Perfect Rigour

by Masha Gessen

  “Why not?”

  “He has his principles.”

  “What principles?”

  “Because Clay is a nothing, from his point of view—why should he take his money?”

  “Okay, Clay is a businessman, but it’s Perelman’s colleagues who are making the decision,” I objected, using a word that in Russian meant both “decision” and “solution.”

  “Those colleagues are playing along with Clay!” Gromov was very irritated now. “They are deciding [solving]! He has no use for any of their solutions! He has already solved the theorem, what’s there left to solve? No one is solving anything! He solved the theorem.”

  EPILOGUE

  A few minutes after ten in the morning on June 8, 2010, several hundred people crowded onto the steps and sidewalk in front of the Institut Océanographique in Paris. The event for which they had traveled from as far away as Russia, the United States, Australia and Japan had been planned at the Institut Henri Poincaré next door, but it proved too small for what would certainly be one of the oddest award ceremonies ever held.

  Two months earlier, the Clay Institute had made the long-expected announcement, and Jim Carlson had made the call informing Grisha Perelman that he had been awarded the million-dollar prize. Perelman had been cordial but had made it clear he would not attend the Paris ceremony. Nor would he make anyone’s life easier by announcing ahead of the ceremony whether or not he planned to accept the prize. In the end, to celebrate Perelman’s accomplishment and the first Millennium Prize award, the Clay Institute had planned two full days of lectures and an event that the printed program called, simply, The Ceremony. Something—it was not certain what—would happen in the presence of some of the best mathematical minds of our age.

  The first lecturer was Sir Michael Atiyah, the British mathematician who had spoken on the Poincaré Conjecture at the original Millennium Meeting almost exactly ten years earlier. Back then he had correctly predicted that the proof of the Poincaré would have to employ tools found outside of topology. Now he gave a talk outlining a history of mathematics from the point of view of dimensions: In the nineteenth century mathematicians studied two dimensions, the twentieth century in mathematics was spent in three dimensions, and the twenty-first, thrown open by Perelman’s work, would conquer the fourth dimension. John Morgan followed Atiyah with an overview of the history of the Poincaré.

  One after another, the greatest names in mathematics took the stage. Curtis McMullen gave a witty overview of the Geometrization Conjecture, complete with slides picturing bunny rabbits, mushrooms and dinosaurs, all of which represent shapes composed of Thurston’s eight proposed geometries. McMullen also noted that he had once heard Perelman speak and “back then it was already apparent he was immune to commonly accepted ways of thinking.” The audience giggled.

  Thurston himself said what all the mathematicians present could say: “Perelman managed to do what I could not.” Stephen Smale seconded, along with Misha Gromov, who called his protégé’s work the greatest accomplishment of the century. Andrew Wiles, who proved Fermat’s Last Theorem—and who was the only speaker with no personal relationship to the Poincaré—pointed out how surprising it was that Perelman’s solution materialized so soon after the Millennium Problems were announced.

  It was, in other words, a proper and festive occasion. The speakers were all on top form: Atiyah cracked jokes that made the listeners roar with laughter; McMullen showed slides that made the audience gasp; Thurston all but danced around the stage, gesticulating widely, as though the imaginary shapes he was describing were just beyond his reach. All was as it should be except for two conspicuous absences: Perelman and Richard Hamilton were not there.

  In the late afternoon, Landon Clay took the stage, holding an object in his hands. “It gives me great pleasure,” he said, “to award this prize to whoever takes it.” He read aloud the inscription on the glass sculpture in his hands—“The Millennium Prize is awarded to Grigory Perelman for proving the Poincaré Conjecture”—and handed it to Jim Carlson, once again saddling him with the task of actually presenting the award.

  * * *

  A week after the Paris ceremony, Perelman called Carlson himself to inform him that he would not be accepting the million dollars. The Clay Institute’s board would now have to decide how to use the money for the benefit of mathematics. An implied condition was that the money be applied in a way that Perelman himself would not find insulting or inappropriate. As of this writing, this problem remained unsolved.

  January 2011

  PERFECT RIGOR

  ACKNOWLEDGMENTS

  Acknowledgments

  I owe a special debt of gratitude to all the sources in this book. Writing about a man who does not wish to be written about is an unusual undertaking, and the decision to talk to me could not have been easy for some of Perelman’s friends and teachers. In particular, Alexander Golovanov, Viktor Zalgaller, and Sergei Rukshin went to great lengths to try to make me understand their friend and student, and I hope that this book reflects at least some of their insights. I am also very grateful to Jim Carlson, Sergei Gelfand, and especially Leonid Dzhalilov for doing what they could to ensure I wrote about mathematics in a way that made sense. Any mistakes are, of course, still mine. Finally, many thanks to my agent, Elyse Cheney, and my editors Becky Saletan and Amanda Cook, for making this book much better than it would otherwise have been.

