Perfect Rigour

by Masha Gessen

  Politicians joined in the madness too. The St. Petersburg city council considered stationing guards outside the apartment he shared with his mother. It seemed that everyone wanted to give him money. A cabinet member asked to talk to him. Perelman wanted no part of this. Elderly teachers of his, approached by powerful, respected men, agreed to act as intermediaries and called him. He shouted profanities, which the teachers would not repeat. They told me only that he had been rude, very rude. On one occasion, a private Moscow foundation in cooperation with Rukshin cooked up a scheme to give money to his mother, a sort of reward for nurturing a genius son. Perelman overheard her speaking on the phone and ripped the receiver out of her hands, shouting. The once meek, conspicuously well-behaved Jewish boy had, cornered, turned into a domestic tyrant. If the world was not going to respect his seclusion, he would consider the world—the whole world—his enemy.

  A year later, when I asked Rukshin to get a copy of Morgan and Tian’s new book to Perelman, Rukshin demurred; the last time he had tried to pass on a gift from a foreign admirer, he said, Perelman had lobbed the gift—a classical-music CD—at Rukshin’s head.

  PERFECT RIGOR

  THE MILLION-DOLLAR QUESTION

  11

  The Million-Dollar Question

  WHEN JIM CARLSON1 was in elementary school, he found arithmetic tedious; his mind wandered. His mother had to tutor him with flash cards to prevent a failing grade. When Carlson was in his senior year of high school, his mathematics teacher handed him a typewritten sheet of paper and sent him to the back of the room. The sheet contained the names of a dozen books on mathematics that the teacher thought Carlson would find interesting—and he could study them on his own time in the back of the room so long as he got his other work done. The list included Courant and Robbins’s classic What Is Mathematics?, where Carlson read about irrational numbers, among other things, for the first time. When Carlson started college at the University of Idaho in 1963, he planned to major in either physics or psychology. He never took a course in psychology; physics fared a little better, but by the time Carlson was a sophomore, he was doing graduate-level work in mathematics.

  He received his PhD from Princeton in 1971, taught at Stanford and Brandeis, and finally settled at the University of Utah, where he spent a quarter of a century, eventually becoming chair of the mathematics department. Then he left for Cambridge, Massachusetts, to run the Clay Mathematics Institute. He had taken the job for a variety of reasons, including the fact that the schedule suited his personal circumstances, but the mission suited him as well. His job was to promote mathematics. Part of that job was to ensure that children and young people would enter mathematics in more elegant ways than he had—that is, through the back of the mathematics classroom. In a sense, he had to give American mathematics some of the luster and streamlined institutionalization that distinguished Russian mathematics. And one of the tools he was handed for popularizing mathematics was the ambitious and extremely well-funded Millennium Prize project. Though truth be told, Jim Carlson did not expect to be wielding that sort of money; he did not think any of the Millennium Problems would actually be solved in his lifetime.

  Carlson assumed his position as president of the Clay Institute in the summer of 2004, just as the controversy that would eventually surround Perelman’s proof and his prize started to brew. I always had the impression that to be who he was and do what he did, Carlson had to constantly keep at bay great, potentially overwhelming shyness. He was soft-spoken, retiring, exceedingly polite, and the last person one could imagine at the center of a controversy. Fortuitously, when he began his tenure as president of the Clay Institute, he did not know enough to expect the kind of media storm that ultimately surrounded the award. “I heard reports [about Perelman’s preprints],” Carlson recalled when he talked to me. “I actually remember thinking, ‘My goodness, isn’t this fantastic that perhaps there will be a solution to the Poincaré Conjecture?’ And of course I started thinking about the Millennium Prizes. And of course, isn’t this remarkable, it will certainly be the only one in my lifetime that anyone will receive. But you know, one really doesn’t know. I liken it to an earthquake: You know it when it happens. And maybe you could say that tension is building up in the rocks, but no one has successfully been able to predict earthquakes. And nobody knows when somebody will find that breakthrough idea that leads to a solution.”

  That was what Carlson thought a couple of months before he took the reins at the Clay Institute. He knew that Perelman had posted his preprints on the arXiv—not an unusual circumstance these days; many mathematicians post their articles as soon as they submit them to journals, to spur mathematical discussion before the peer-review process is over. But it was becoming apparent that Perelman had not submitted his papers to any journals and had no intention of doing so. What had seemed a perfectly innocuous and self-evident condition of the Millennium Prizes was emerging as a potential sticking point.

  Carlson steered the Millennium boat gracefully and skillfully, funding workshops on Perelman’s proof and on Kleiner and Lott’s and Morgan and Tian’s work explicating it. When he talked to me, he likened Perelman’s work to a “flash of light that allows you to get through the forest.” Sure, “there is a lot of work to be done, you have to cut down a lot of trees and climb over some boulders and stuff, but it’s finding that new way that is so difficult. And if you can’t find that, it doesn’t matter how much work you do, it will be in vain. And this is what Perelman did.” The projects undertaken by those who wrote the explications were clearly much less rewarding than the original solution, and this too filled Carlson with admiration—both for the mathematicians and for the mathematical system, which somehow bent itself to the unusual conditions set by Perelman to deliver the kind of examination and explanation his proof required.

