manifold destiny

8/3/2019 Manifold Destiny

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ANNALS OF MATHEMATICS

MANIFOLD DESTINY

On the evening of June 20th, sev-eralhundred physicists, includinga Nobellaureate, assembled in an audi-torium at the Friendship Hotel in Bei-jing for a lecture bythe Chinese math-ematician Shing-Tung Yau. In the latenineteen-seventies, when Yau was inhis twenties, he had made a series of,breakthroughs that helped launch thestring-theory revolution in physics andearned him, in addition to a FieldsMedal-the most coveted award inmathematics---:a reputation in bothdisciplines as a thinker of unrivalledtechnical power.Yau had sincebecome a professor ofmathematics atHarvard and the direc:-tor ofmathematics institutes in Beijingand Hong Kong, dividing his time be-tween the United States and China.His lecture at the Friendship Hotel waspart of all international 'conference onstring theory, which he had organizedwith the support of the Chinese gov-ernment, in partto promote the coun-try's recent advances in theoreticalphysics. (More than six thousand stu-dents attended the keynote address,which was delivered by Yau's closefriend Stephen Hawking, in the GreatHall of the People.) The subject ofYau's talk was something that few inhis audience knew much about: thePoincare conjecture, a century-old co-nundrum about the characteristics ofthree-dim,ensional spheres, which, be-cause it has important implications formathematics and cosmology and be-causeit has eluded all attempts at solu-tion, is regarded bymathematicians asa holygrail.Yau, a stocky'man of fifty-seven,stood at a lectern in shirtsleeves andblack-rimmed glasses and, with hishands in his pockets, described howtwo of his students, Xi:-Ping Zhu andHuai-Dong Cao, had completed aproof of the Poincare conjecture a fewweeks earlier. "I'm very positive about44 THE NEWYOR.KER..AUGUST 28, 2006

A legendary problem and the battle over who solved it.BY SYLVIA NASAR AND DAVID GRUBER

fessional association. The meeting,which took place at a conference centerin a stately mansion overlooking theN evaRiver,was highlyunusual. At theend ofMay, a committee ofnine prom-inent mathematicians had voted toaward Perelman a FieldsMedal for hiswork on the Poincare, and Ball hadgone to St. Petersburg to persuade himto accept the prize in a public ceremonyat the I.M.U.'s quadrennial congress, inMadrid, on August 22nd.The Fields Medal, like the NobelPrize, grew, in part, out of a desire toelevate science above national animos-ities. German mathematicians were ex-cluded from the first I.M.U. congress,in 1924, and, though the ban was liftedbefore the next one, the trauma itcaused led, in 1936, to the establish-ment of the Fields, a prize intended tobe "as purely international and imper-sonal as possible."However, the Fields Medal, whichis awarded everyfour years, to betweentwo and four mathematicians, is sup-posed not only to reward past achieve-ments but also to stimulate future re~'search; for this reason, it is given onlyGrigory Perelman is indeed reclu~ to mathematicians aged forty andsive.He left hisjob asa researcher younger. In recent decades, asthe num-at the Steklov Institute of Mathemat- ber of professional mathematicians has

ics, in St. Petersburg, last December; grown, the Fields Medal has becomehe has few,friends;and he liveswith his increasingly prestigious. Only forty-mother in an apartment on the out- four medals have been awarded inskirts of the city. Although he had" nearly seventy years-including threenevergranted an interviewbefore, he " forworkcloselyrelatedto the Poincare .was cordial and frank when we visited conjecture--and nomathematician hashim, in late June, shortly after Yau's ever refused the prize. Nevertheless,conference in Beijing, taking us on a Perelman told Ball that he had no in-long walking tour ofthe city."I'm look- tention of accepting it. "I refuse," heing for some friends, and they don't said simply.have to be mathematicians," he said. Over a period of eight months, be-The week before the conference, Perel- ginning inNovember, 2002, Perelmanman had spent hours discussing the posted a proof of the Poincare on thePoincare conjecture with Sir John M. Internet in three installrrients. Like aBall, the fifty-eight-year-"bld president sonnet or an aria, a mathematical proof ~of the International Mathematical has a distinct form and set of conven- ~Union, the discipline'sinfluentialpro- . tions. It begins with axiom$,or ac- ~

