Perfect Rigour

by Masha Gessen

  “I did ask him what he was working on”: Bernhard Leeb, e-mail to the author, July 7, 2008.

  “I’ve just read your paper”: Grigory Perelman, e-mail message to Michael Anderson, February 28, 2000.

  “Dear Grisha”: Michael Anderson, e-mail to Grigory Perelman, February 29, 2000.

  He asked if Perelman had looked at his other two papers on related topics: Michael Anderson, e-mail to Grigory Perelman, March 2, 2000.

  Perelman responded by saying he could not open the file: Grigory Perelman, e-mail to Michael Anderson, March 20, 2000.

  8. The Problem

  “The very possibility of mathematical science seems”: Henri Poincaré, “Science and Hypothesis,” in The Value of Science: Essential Writings of Henri Poincaré (New York: Modern Library, 1999), 9.

  “We know what it is to be in love or to feel pain”: Donal O’Shea, The Poincaré Conjecture: In Search of the Shape of the Universe (New York: Walker and Company, 2007), 46.

  “that which has no parts, or which has no magnitude”: Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books; with Notes, an Appendix, and Exercises (New York: Adamant Media Corporation, 2003), 1.

  “such as are in the same plane, and which being produced ever so far do not meet”: Ibid., 5.

  “things that are equal to the same thing are equal to one another”: Ibid., 6.

  postulate 1 and interpretation: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/post1.html, accessed June 18, 2008.

  postulates 2 and 3: Todhunter, 5.

  postulates 4 and 5: Book I of Euclid’s Elements, http://www.mathsisgoodforyou.com/artefacts/EuclidBook1.htm, accessed June 18, 2008.

  “I had been told that Euclid proved things”: Bertrand Russell, The Autobiography of Bertrand Russell (New York: Routledge, 1998), 31. (My attention was drawn to this passage by William Dunham’s Journey Through Genius: The Great Theorems of Mathematics.)

  “Since the Euclidean system is rather simpler to deal with”: 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), 223.

  The geometry is called elliptic: Ibid., 224–27.

  Euler and the invention of topology: George Szpiro, Poincaré’s Prize: The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles (New York: Dutton, 2007), 54–56; J. J. O’Connor, E. F. Robertson, “A History of Topology,” http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Topology_in_mathematics.html, accessed June 20, 2008.

  proof of the conjecture for seven dimensions or more: J. Stallings, “Polyhedral Homotopy Spheres,” Bulletin of the American Mathematical Society 66 (1960): 485–88.

  just a year after he received his PhD from Princeton: Mathematics Genealogy Project, http://www.genealogy.math.ndsu.nodak.edu/id.php?id=452, accessed June 29, 2008.

  he, however, proved the conjecture for dimensions five and higher: S. Smale, “Generalized Poincaré’s Conjecture in Dimensions Greater than Four,” Annals of Mathematics 74 (1961): 391–406.

  extended Stallings’s proof to dimensions five and six: E. C. Zeeman, “The Poincaré Conjecture for n ≥ 5,” in Topology of 3-Manifolds and Related Topics (Englewood Cliffs, NJ: Prentice Hall, 1962).

  published a proof essentially similar to Smale’s: A. Wallace, “Modifications and Cobounding Manifolds,” II, Journal of Applied Mathematics and Mechanics 10 (1961): 773–809.

  There was also a Japanese mathematician: Szpiro, 163.

  John Stallings: Stallings’s website, http://math.berkeley.edu/~stall/, accessed June 29, 2008.

  “I have committed—the sin”: John R. Stallings, “How Not to Prove the Poincaré Conjecture,” http://math.berkeley.edu/~stall/notPC.pdf, accessed June 29, 2008.

  Michael Freedman published a proof of the conjecture for dimension four: M. H. Freedman, “The Topology of Four-Dimensional Manifolds,” Journal of Differential Geometry 17 (1982): 357–453.

  The accomplishment was hailed as a breakthrough: Szpiro, 169–71.

  John Morgan: John Morgan, interview with the author, New York City, November 6, 2007.

  9. The Proof Emerges

  He had told Anderson at the outset: Grigory Perelman, e-mail message to Michael Anderson, November 20, 2002.