  NOTES

  NOTES

  Notes

  Prologue

  Millennium Meeting descriptions and quotes: The CMI Millennium Meeting, documentary, directed by François Tisseyre (New York: Springer, 2002).

  1. Escape into the Imagination

  “The mathematician needs no laboratories or supplies”: A. Ya. Khinchin, “Matematika,” in F. N. Petrov, ed., Desyat Let Sovetskoy Nauke (Moscow: N.P., 1927).

  “Mathematics is uniquely suited to teaching”: “Sudby matematiki v Rossii,” a lecture by Mikhail Tsfasman, http://www.polit.ru/lectures/2009/01/30/matematika.html, accessed February 1, 2009.

  The movement’s slogans were based on Soviet law: Alexander Yesenin-Volpin, interview, http://www.peoples.ru/family/children/alexander_esenin-volpin/, accessed January 31, 2009.

  His article on linguistics: I. V. Stalin, “Marxism i voprosy yazykoznaniya,” Pravda, June 20, 1950, http://www.philology.ru/linguistics1/stalin-50.htm, accessed January 31, 2009.

  Stalin personally promoted: V. D. Yesakov, “Novoye o sessii VASKhNIL 1948 goda,” http://russcience.euro.ru/papers/esak94os.htm, accessed January 31, 2009.

  One of them, Dimitri Egorov: J. J. O’Connor and E. F. Robertson, “Dimitri Fedorovich Egorov,” www-history.mcs.st-andrews.ac.uk/Biographies/Egorov.html, accessed December 27, 2007.

  Luzin case descriptions and quotes: S. S. Demidov, V. D. Yesakov, “‘Delo akademika N. N. Luzina’ v kollektivnoy pamyati nauchnogo soobshestva,” Delo Akademika N. N. Luzina (St. Petersburg: RKhGI, 1999).

  As a result, Soviet and Western mathematicians: Dennis Shasha and Cathy Lazere, Out of Their Minds: The Lives and Discoveries of Fifteen Great Computer Scientists (New York: Springer, 1998), 142.

  A top Soviet mathematician: Lev Pontryagin’s entire memoir is devoted to the backstabbing and intrigue in which this outstanding mathematician personally took part. Lev Pontryagin, Zhizneopisaniye Lva Semenovicha Pontryagina, matematika, sostavlennoye im samim (Moscow: Komkniga, 2006), 134.

  “It was in the 1960s”: Sergei Gelfand, interview with the author, Providence, RI, November 9, 2007.

  Three weeks later, the Soviet air force was gone: Richard Overy, Russia’s War: A History of the Soviet War Effort: 1941–1945 (New York: Penguin, 1998), 73–85.

  The greatest Russian mathematician of the twentieth century, Andrei Kolmogorov: This work by Andrei Kolmogorov is classified, so the p
ublished results, apparently called Strelyaniy sbornik, are not available. The information comes from Alexander Abramov, his student and biographer, interview with the author, Moscow, December 5, 2007, and from Etikh strok begushchikh tesma, ed. A. N. Shiryaev (Moscow: Fizmatlit, 2003), 355, 500.

  by the end of his life he had served as an adviser on seventy-nine dissertations: Mathematics Genealogy Project, http://genealogy.math.ndsu.nodak.edu/id.php?id=10480, accessed January 22, 2008.

  it was to promise the people of his country that the Soviet Union would surpass the West: Cited in Roger S. Whitcomb, The Cold War in Retrospect: The Formative Years (Westport, CT: Praeger Publishers, 1998), 71.

  The effort to assemble an army of physicists and mathematicians: Zhores A. Medvedev, Soviet Science (New York: Norton, 1978), 46.

  Estimates of the number of people engaged in the Soviet arms effort: Clifford G. Gaddy, The Price of the Past: Russia’s Struggle with the Legacy of a Militarized Economy (Washington DC: Brookings Institution Press, 1998), 24–25.

  official mathematicians and other scientists could shop at specially designated stores: Etikh strok, 293, 467.

  Sergei Novikov, was not allowed to travel to Nice to accept his award: Pontryagin, 169.

  Leonid Levin, describes being ostracized: Leonid Levin, “Kolmogorov glazami shkolnika i studenta,” in Kolmogorov v vospominaniyakh, 168–69.