Zhu and Cads work,"Yau 'said."Chi-nese mathematicians shouldhave everyreason to beproud ofsuch a big successin completely solving the puzzle." Hesaid that Zhu and Cao were'indebtedto his longtime American collaboratorRichard Hamilton, who deservedmostof the credit for solving the Poincare.He alsomentioned Grigory Perelman,a Russian mathematician who, he ac-knowledged, had made an importantcontribution. Nevertheless, Yau said,"in Perelman's work, spectacular as itis, many key ideas of the proofs aresketched or outlined, and complete de-tails are often missing."He added, 'Wewould like to get Perelman to makecomments. But Perelman resides inSt. Petersburg and refuses to commu-nicate with other people."For ninety minutes, Yau discussedsome ofthe technical details of his stu-dents' proof When hewas finished, noqne asked any questions. That night,however, a Brazilian physicist posted areport of the lecture on his blog. "Lookslike China soon will take the lead alsoin mathematics," hewrote.


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, .cepted truths, and employs a series oflogical statements to arrive at a conclu-sion. If the logic is deemed to bewater-tight, then the result is a theorem. Un-like proof in law'or science, which isbased on evidenceand therefore subjectto qualification and revision, a proofof a theorem is definitive. Judgmentsabout the accuracyof a proof are medi-ated by peer-reviewed journals;' to in-sure fairness, reviewersare supposed tobe carefullychosen by journal editors,and the identity of a scholar whose pa-per is under consideration is kept se-cret. Publication implies that a proofiscomplete, correct, and original.Bythesestandards,Perelman's proofwas unorthodox. It was aStonishingly

brief for such an ambitious piece or"work; logic sequences that could havebeen elaborated over many pages wereoften severclycompressed. Moreover,the proof made no direct. mention ofthe Poincare and included many ele-gant results that were irrelevant to thecentral argument. But, four years later;at least two teams of experts had vettedthe proof and, had found no signifi-cant gaps or errors in it. A consensu~ .was emerging in the math community:Perelman had solved the Poincare.Even so, the proof's complexity-and. Perelman's use of shorthand in makingsome of his most important daims-made it vulnerable to challenge. Fewmathematicians had the expertise nec-essaryto evaluateand defend it.

After giving a series of lectures onthe proofin the United States in 2003,Perelman returned to St. Petersburg.Since then, although he had continuedto answer queries about it bye-mail,he had had minimal contact with col-leagues and, for reasons no one under-stood, had not tried to publish it. Still,there was little doubt that Perelman,who turned forty on June 13th, de-serveda Fields Medal. As Ballplannedthe I.M.U.'s 2006 CQngress,he beganto conceiiTeof it as a historic event.More than three thousand mathemati-cians would be attending,- and KingJuan Carlos of Spain had agreedto pre-side over the awards ceremony. ThelM.U.'snewsletter predicted that thecongresswould be remembered as "theoccasion when this conjecture becamea theorem." Ball, determined to make.sure that Perelman would be there, de-. cided to g9 to St. Petersburg.

Ball wanted to keep his visit a se-cret-the names of Fields Medal re-cipients are announced officiallyat theawards ceremony-and the conferencecenter where h~ met with Perelmanwas deserted. For ten hours over twodays, he tried to persuade Perelman toagree to accept the prize, Perelman, aslender, balding manwith a curlybeard,bushy eyebrows, and.blue-green eyes,listened politely. He had not spokenEnglish for three years, but he fluentlyparried Ball's entreaties, at one pointtaking Ball on a long walk-one of

"Should we ha!fheartedly try to relate?"

Perelman's favorite activities. As hesummed up the conversationtwo weekslater: "He proposed to me three alter-natives: accept and come; accept anddon't come, and we will send you themedal later; third, I don't accept theprize. From the very beginning, I toldhim I have chosen the third one." TheFields Medal held no interest for him,Perelman explained."It wascompletelyirrelevant for me," he said. "Everybodyunderstood that if the proof is correctthen no other recognition is needed." .Proofs ofthe Poincare havebeen an-nounced nearlyeveryyear since theconjecture was formulated, by HenriPoincar6, more than a hundred yearsago. Poincare was a cousinofRaymondPoincare, the President of France dur-ing the First World War, and one ofthe most creative mathematicians ofthe nineteenth century. Slight, myopic,and notoriously absent-minded, heconceived hisfamous problem in 1904,eight years before he died, and tuckedit as anoflhand question into the endof 3;sixty-five-page paper.Poincare didn't makemuch progresson proving the conjecture. "Cetteques-./ion nous'entrainerait trap loin" ("Thisquestion would take us too far"), hewrote. He was a founder of topology,also known as"rubber-sheet geometry,"forits focuson the intrinsicproperties ofspaces. From a topologist's perspective,there is no difference between a bageland a coffeecupwith a handle. Each hasa singlehole and can be manipulated toresemble the other without being tornor cut. Poincare used the term "mani-fold" to describe such an abstract topo-logicalspace.The simplestpossibletwo-dimensional manifold is the surfaceof asoccer ball, which, to a topologist, is asphere-even when it is stomped on,stretched, or crumpled. The proof thatan objectis a so-calledtwo-sphere, sinceit can take on any number of shapes, isthat it is "simply connected," meaningthat no holespuncture it. Unlike a soc-cer ball, a bagel is not a true sphere. Ifyou tie a slipknot around a soccer ball,you caneasilypull the slip~ot closedbysliding it along the surface of the ball.But ifyou tie a slipknot around a bagelthrough the hole in its middle you can-not pull the slipknot closed withouttearing the bagel.