  He handled the U.S. visa formalities: Ibid., March 31, 2003.

  he even added a footnote to that effect to his first preprint: The footnote read, in part: “I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993–95. I’d like to thank everyone who worked to make those opportunities available to me.” Grisha Perelman, “The Entropy Formula for Ricci Flow and Its Geometric Applications,” http://arxiv.org/PS_cache/math/pdf/0211/0211159v1.pdf, accessed August 29, 2008.

  He submitted the second of his three preprints: Grisha Perelman, “Ricci Flow with Surgery on Three-Manifolds,” http://arxiv.org/abs/math/0303109, accessed August 28, 2008.

  the New York Times published an article: Sara Robinson, “Russian Reports He Has Solved a Celebrated Math Problem,” New York Times, April 15, 2003.

  an intentional revolt: Mikhail Gromov, interview with the author, Paris, June 24, 2008.

  he was happy to let the organizers: Grigory Perelman, e-mail to Michael Anderson, April 2, 2003.

  Perelman’s screaming at his mentor had been heard: Mathematician Nikolai Mnev, interview with the author, St. Petersburg, April 22, 2008.

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

  they were incapable of seeing his approach to footnoting as anything but: Gromov interview; Viktor Zalgaller, interview with the author, Rehovot, Israel, March 16, 2008; Yuri Burago, phone interview with the author, February 26, 2008.

  “as modest as possible”: Grigory Perelman, e-mail to Michael Anderson, March 31, 2003.

  exhibited fantastic clarity in his lectures and unparalleled patience during the discussions: Anderson interview.

  the New York Times published another article: George Johnson, “The Nation: A Mathematician’s World of Doughnuts and Spheres,” New York Times, April 20, 2003.

  “one should never force oneself on anyone”: Grigory Perelman, telephone conversation with Abramov in 2007, in which Perelman told Abramov that this was one of his principles.

  to attend a daylong math competition at a physics-and-math school: Andrei Minarsky, interview with the author, St. Petersburg, October 23, 2008.

  He submitted the third and last in his Poincaré series of preprints: Grisha Perelman, “Finite Extinction Times for the Solutions to the Ricci Flow on Certain 3-Manifolds,” http://arxiv.org/abs/math/0307245, accessed August 31, 2008.

  Kleiner and his University of Michigan colleague John Lott: The product of that website is now posted on the arXiv, http://arxiv.org/PS_cache/math/pdf/0605/0605667v2.pdf, accessed August 31, 2008.

  a joint workshop on the first preprint: Allyn Jackson, “Conjectures No More? Consensus Forming on the Proof of the Poincaré and Geometrization Conjectures,” Notices of the AMS 53, no. 8 (September 2006): 897–901.

  10. The Madness

  When Perelman spoke to two New Yorker writers: Sylvia Nasar and David Gruber, “Manifold Destiny: A Legendary Problem and the Battle Over Who Solved It,” New Yorker, August 28, 2006.

  The arrangement with camp officials: Sergei Rukshin, interview with the author, St. Petersburg, October 17 and October 23, 2007, and February 13, 2008.

  The entire mathematics contingent broke out lau
ghing: Boris Sudakov, interview with the author, Jerusalem, December 31, 2007.

  it was the Soviet child psychiatrist Grunya Sukhareva: V. Ye. Kogan, “Preodoleniye: Nekontaktniy rebyonok v semye,” http://www.autism.ru/read.asp?id=29&vol=2000, accessed March 3, 2008. Tony Attwood, in The Complete Guide to Asperger’s Syndrome (London: Jessica Kingsley Publishers, 2006), 36, erroneously identifies the psychiatrist as Ewa Ssucharewa.

  Hans Asperger observed that these children’s social maturity: Attwood, 13.

  British psychologist named Simon Baron-Cohen: Simon Baron-Cohen, telephone interview with the author, February 18, 2008.

  “the extreme male brain”: Simon Baron-Cohen, The Essential Difference: Male and Female Brains and the Truth about Autism (New York: Basic Books, 2003).