  Carlson opened his MacBook Air to read aloud a passage he had found particularly striking, from Kleiner and Lott’s published notes on Perelman’s proof: “Here it is. ‘We did not find any serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman.’ I think that is a very accurate statement of what happened. You know, there was a very substantial amount of work to be done to ensure this was correct and complete. But the key thing is that there were no ‘serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman.’ And there were many methods and ideas. It’s always hard to communicate these to a general audience, but I hope you can do that when you write your book.” What he wanted me to say, in other words, was that Perelman was the indisputable author of the proof, and that Kleiner and Lott had affirmed this in a way that Carlson greatly admired.

  The months leading up to the ICM in Madrid, with the Cao and Zhu paper and the unfamiliar media attention, had been nerve-racking. But the ICM seemed to settle the score, and the evidence of plagiarism that emerged in the fall of 2006 rendered the issue of authorship entirely moot. The publication of Morgan and Tian’s book on the proof followed; the Clay Institute commenced the two-year waiting period required by the rules of the Millennium Prizes. At the end of that time it will appoint a committee, which could make its recommendations by fall 2009. Barring the emergence of an error in the proof or some other unforeseen and highly improbable disaster, the committee will recommend that the million-dollar prize be given to Grigory Perelman. Which leaves only one question: Then what?

  If Perelman’s reasoning on prizes, awards, and honors were consistent, he might accept the Clay million if it was offered to him. After all, his stated objection to the European prize had been that it would have been given for work he did not consider complete. Nothing of the sort could be said of the Poincaré proof. Not only did other mathematicians consider it complete but Perelman himself clearly believed he had completed his project this time. His objection to the Fields Medal, though never stated as clearly, seemed to have been twofold: first, he no longer considered h
imself a mathematician and hence could not accept a prize intended for the encouragement of midcareer researchers; and second, he wanted no part of the ICM, with all the attendant publicity, speeches, ceremony, and king of Spain.

  The Clay prize, however, was designed to be awarded for a particular achievement; there was no stipulation that the recipient had to continue practicing mathematics. Nor did it necessarily require any ceremony. It was an honor bestowed on a mathematician by his colleagues, with no nonmathematical royalty involved. And it was different from both the European prize and the Fields Medal in another very important respect: it represented recognition of Perelman’s singular achievement. He could not be compared with any other recipients, concurrent or past—indeed, there was some likelihood that no one alive today would see another Millennium Prize bestowed.

  “I think he might have a plan,” Alexander Abramov, Perelman’s former olympiad coach, told me. “He may have decided that when he is awarded the Clay prize, he will actually accept it because it will be a sign of total recognition and then he could live however he wants to live and not be dependent on anyone.” Abramov paused. “But you see, that’s just because one needs to come up with some sort of reasonable hypothesis here.” That is, one needed to contemplate happy-ending scenarios for Perelman because otherwise, if one cared about Perelman, one might be scared for him, as Abramov was. “I fear this is a situation that will end badly,” he said. “He is too full of stuff and too alone.” Abramov was yet another person who gave up on calling Perelman after Perelman had grown abrasive on the phone. Before that happened, Abramov had called occasionally, offering support, both moral and financial. For example, he had suggested that if Perelman wanted no part of prizes, he could write an article for Kvant, the popular-science magazine founded by Kolmogorov and at which Abramov was now an editor, and receive money for it. Perelman turned down all offers, including Abramov’s offer of his friendship. “He told me,” Abramov recalled, “that one of his principles was ‘One should not force one’s friendship on anyone.’ So I asked him if he knew the story of Kolmogorov and Pavel Alexandrov’s friendship, and he showed a sudden interest in this topic and we talked about it for about ten minutes. He was most interested in the story of Kolmogorov slapping Luzin”—the time Kolmogorov attacked his and Alexandrov’s former teacher after Luzin failed to cast a promised vote to induct Alexandrov into the Academy of Sciences. Happy to locate any common ground with his former student, Abramov offered to send Perelman a book on Kolmogorov and Alexandrov. “I’m not reading anything,” said Perelman, using the excuse he used to reject all offers of books, including books on his own proof. Abramov was inclined to see some hope in the exchange he had had with Perelman: “At least he has not lost all interest in all things.” But I am inclined to interpret it differently. It seemed that Perelman was then getting ready finally to end his last remaining close personal relationship outside of his mother, that with Rukshin. Sometime in the winter or spring of 2008, Perelman cut off all contact with his former teacher.

