish his work in a refereed journal. He did not agree to vet or even review the explications of his proof written by others. He refused job offers from the world’s best universities. He refused to accept the Fields Medal, mathematics’ highest honor.” He even withdrew from the world of mathematics. And if the Clay Institute offers the million dollars that comes with the Millennium Prize, he probably won’t move to collect it.

With all of the above in mind, Failure took the opportunity to question Gessen about “Perfect Rigor” and her remarkable subject.

What is the Poincaré Conjecture?

It is no more, actually. Now that Perelman has proved it, it’s a theorem—a classic theorem of topology, one of the most wonderfully weird mathematical disciplines. Much of topology is concerned with things that are essentially the same as other things, even if at particular moments in time they happen to look different. For example, if you have a blob that can be reshaped into a sphere, then the sphere and the blob are essentially similar, or homeomorphic, as topologists say. Poincaré asked, in essence, whether all three-dimensional blobs that are not twisted and have no holes in them are homeomorphic to a three-dimensional sphere. It took more than a hundred years to prove that yes, they are.

What is the significance of this discovery?

A discovery like this generally has far-reaching repercussions that are rarely evident at the moment of the breakthrough. It will almost certainly have profound consequences for our understanding of space—the universe we inhabit.

How were you able to write Perelman’s biography without ever talking to the man?

It was the only way to do it. When I first began researching the book, the only person he was speaking with was Sergei Rukshin—his lifelong math tutor, his competition coach, and in many ways, the architect of his life. But sometime in the last couple years, Perelman stopped talking to Rukshin as well. As far as I know, the only person with who he is still in contact with is his mother, with whom he shares an apartment on the outskirts of St. Petersburg.

While I had no access to Perelman, I talked to virtually all the people who have been important in his life: Rukshin, his classmates, his math club mates, his high school math teacher, his competition coaches and teammates, his university thesis advisor, his graduate school advisor, and those who surrounded him in his postdoctoral years. I think these people were motivated to speak with me because Perelman himself wouldn’t—and because they felt his story had been misinterpreted in so many ways.

It almost sounds as if not talking to Perelman was an advantage.

In some ways, yes. When you write a biography, you are in constant negotiation with that person’s view of himself. So you are always balancing your own perceptions against the subject’s aspirations, and this can be painful for all involved. All I had was my research material and my own perceptions, so it was a little like writing a novel. I was constructing a character.

What made you believe you could pull this off?

Actually, I made two erroneous assumptions. First, I assumed that the journalists who wrote about Perelman back when he turned down the Fields Medal [in 2006] were wrong. I figured he was not as weird as they made him sound. I expected he was a familiar type of Russian scientist—entirely devoted to his field, not at all attuned to social niceties and bureaucratic customs, and given to behaviors that can be misinterpreted, especially by foreign journalists.

My second assumption, related to the first, was that my background gave me the tools to describe him. I’m Perelman’s age, and I come from the same kind of family—socially, educationally, and economically. That is, Jewish engineers with two children living on the outskirts of Leningrad in his case and Moscow in mine. That was barely a start, though, because Perelman turned out to be much stranger than I expected.

What is it about Perelman that allowed him to solve perhaps the most difficult mathematical problem ever solved?

Perelman has a mind that is capable of taking in more information than any mathematical mind that has come before. His brain is like a universal math compactor. He grasps complex problems and reduces them to their solvable essence. The problem is that he expects human beings to be similarly subject to reduction. He expects the world to function in accordance with a set of strictly laid out rules, and he cannot take in anything that does not conform to those rules. And because the world is so unruly, Perelman has had to cut off successive chunks of it. All that is left for him now is the apartment he shares with his mother.

Has Perelman ever experienced failure in a mathematical setting?

His single known failure was not really a failure. As a 14-year-old he took second place in the national math competition in the Soviet Union. He had never placed second before and apparently resolved never to finish second again. He succeeded, as is his habit.

What do you think the future holds for Perelman?

Some people who are very fond of him have speculated that when he is finally awarded the Millennium Prize, he will come out of hiding, claim his just reward, and perhaps reveal that he never really abandoned mathematics. It’s a wonderful but unlikely scenario. The commercialization of mathematics offends him. He was deeply hurt by the many generous offers he received from U.S. universities after he published his proof. He apparently felt he had made a contribution that was far greater than any amount of money—and rather than express their appreciation in appropriately mathematical ways, by studying his proof and working to understand it—they were trying to take a shortcut and basically pay him off. By the same token, the million dollars will probably offend him. I don’t think we will be hearing from Perelman again.

The seven Millennium Problems of the Clay Mathematics Institute