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Two-dimensional manifolds werewell understood by the mid-nineteenthcentury.But it remainedunclearwhetherwhat was true for two dimensions wasalso true for three. Poincare proposedthat allclosed, simplyconnected, three-dimensional manifolds-those whichlackholes and areof finite extent-werespheres.The conjecture was potentiallyimportant for scientists studying thelargestknown three-dimensional mani-fold:the universe.Proving it mathemat-ically,however,was far from easy.Mostattemptsweremerely embarrassing, butsome led to important mathematicaldiscoveries,including proofs ofDehn'sLemma, the Sphere Theorem, and the -Loop Theorem, which are now funda-mental concepts in topology.By the nineteen-sixties, topologyhad becomeone of the most pro.ductiveareasof mathematics, and young topol-ogists were launching regular attackson the Poincare. To the astonishmentof most mathematicians, it turned outthat manifolds of the fourth, fifth, andhigher dimensions were more tractablethan those of the third dimension. By1982, Poincare's conjecture had peenproved in -alldimensions except the- third. In 2000, the Clay MathematicsInstitute, aprivate foundation that pro-motes mathematical research, namedthe Poincare one ofthe sevenmost im-portant outstanding problems in math-ematics and offered a million dollars toanyonewho could prove it.

"My whole life as a mathematicianhas been dominated by the -Poincareconjecture," John M


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Poincare"definitelyswings open doors,"BarryMazur, a mathematician at Har-vard, said.The implications of the con-jectures for other disciplinesmaynot beapparent for years, but for mathemati-cians the problems are fundamental."This is a kind of twentieth-centwy Py-thagorean theorem," Mazm added. "Itchanges the landscape."In 1982,Thurston wonaFieldsMedalfor his contributions to topology. Thatyear,RichardHamilton, amathematicianatComel1,publisheda paper on anequa-tion calledthe Ricciflow,which he sus-pectedcouldbe relevantfor solvingThur-ston's conjectureand thus the Poincare.Like a heat equation, which describeshowheatdistributesitselfevenlythrougha substance-flowing from hotter tocooler parts of a metal sheet, for exam-ple--to createa more uniform.tempera-ture, the Ricci flow, by smoothing outirregularities,givesmanifoldsamoreuni-formgeometry.Hamilton, the son of aCincinnatidoctor,defiedthemath profession'snerdystereotype.Brash.and irreverent,he rodehorses,windsurfed,and had a successionof girlfiiends.He treated math asmerely

. oneof life'spleasures.At forty-nine,hewasconsideredabrilliantlecturer,but he.hadpublishedrelativelyittlebeyonda se-ries of seminal articleson the Ri


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Yau's entrepreneurial drive ex-tended to collaborations with col""leagues and students, and, in additionto conducting his own research, hebegan organizing seminars. He fre-quently allied himself with brilliantlyinventive mathematicians, includingRichard Schoen and William Meeks.But Yau was especially impressed byHamilton, asmuch for his swagger asfor his imagination. "I can have funwith Hamilton," Yau told us duringthe string-theory conference in Bei-jingo"I can go swimming with him. Igo out with him a,nd his girlfriendsand all that." Yau was convinced thatHamilton could use the Ricci-flowequation to solve the Poincare andThurston conjectures, and he urgedhim to focuson the problems. "Meet-ing Yau changed his mathematicallife," a friend ,ofboth mathematicianssaid of Hamilton. "This was the firsttime he had been on to something ex-tremely big. T a1kingto Yau gave himcourage and direction."