  But before Perelman stopped speaking to Rukshin, the two spent some time talking about the million-dollar prize, and apparently they jointly worked out their approach to it. Just like the rest of the world of mathematics, they believed, the Clay Institute had betrayed Perelman. Rukshin even suggested to me that Clay had changed its rules along the way, introducing the refereed-publication requirement and the two-year waiting period just to delay giving Perelman the money, or possibly to avoid giving it to him altogether. There is in fact no evidence of any changes being made to the Clay Millennium rules after the prizes were instituted, in 2000. Indeed, someone in Jim Carlson’s position might find himself wishing for a way to postpone the decision and the subsequent probable failure to convince Perelman to accept it and then the uneasy publicity that would accompany the award. This series of events would certainly not be the story of mathematical triumph and glory that the Clays had envisioned, and while it would fulfill the stated goal of attracting the public’s attention to mathematics, it would hardly qualify as the fairy tale meant to inspire droves of young people to pursue mathematical careers. Jim Carlson might well have wished to put off navigating this tricky terrain. But there is no evidence that he did. In fact, he did everything in his power to speed up the process, driven mostly by the desire to fulfill his weighty mission by helping to affirm Perelman’s achievement, but also a little bit by the hope of meeting Perelman himself.

  In the spring of 2008, Carlson was planning a trip to Europe. He decided to take a detour to St. Petersburg. It seemed as good a time as any: the controversy had died down, there was no lingering doubt about Perelman’s proof, and the moment when someone — probably Carlson himself—would have to ask Perelman to accept a million dollars was very clearly approaching. It was time to start talking to Perelman.

  Carlson was perhaps hoping for a conversation much like the one John Ball had had with Perelman—long and in-depth, if fruitless. He had little reason to expect the conversation would end any differently, but he had to hope for this nonetheless.

  Carlson called Perelman from his hotel room on his first day in St. Petersburg. He introduced himself and proceeded to explain the Clay prize timetable to Perelman. He repeated all the things Perelman surely knew—that two years had to pass following refereed publication, and that Morgan and Tian’s book had provided the starting point for the countdown. He said the committee would likely be appointed as soon as May 2009 and might report back in August 2009.

  Perelman listened politely.

  Carlson did not ask whether Perelman would accept the money if offered. “The way the conversation was going,” he explained to me, “I didn’t think it was appropriate.” A wave of shyness, held back for so long, may have broken through. Or perhaps Carlson simply wanted to hold off on asking the question, allowing himself another year of slim hope that Perelman might accept the prize. “I didn’t get the sense that the door is completely closed,” Carlson told me.

  At the end of the conversation, Perelman said, “I don’t see any point in our meeting.”

  The next day, I found Carlson at the Steklov, visiting with his old friend Anatoly Vershik, chairman of the St. Petersburg Mathematical Society and the man who once nominated Perelman for the European prize he later turned down. Vershik and Carlson were having tea. Yau’s name came up; he was apparently holding a conference to celebrate his fifty-ninth birthday.2 “I don’t understand it,” Vershik grumbled. “I know Gian-Carlo Rota held a conference to celebrate his sixty-fourth birthday,3 but sixty-four is two to the sixth power—and what is fifty-nine? A prime number!” This was mathematicians gossiping.

  Carlson spent the rest of his three-day visit seeing old mathematics friends, practicing his cello—a special, highly geometrical travel model—in his hotel room, and thinking about Perelman and the prize. He concluded that no matter what Perelman decided, the Clay prize could be used for the benefit of mathematics. In fact, it already had been. “It’s good to explain to the public that there are unsolved mathematical problems,” he told me when we went out for some exotic midday vodka at a café called the Idiot. “Surprisingly, a lot of people don’t know that.”

  It is true, Carlson admitted, that a lot of mathematicians criticize monetary prizes for their superficiality; some find it offensive. His friend Vershik had published a piece4 criticizing the Clay Millennium Prize on these exact grounds. But Carlson told me he had many conversations with undergraduates who wanted to know what these million-dollar problems were. In a way, the buildup to the prize had brought unexpected benefits: “To spend no money to get mathematics in the public eye is not a bad accomplishment,” Carlson boasted. Perelman had been his unwitting accomplice: “There is more interest in the public eye in a person who has no interest in the money.”

  Carlson was not simply putting on a brave face, though he was certainly doing that. He clearly felt that, in an awkward way, he was helping to draw attention
to an accomplishment that deserved it. In all my conversations with Carlson, I never perceived any resentment of Perelman, which set him slightly apart from other mathematicians I interviewed: unlike Kleiner, Carlson had not had to cede any of his professional ambition to Perelman’s achievement; unlike Tian, he had not been personally slighted by Perelman. He did not understand Perelman—or claim to understand him. All he had was abiding respect for him.

  The only person who not only claimed to understand Perelman but at times seemed to channel him was Gromov.

  “Do you think he’ll accept the million dollars?” I asked Gromov.

  “I don’t think so.”