  Cook and Levin, who became a professor at Boston University, are considered coinventors: Shasha and Lazere, 139–56; Leonid Levin’s homepage at Boston University, http://www.cs.bu.edu/~lnd/, accessed January 29, 2008; description of the P versus NP problem http://www.claymath.org/millennium/P_vs_NP/, accessed January 29, 2008.

  One of the people who came for an extended stay was Dusa McDuff: Dusa McDuff, “Advice to a Young Mathematician,” in Princeton Companion to Mathematics, ed. Timothy Gowers, June Barrow-Green, and Imre Leader (Princeton, NJ: Princeton University Press, 2008), 1007.

  “It was a wonderful education”: Dusa McDuff, “Some Autobiographical Notes,” http://www.math.sunysb.edu/~tony/visualization/dusa/dusabio.html, accessed March 19, 2009.

  Mathematicians called it “math for math’s sake”: Vladimir Uspensky, “Apologiya matematiki, ili O matematike kak chasti duhovnoy kultury,” Noviy Mir 11, 2007.

  “If I had been free to choose any profession”: Grigory Shabat, professor at the Russian State Humanities University, interview with Katerina Belenkina, Moscow, April 2007.

  2. How to Make a Mathematician

  Alexander Golovanov: Alexander Golovanov, interview with the author, St. Petersburg, October 18 and October 23, 2008.

  Three other boys beat Grisha in competitions: According to Rukshin, these three were Nikolai Shubin, who went on to become a chemist, and Alexander Vasilyev and Alexander Levin, both of whom became computer scientists.

  Boris Sudakov: Boris Sudakov, interview with the author, Jerusalem, December 31, 2007.

  he hummed, moaned, threw a Ping-Pong ball against the desk: Sergei Rukshin, interview with the author, St. Petersburg, October 17 and October 23, 2007, and February 13, 2008; Alexander Abramov, interview with the author, Moscow, December 5, 2007.

  he never dazzled colleagues with his geometric imagination, but he almost never failed to impress them: John Morgan, interview with the author, New York City, November 9, 2007; Yuri Burago, phone interview with the author, February 26, 2008.

  loudmouthed man named Sergei Rukshin: Rukshin interview.

  I observed practice sessions: I visited the Mathematics Education Center in St. Petersburg on February 13, 2008.

  Mathematicians know this as the Party Problem: http://mathworld.wolfram.com/PartyProblem.html, accessed March 19, 2009.

  the Ramsey theory, a system of theorems: Ronald Graham, Bruce Rothschild, Joel Spencer, Ramsey Theory (New York: John Wiley and Sons, 1990).

  When they grew older, Rukshin hounded: Golovanov interview.

  3. A Beautiful School

  Steven Pinker observed: Steven Pinker, The Stuff of Thought: Language as a Window into Human Nature (New York: Viking, 2007), 177.

  “A layer or a slab has two primary dimensions”: Ibid., 179–80.

  words like end and edge are used: Ibid., 180.

  the Möbius strip . . . is among the earliest known objects of topological inquiry: Richard Courant and Herbert Robbins, What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd ed., revised by Ian Stewart (New York: Oxford University Press, 1996), 235.

  His students always wondered why: V. M. Tihomirov, “Geniy, zhivushchiy sredi nas,” in Alexander Abramov, ed., Yavleniye chrezvychaynoye: Kniga o Kolmogorove (Moscow: FAZIS, 1999), 73. Tihomirov notes that Ivan Vinogradov, Nikolai Luzin, and Pavel Alexandrov also avoided being drafted into top-secret work but explains that their research had no apparent military application at the time; the same could not be said of Kolmogorov’s.

  with whom he shared a home starting in 1929: Andrei Kolmogorov, “Vospominaniya o P. S. Alexandrove,” in A. N. Kolmogorov, Matematika v yeyo istoricheskom razvitii (Moscow: LKI, 2007), 141.

  they generally requested academic appointments together: The donation, food for people held in the siege of Leningrad, is described in Etikh strok begushchikh tesma, ed. A. N. Shiryaev (Moscow: Fizmatlit, 2003), 332. Issues of joint appointments and accommodations appear throughout the correspondence published in Etikh strok, for example, on page 80.

  Kolmogorov asked the filmmaker to use Johann Sebastian Bach’s Double Violin Concerto: “Posledneye interview,” in Yavleniye, 205.

  “Through the woods or along the shore of the Klyazma River”: R. F. Matveev, “Vspominaya Kolmogorova . . . ,” in Albert Shiryaev, ed., Kolmogorov v vospominaniyakh uchenikov (Moscow: MTsNMO, 2006), 170.