, Yau believedthat if he couldhelpsolve the Poincare it would be a vic-tory not just for him but also forChina. In the mid-nineties, Yau andseveral other Chinese scho1aIs,beganmeeting with PresidentJiafig Zeminto discuss how to rebuild the country'sscientific institutions, which had beenlargely destroyed during the CulturalRevolution. Chinese universities werein dire condition. According to SteveSmale, who won a Fields for provingthe Poincare in higher dimensions,and who, after retiring from Berkeley,taught in Hong Kong, Peking Uni-versity had "halls filled with the smellof urine, one common room, oneoffice for all the assistant professors,"and paid its faculty wretchedly lowsalaries.Yau persuaded a Hong Kongreal-estate mogul to help finance amathematics institute at the ChineseAcademy of Sciences, in Beijing, andto endow a Fields-style medal forChinese mathematicians under theage of forty-five. On his trips toChina, Yau touted Hamilton andtheir joint work on the Ricciflow andthe Poincare as a model for youngChinese mathematicians. As he put itin Beijing, "They always say that the

,wholecounttyshouldlearnfromMaoor some big heroes. So I made a joke

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to them, but I was half serious. I saidthe whole country should learn fromHamilton."

Grigory Perelman was learningfrom Hamilton already.In 1993,he began a two-year fellowship atBerkeley. While he was there, Hamil-ton gave several talks on campus, andin one he mentioned that he wasworking on the Poincare. Hamilton'sRicci-flow strategy was extremelytechnical and tricky to execute. M-ter one of his talks at Berkeley, hetold Perelman about his biggest ob-stacle. As a space is smoothed underthe Ricci flow, some regions deforminto what mathematicians refer to as"singularities." Some regions; called"necks," become attenuated areas ofinfinite density. More troubling toHamilton was a kind of singularity hecalled the "cigar." If cigars formed,Hamilton worried, it might beimpos-

. .sible to achieve uniform geometry.Perelman realized that a paper he hadwritten on Alexandrov spaces mighthelp Hamilton prove Thurston's con-jecture-and the Poincare-onceHamilton solved the cigar problem."At some point, I asked Hamilton ifhe knew a certain collapsing resultthat I had proved but not published-which turned out to be very useful,"Perelman said. "Later, I realized thathe didn't understand what I was talk-ing about." Dan Stroock, of M.LT.,said, "Perelman may havelearned stufffrom Yau and Hamilton, but, at thetime, they were not learning fromhi "m. .By the end ofhis first year at Berke-ley, Pere1man hadwritten several strik-

ingly original papers. He was asked togive a lecture at the 1994 I.M.D. con-gress, in Zurich, and invited to applyforjobs at Stanford, Princeton, the In-stitute for Advanced Study, and theTHE NEW YOI\KI\, AUGUST 28, 2006 49


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ve, which hasprotected wetlands and ten miles of natural coast.

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University ofTel Aviv. Like Yau,Perelman was a formidable problemsolver. Instead of spending years con-0structing an intricate theoretical :fi:ame-work, or defining new areas of re-search, he focussed on obtaining par-ticular results. According to MikhailGromov, a renowned Russian geome-ter who has collaborated with Perel-man, he had been trying to overcome atechnical difficulty relating to Alexan-drov spaces and had apparently beenstumped. "He couldn't do it,"Gromovsaid. "It was hopeless."Perelman told us that he liked towork on several problems at once. AtBerkeley, however, he found himselfreturning again and again to Hamil-ton's RiccHlowequation and the prob-lem that Hamilton thought he couldsolve with it. Some of Perelman'sfriends noticed that he was becomingmore and more ascetic. Visitors fromSt. Petersburg who stayedin his apart-ment were struck by how sparselyfur-nished it was. Others worried that heseemed to want to reduce life to a setof rigid axioms; When a member of ahiring committee at Stanford askedhim-for a C.V. to include with requestsfor'letters of recommendation, Perel-man balked. "If they know my work,they don't need my C.V.," he said. "Ifthey need my C.V., they don't knowmy work."Ultimately, he received severaljoboffers. But he declined them all,and inthe summer of 1995 returned to St.Petersburg, to his,old job ,atthe Stek-lov Institute, where he was paid lessthan a hundred dollars a month. (Hetold a friend that he had savedenoughmoney in the United States to live onfor the rest of his life.) His father hadmoved to Israel two years earlier, andhis younger sister was planning tojoinhim there after she finished college.His mother, however, had decided toremain'in St. Petersburg, and Perel-man moved in with her. "I realize thatin Russia I work better," he told col-leagues at the Steklov.At twenty-nine, Per elm an wasfinnly established as a mathematician ~. zand yet largely unburdened by profes- S'sional responsibilities. He was free to ~

pursue whatever problems he wanted ~to, and he knew that his work, should ghe choose to publish it, would be ~THE NEW YOR.KER.,AUGUST 28, 2006 51