  When he tested this theory on a population of Cambridge University undergraduates: Simon Baron-Cohen, Sally Wheelwright, Amy Burtenshaw, and Esther Hobson, “Mathematical Talent Is Linked to Autism,” Human Nature 18, no. 2 (June 2007): 125–31.

  mathematicians scored higher than other scientists: Simon Baron-Cohen, Sally Wheelwright, Richard Skinner, Joanne Martin, and Emma Clubley, “The Autism-Spectrum Quotient (AQ),” Journal of Autism and Developmental Disorders 31 (2001): 5–17.

  once he had received the information he sought, he had no further use for communication: Lev Pontryagin, Zhizneopisaniye Lva Semenovicha Pontryagina, matematika, sostavlennoye im samim (Moscow: Komkniga, 2006), 22.

  Kolmogorov . . . was accosted in a hallway by a man: Alexander Abramov, interview with the author, Moscow, December 5, 2007.

  what they called his “temper”: Ibid.

  “theory of mind”: Simon Baron-Cohen, Alan M. Leslie, and Uta Frith, “Does the Autistic Child Have a ‘Theory of Mind’?” Cognition 21 (1985): 37–46.

  a child who sketched a picture: Attwood, 115–16.

  “I suspect that many ‘whistle-blowers’ have Asperger syndrome”: Ibid., 118.

  the founders of the dissident movement in the Soviet Union: “Yesenin-Volpin Alexander Sergeevich,” Novoye zerkalo hronosa, http://www.hrono.ru/biograf/bio_we/volpin.html, accessed February 23, 2008.

  a nuisance forced upon them by the incomprehensible world of social mores: Michelle G. Winner, founder and director of the Center for Social Thinking in San Jose, CA, telephone interview with the author, February 1, 2008.

  “He was very patient”: Yelena Vereshchagina, interview with the author, St. Petersburg, February 13, 2008.

  “weak central coherence”: Francesca Happé and Uta Frith, “The Weak Coherence Account: Detail-Focused Cognitive Style in Autism Spectrum Disorders,” Journal of Autism and Developmental Disorders 36 (January 2006): 5–25.

  “The most interesting facts are those which can be used several times”: Henri Poincaré, Science and Method, trans. Frances Maitland, unabridged republication of the 1914 edition (Mineola, NY: Dover Publications, 2003), 17.

  “jigsaw puzzle of 5000 pieces”: Attwood, 92.

  socialization seemed to rob the person: John Elder Robison, Look Me in the Eye: My Life with Asperger’s (New York: Crown, 2007).

  “He didn’t think he needed it”: Perelman’s last few years at the Steklov described primarily by Sergei Kislyakov, Steklov Institute director, interview with the author, St. Petersburg, April 21, 2008.

  he was unable to file his expense report: Tamara Yakovlevna, Steklov accountant, interview with the author, St. Petersburg, April 22, 2008.

  The journal’s entire three hundred pages were devoted to an article: Huai-Dong Cao and Xi-Ping Zhu, “A Complete Proof of the Poincaré and Geometrization Conjectures—Application of the Hamilton-Perelman Theory of the Ricci Flow,” Asian Journal of Mathematics 10, no. 2 (June 2006): 165–492.

  telling Science magazine that he thought: Dana Mackenzie, “Mathematics World Abuzz Over Possible Poincaré Proof,” Science, April 18, 2003.

  Yau held a press conference: Nasar, Gruber.

  Yau used the occasion to announce Cao and Zhu’s putative breakthrough: Nasar, Gruber; George Szpiro, Poincaré’s Prize: The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles (New York: Dutton, 2007), 238.

  “In the last three years, many mathematicians have attempted to see whether the ideas”: Shing-Tung Yau, “Structure of Three-Manifolds—Poincaré and Geometrization Conjectures,” http://doctoryau.com/papers/yau_poincare.pdf, accessed October 4, 2008. Date of publication obtained from http://www.mcm.ac.cn/Active/yau_new.pdf.

  Yau rushed the Cao-Zhu paper through to publication: Following much criticism, Yau described the process himself in a letter to the newsletter of the American Mathematics Society. He wrote that he had unilaterally reviewed and approved the paper for publication in his journal. Shing-Tung Yau, “The Proof of the Poincaré Conjecture,” Notices of the AMS, April 2007, 472–73, http://www.ams.org/notices/200704/commentary-web.pdf, accessed June 13, 2009.

  stated clearly, at the outset, that the proof explicated was Perelman’s: Bruce Kleiner and John Lott, “Notes on Perelman’s Papers,” http://arxiv.org/PS_cache/math/pdf/0605/0605667v2.pdf, accessed October 4, 2008.