  Another of Kolmogorov’s students wrote in his memoir: M. Arato, “A. N. Kolmogorov v Vengrii,” in Kolmogorov v vospominaniyakh, 31.

  a math problem he authored at the age of five: Alexander Abramov, interview with the author, Moscow, December 5, 2007.

  two professional mathematicians: These are Alexander Abramov and Vladimir Tihomirov.

  In 1922, Kolmogorov: “Avtobiografiya Andreya Nikolayevicha Kolmogorova,” in Matematika, 21.

  The Dalton Plan: http://www.dalton.org/philosophy/plan/, accessed January 23, 2008.

  “So every student spent most of his school time at his desk”: “Posledneye interview,” 186.

  “In just three hours at an elevation of 2400 meters”: Vladimir Arnold, “Ob A. N. Kolmogorove,” in Kolmogorov v vospominaniyakh, 40.

  the pair spent the 1930–1931 academic year abroad: Kolmogorov, “Vospominaniya,” 143.

  all culture, and gay culture in particular: Harry Oosterhuis, Homosexuality and Male Bonding in Pre-Nazi Germany: The Youth Movement, the Gay Movement, and Male Bonding Before Hitler’s Rise (New York: Haworth Press, 1991).

  “Interesting that this idea”: Etikh strok, 63.

  “The wife will always have pretensions to that role”: Ibid., 430.

  after Alexandrov’s death, Kolmogorov: A. V. Bulinsky, “Shtrihi k portretu A. N. Kolmogorova,” in Kolmogorov v vospominaniyakh, 114–15.

  At the age of forty, Kolmogorov wrote up a plan: Albert Shiryaev, ed., Zvukov serdtsa tihoe eho: Iz dnevnikov (Moscow: Fizmatlit, 2003), 110–11.

  In 1935, Kolmogorov and Alexandrov organized: B. V. Gnedenko, “Uchitel i drug,” in Kolmogorov v vospominaniyakh, 131. Kolmogorov did not invent the format; the first competition actually occurred a year earlier, in Leningrad. He was, however, instrumental in taking the competitions national. See N. B. Vasilyev, “A. N. Kolmogorov i matematicheskiye olimpiady,” in Yavleniye, 168.

  Kolmogorov teamed up with Isaak Kikoin: “Istoriya olimpiady,” http://phys.rusolymp.ru/default.asp?trID=118, accessed January 24, 2008. />
  The Soviet of Ministers issued a decree: Abramov interview.

  That August, Kolmogorov organized: Alexander Abramov, “O pedagogicheskom nasledii A. N. Kolmogorova,” in Yavleniye, 105.

  nineteen boys were chosen: A. A. Egorov, “A. N. Kolmogorov i kolmogorovskiy internat,” in Yavleniye, 163.

  Lectures in mathematics: Abramov, 107.

  what he called “a spark from God”: Egorov, 164.

  a high-school course in the history of antiquity: Alexander Prohorov, interview with the author, Moscow, December 8, 2007.

  more hours of physical education instruction: Abramov, 111.

  Kolmogorov himself lectured the students in music: Egorov, 165.

  He also took the boys on boating, hiking, and skiing trips: Gnedenko, 149.

  “And few of us understood the music”: L. A. Levin, “Kolmogorov glazami shkolnika i studenta,” Kolmogorov v vospominaniyakh, 167.

  He oversaw a curriculum-reform effort: A. S. Monin, “Dorogi v Komarovku,” in ibid., 182.

  Kolmogorov sought to revamp the secondary-school understanding of geometry: Alexander Abramov, “O polozhenii s matematicheskim obrazovaniyem v sredney shkole” (1978–2003) (Moscow: FAZIS, 2003), 13.

  “These things can provoke nothing but disgust”: Ibid., 40.

  authors of the curriculum reform were exposed: R. S. Cherkasov, “O nauchno-metodicheskom vklade A. N. Kolmogorova,” in Yavleniye, 156.

  The New Math movement brought actual mathematicians: David Klein, “A Brief History of American K-12 Mathematics Education in the 20th Century,” from James Royer, ed., Mathematical Cognition. Preprint version at http://www.csun.edu/~vcmth00m/AHistory.html, accessed January 25, 2008; Patrick Suppes and Shirley Hill, “Set Theory in the Primary Grades,” New York State Mathematics Teachers’ Journal 13 (1963): 46–53.

  “the effect of freshening [the student’s] eye”: Quoted in Klein.