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~-"Over here,' Billingsley. "

shown serious consideration. YakovEliashberg, a mathematician at Sta,rhford who krlew Perelman at Berkeley,thinks that Perelman returned to Rus-sia in order to work on the Poincare."Why not?" Perelman said when weaskedwhether Eliashberg's hunch wascorrect. . 'The Internet made it possible forPerelman to work alonewhile continu-,

ing to tap a common pool of knowl-edge. Perelman searched Hamilton'spapers for clues to his thinking andgaveseveralseminars on his work. "Hedidn't need any help," Gromov said."He likes to be alone. He reminds meof Newton-this obsession with anidea,working byyourse1f,the disregardfor other people's opinion. Newton wasmore obnoxious. Perelman is nicer, but,very ob~essed."In 1995, Hamilton published apaper inwhich he discussed afewof hisideas for completing a proof of thePoincare. Reading the paper, Perelrrianrealized tHat Hamilton had made noprogress on overcoming hisobstacles-the necks and the cigars."I hadn't seenany evidence of progress after early1992," Perelman told us. "Maybe hegot stuckeven earlier."However, Perel-52 TI-fE NEW YOI\KEI\ , AUGUST' 28 . 2006

. .man thought he saw a way around theimpasse. In 1996, hewrote Hamilton along letter ourlinihghis notion, in thehope of collaborating. "He did not an-swer,"Perelman said. "So I decided towork alone.""'\Tau had no idea that Hamilton's~ work on the Poincare had stalled.He was increasingly anxious about hisown standing in the mathematics pro-fession, particularly in China, where,heworried, ayounger scholar could tryto supplant him asChern's heir. Morethan a decade had passed since Yauhad proved his lastmajor result, thoughhe continued to publish prolifically."Yau wants to be the king of geome-try,"Michael Anderson, a geometer atStony Brook, said. "He believes thateverything should issue from him, thathe should have oversight. He doesn'tlike people encroaching on his terri-tory." Determined to retain controlover his field,Yau pushed his studentsto tackle big problems. At Harvard, heran a notorious'ly tough seminar ondifferential geometry, which met forthree hours at a time three times aweek. Each student was assigned a re-centlypublished proof and asked to re-

construct it, fixingany errors and fillingin gaps. Yau believed that a mathema-ticianhas an obligation to be explicit,and impressed on his students the im-portance of step-by-step rigor.There are two ways to get credit foran original contributiori in mathemat-

ics. The first is to produce an originalproo The second is to identifYa sig-nificant gap in someone else's proofana supply the missing chunk. How-ever, only true mathematical gaps-missing or mistaken arguments-canbe the basis for itclaim of originality.Filling in gaps in exposition-shortcutsand abbreviations used to make a proofmore efficierit-does not count.When,in 1993, Andrew Wiles revealedtkt agap had been fotind in his proof ofF er-mat's last theorem, the problem be-came fair game for anyone, until, thefollowing year, Wiles fixed the error.Most mathematicians would agreethat, by contrast, if a proof's implicitsteps can be made explicit by an expert,then tj:legap is merely one of exposi-tion, and the proof should be consid-ered complete and correct.,Occasionally, the difference be:" .tween a mathematical gap and a gap inexpositioITcan be' hard to discern. Onat least one occasion, Yau and his stu-dents have seemed to confuse the two',making claims of originality that othermathematicians believe are unwar-ranted. In 1996, a young geometer atBerkeley named Alexander Giventalhad proved a mathematical conjec-ture about mirror symmetry, a conceptthat is fundamental,to string theory.Though other mathematicians foundGivental's proof hard to follow, theywere optimistic that he had solved theproblem. As one geometer put it, "No-body at the time saidit Wasincompleteand incorrect." , 'In the fall of 1997, Kefeng Liu, a

former student ofYau's who taught at'Stanford, gave, a talk at Harvard onmirror symmetry. According to twogeometers in the 'audience, Liu pro-ceeded to present a proof strikinglysimilar to Givental's, describing it as a.paper that 4e had co-authored withYauand, another student of Yau's."Liu mentioned Givental but only asone of a long list of people who hadcontributed to the field," one of the'geometers said. (Liu maintains that