  “for his contributions to geometry and his revolutionary insights”: http://www.icm2006.org/dailynews/fields_perelman_info_en.pdf, accessed October 4, 2008.

  “It was so much fun”: Sergei Gelfand, interview with the author, Providence, RI, November 9, 2007.

  ICM newsletter published back-to-back interviews: ICM 2006 Daily News, Madrid, August 29, 2006.

  Yau engaged a lawyer: “Harvard Math Professor Alleges Defamation by New Yorker Article; Demands Correction,” press release, September 18, 2006, www.doctoryau.com, accessed September 9, 2008.

  The committee drafted a carefully worded invitation: Jeff Cheeger, New York University professor, interview with the author, New York City, April 1, 2008.

  “A Fields Medal is awarded to Grigory Perelman”: International Congress of Mathematics 2006, opening ceremony, http://www.icm2006.org/proceedings/Vol_I/2.pdf, accessed September 11, 2008.

  John Lott gave what would ordinarily have been the laudation: John Lott, “The Work of Grigory Perelman,” talk at the 2006 ICM, http://www.icm2006.org/v_f/AbsDef/ts/Lottlight-GP.pdf, accessed September 11, 2008.

  Two hours later, Richard Hamilton: ICM 2006 schedule, http://www.icm2006.org/v_f/fr_Resultat_Cos.php?Titol=O, accessed September 12, 2008.

  The announcement of this session in the program: Ibid.

  The Clay Institute would now use the ICM: James Carlson, interview with the author, Boston, August 27, 2007.

  a pdf file started circulating: http://www.cds.caltech.edu/~nair/pdfs/CaoZhu_plagiarism.pdf, accessed September 12, 2008.

  Cao and Zhu claimed they had forgotten they had copied the material: Denis Overby, “The Emperor of Math,” New York Times, October 17, 2006.

  “In this paper, we provide an essentially self-contained”: Huai-Dong Cao and Xi-Ping Zhu, “Hamilton-Perelman’s Proof of the Poincaré Conjecture and the Geometrization Conjecture,” http://arxiv.org/PS_cache/math/pdf/0612/0612069v1.pdf, accessed September 12, 2008.

  Channel 1 . . . reported that Perelman: “Rossiyskiy matematik razgadal zagadku, kotoraya muchayet uchenykh uzhe 100 let,” transcript of television broadcast, http://www.1tv.ru/owa/win/ort6_main.print_version?p_news_title_id=92602, accessed September 12, 2008.

  he did not have the money to buy a ticket: “Perelman igraet v pryatki,” MK v Pitere, August 30, 2006, http://www.mk-piter.ru/2006/08/31/022/, accessed September 12, 2008.

  Alexander Abramov, his old coach, contributed: Alexander Abramov, “Zagadki Perelmana net,” Moskovskiye Novosti, September 1, 2006.

  “You could say I’m engaged in self-education”: http://www.youtube.com/watch?v=jG-DGAdughs, accessed September 12, 2008.

  11. The Million-Dollar Question


  Jim Carlson: James Carlson, interviews with the author on numerous occasions, including Boston, August 27, 2007, and St. Petersburg, May 24 and May 25, 2008.

  he was apparently holding a conference to celebrate his fifty-ninth birthday: “International Conference in Honor of the 59th Birthday of Shing-Tung Yau!” http://qjpam.henu.edu.cn/home.jsp, accessed October 5, 2008.

  “I know Gian-Carlo Rota held a conference to celebrate his sixty-fourth birthday”: That conference was actually organized by Rota’s students. A mention is contained in a Rota obituary, http://www.math.binghamton.edu/zaslav/Nytimes/+Science/+Math/+Obits/rota-mit-obit.html, accessed October 5, 2008.

  Vershik had published a piece: Anatoly Vershik, “What Is Good for Mathematics? Thoughts on the Clay Millennium Prizes,” Notices of the AMS, January 2007, http://www.ams.org/notices/200701/comm-vershik.pdf, accessed October 5, 2008.