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his proofwas significantly differentfrom Givental's.), Aroundthe sametime,Giventalre-ceivedan e-mail signed byYau and hiscollaborators, explaining that they hadfound his arguments impossible to fol-low and his notation bafEing, and hadcome up with a proof of their own.They praised Givental for his "brilliantidea" and wrote, "In the final version ofour paper your important contributionwill be acknowledged."A fewweeks later, the paper, "Mir-ror Principle I," appeared in the Asian

Journal ofMathematics, which is co-edited by Yau. In it, Yau' and his co-authors describe their result as "the first .complete proof" of the mirror conjec-ture.They mention Givental's workonly in passing. "Unfortunately," theywrite, his proof, "which has been readbymany prominent experts, is incom- .plete." However, they did not identifYa specificmathematical gap.GiventalwastakenabaCk. I wantedto knowwhat their objection was," hetold us. "Not to exposethem. or defendmysel" In March, 1998, he publisheda paper that included a three-pagefootnote in which he pointed out anumber of similarities between Yau'sproof and his own. Several monthslater, a young mathematician' at theUniversity of Chicago who was askedby senior colleagues to investigate thedispute concluded that Givental'sproofwas complete. Yau says that he hadbeen working on the proof for yearswith his students and that they achievedtheir result independently of Givental.'We had our own ideas, and we wrotethem up,"he says. .Around this time,Yau had his firstseriousconflictwithChem and theChi-nese mathematical establishment. Foryears,Chem had been hoping to bringthe LM.U.'s congress to Beijing. Ac-cording to severalmathematicians whowere activein the LM.U. at the time,Yau made an eleventh-hour effort tohave the congress take place in HongKong instead.But he failed to persuadea sufficientnumber of colleaguesto goalongwith his proposal, and the LM.U.ultimatelydeCidedto hold the 2002 con-gressinBeijing.(Yaudeniesthat he triedto bring the congress to Hong Kong.)Among the delegates the I.M.U. ap-pointed to agroup that would be choos-

ing speakers for the congresswasYau'smost successful student, Gang Tian,who had been at N.Y.U.with Perelmanand was now a professor atM.LT. The'host committee in Beijing also askedTian to givea plenary address.Yau was caught by surprise. InMarch, 2000, he had published a surveyof recent research in his field studdedwith glowing references to Tian and totheir joint projects. He retaliated byor-ganizing his first conference on stringtheory, which opened in Beijing a fewdaysbefore the math congressbegan, inlate August, 2002. He persuaded Ste-phen Hawking and severalNobellau-reates to attend, and for days the Chi-nese newspapers were full ofpicturesoffamous scientists.Yau evenmanaged toarrange for his group to have an audi-ence with Jiang Zemin. A mathemati-cian who helped organize the mathcongress recalls that along the highwaybetween.Beijing and the airport there'were "billboards with pictures of Ste-phen Hawking plastered everywhere."That summer, Yau wasn't thinkingmuch about the Poincare. He had

confidence in Hamilton, despite hisslow pace. "Hamilton is a very goodfriend," Yau told us in Beijing. "He is

more than a friend. He is a hero. He isso original. We were woiking to finishour proo Hamilton worked on it fortwenty-five years. You work, you gettired. He probably got a little tired-

, and you want to take a rest."Then, on November 12, 2002, Yaureceived an e-mail message from aRussian mathematician whose namedidn't immediately register. "May Ibring to your attention my paper," thee-mail said.

On November 11th, Perelman hadposted a thirty-nine-page paperentitled "The Entropy Formula for theRicci Flowand Its Geometric Applica-tions," on arXiv.org, aWeb siteused bymathematicians to post preprints-ar-ticles awaiting publication in refereedjournals. He then e-mailed an abstractdhis p~perto a dozen mathematiciansin the United States-including Ham-ilton, Tian, and Yau-none of whomhad heard ITom him for years. In theabstract, he explained that he had writ-ten "a sketch of an eclectic proof" ofthe geometrization conjecture.Perelman had not mentioned theprQofor shown it to anyone~"I didn'thave any friends with whom I could


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;}!/~~, tIlli," he ,aid in St. P,t=buq;."I didn t want to discuss my work Withsomeone I didn't trust." Andrew Wileshad alsokeptthe fact that he waswork-ing on Fermat's last theorem a secret,but he had had a colleaguevet the proofbefore making it public. Perelman,bycasuallyposting a proof on the Internetof one of the most famous problems inmathematics, was notjust flouting aca-demic convention but taking a consid-erable risk. If the proof was flawed, hewould bepubliclyhumiliated, and therewould be no way to prevent anothermathematician from fixing any errorsand claimingvictory.ButPerelman saidhewas not particularly concerned. "Myreasoning was: if I made an error andsomeone used my work to construct acorrect proof! would be pleased," hesaid."I never set out tobe the solesolverof the Poincare.", Gang Tian was in his office atM.I.T. when he received Perelman'se-mail. He and Perelman had beenmendly in 1992, when they were bothat N.Y.U. and had attended the sameweekly math seminar in Princeton. "Iimmediately realized its importance,"Tian said of Perelman's paper. Tianbegan to read the paper and discussit with colleagues, who were equallyenthusiastic.On November 19th, Vitali Kapo- Hamilton and Yau were stunned byvitch, a geometer, sent Perelman an Perelman's announcement. "Wee-mail: felt that nobody else would be able toHi Grisha,Sorryto botheryou buta lot discover the solution," Yau told us inofpeopleareaskingmeaboutyourpieprint Beijing. "But then, in 2002, Perelman"TheentrPJ:formulaforthe ~cci. . ."Do said that he published something. HeI understand It correctly that while you tan: .. -..;--- . di.d h . .th d .not yet do all the steps in the Hamilton baslcally a s ortcut WI out omgprogram you can do enough so that. using all the detailed estimates that we did."

some ~oIl~ps~ng.e~ults you can prove ge- .Moreover, Yau complained, Perelman'smetrlZanon. Vltali. f " . . ch'. proo waswntten m su a messywayPerelman's response, the next day, that we didn't understand." .was terse: "That's correct. Grisha." Perelman's April lecture tour wasIn fact, what Perelman .had posted treated by mathematicians and by the

on the Internet was only the first in- press as a major 'event.Among the au-stallment of his proof. But it was dience at his talk atPrinceton weresUfficient for mathematicians to see John Ball, Andrew Wiles, John Forbesthat he had figured out how to solvethe Nash,Jr., who had provedtheRiemann-Poincare. Barry Mazur, the Harvardmathematician, uses the image of adented fender to describe Perelman'sachievement: "Suppose your car has adented feader and you calla mechanicto ask how to smooth it out. The me-chanic would ha~e a hard' time tellingyou what to do over the phone. Youwould haveto bring the carinto the ga-54 TI-iE NEW YOI\KEI\ , AUGUST 28. 2006

rage for him to examine~Then he couldtell you where to give it a few knocks.'What Hamilton introduced andPetel-man completed is a procedure that isindependent of the particularities of theblemish. If you apply the Ricci flow toa 3-D space, it will begin to undentit and smooth it out, The mechanicwould not need to even see the car-just apply the equation." Perelmanproved that the "cigars"that had trou-bled Hamilton could not actuallyoccur,and he showed that the "neck"problemcould be solvedby performing an intri-cate sequence of mathematical surger-ies: ci1ttingoutsingularities and patch-ing up the raw edges. "Now we have aprocedure to smooth things and, atcrucial points, control the breaks,",Mazur said.Tian wrote to Perelman, askinghimto lecture on his paper at M.I.T. Col-leagues at Princeton and Stony Brookextended similar invitations. Perelman

accepted them all and wasbooked for amonth of lectures beginning in April,2003. "Why not?" he told us with ashrug. Speaking of mathematiciansgenerally, Fedor Nazarov, a mathema-'tician at Michigan State University,said, "After youve solved a problem,you have a great urge to talk about it."

ian embedding theorem, and JohnConway, the inventor of the cellularautomaton game Life. To the astonish-ment of many in the audience, Perel-man said nothing about the Poincare."Here is a guy who proved aworld-fa-mous theorem imddidn't evenmentionit," Frank Quinn, a mathematician atVirgirua T ech, said; "He stated somekey points and special properties, andthen answered questions. He was es-tablishingcredlbility. If he had beatenhis chest and said, 'I solvedit,' hewouldhave got a huge amount of resistance."He added, "People were expecting astrange sight. Perelman Wasmuch morenormal than they expected."ToPerelman's disappointment,Hamilton did not attend t):latlecture orthe nextt:l~s, at Stony Brook. "fm adiscipleof Hamilton's, though I haven't'received his authorization," Perelmantold us. But John Morgan, atColum-bia, where Hamilton now taught, wasin the audience at Stony Brook, andafter a lecture he invited Perelman tospeak at Columbia.. Perelman, hopingto see Hamilton, agreed. The lecturetookplaceon a Saturdaymorning.Ham-ilton showedup late and asked no ques-tions during either the long discussionsession that followed the talk or thelunch after that. "I had the impressionhe had read only the first part of my .paper," Perelman said.In the Apm18, 2003, issue of Sci-ence, Yau was featured in an article

about Pei"elman's proof: "Many ex-perts, although not all, seem convincedthatPerelman has stubbed out theci-gars and tamed the narrow necks. Butthey are less confident that he cancontrol the number of surgeries. Thatcould prove a fatal flaw, Yau warns,noting. that many other attemptedproofs 'of the Poincare conjecture havestumbled over similar missing steps."Proofs should be treated with skepti-cism until mathematicians have had achance to reviewthern thoroughly,Yau told us. Until then, he said, "it'snotmath-it's religion."By mid-July, Perelman had posted'the final two installments of his proofon the Internet, and mathematicians

had begun the work offormal explica-tion, painstakingly retracing his steps.In the United States, at least two teamsof experts had assigned them~elvesthis


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task GangTian (Yau'srival) and John ,', " 'Morgari.;andapairofresearchersatthe """",',',',,', ,":


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.'. "

other student, Huai- Dong Cao,a pro-fessor at Lehigh University, to under-take an explicationofPerelman's proofZhuand Cao had studied the Ricciflow under Yau,'who considered Zhu,in particular, to be a mathematician ofexceptionalpromise. "We have to figureout whether Perelman's paper holds to-gether," Yau told them. Yau arrangedfor Zhu to spend the 2005-06 academicyear at Harvard, where he gavea semi-nar on Perehnan's proof and continuedto work on his paper with


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a century-Iang prablem, which willprobablyhaveanather few century-Iangimplicatians. If yau can attach yaurname in anyway, it is a contributian."

E T. Bell, the authar .of "Men .of.Mathematics," a witty histary .ofthe disciplinepublished in 1937, .oncelamented "the squabbles aver priaritywhichdisfigurescientifichistary."But inthe daysbefare e-mail, blags, and Websites, a certain decarum usually pre-'vailed.ln 1881,Paincare, whawas then, at theUniversityofCaen,had analter-catianwith a German mathematician inLeipzig named Felix Klein. Paincarehadpublishedseveralpapersin which helabelled certain functians "Fuchsian,'"after anather mathematician. Klein'wrate ta Paincare, painting aut that heand .othershaddane significant wark anthese functians,too.An exchange .ofpa-lite letters between Leipzig and Caenensued. Paincare's lastward an the sub-ectwas a quate ramGaethe's "Faust";"Name ist Scha//und Rauch." Laaselytranslated, that carrespands ta Shake-speare's'What's ina name?"This, essentially, is what Yau'sfriends are asking themselves. "I findmyself getting annayed with Yau thathe seems ta feel the need far marekudos," Dan Straack, of M.I.T., said."This is a guy who did magnificentthings, for which he was magnificentlyrewarded. He wan every prize ta bewon. I find it a little mean .ofhim taseem ta be trying to'get a share of thisas well." Stroack painted aut that~twenty-five years ago, Yau was ina sit-uatian very similar ta the .onePerelmanisin today. His mast famaus result, onCalabi-Yau manifalds, was hugely im-rtant far theoretical physics. "Calabioutlined a program," Stroock said, "Inreal sense, Yau was Calabi's Perel-~ he'son the other side. He'sd na compunctian atallin taking theon's share .ofcredit for Calabi- Yau.nd now he seems ta be resentingelman getting credit far completingamiltan's program. I dan't knaw ife analogy has everoccurred ta him."Mathematics, more than manyer fields, depends on callabaratian.st problems require the insights ofveral mathematicians in order to belved, and the professian has evalvedstandard for crediting individual con-

tributians that is as stringent as therules gaverning math itself As Perel-man put it, "If everyoneis hanest, iUsnatural to share ideas." Many mathe-maticians viewYau's canductaver thePoincare as a violation .of this basicethic, and worry about the damage ithas caused the professian. "Politics,pawer, and cantral have no legitimaterale in our cammunity, and theythreaten the integritY .of.our field,"Phillip Grffiiths said.Pere1man likes to attend opera per-formances at the Mariinsky The- ,atre, in St. Petersburg. Sitting high upin the back of the house, he can't makeout the singers' expressions or see thedetails .oftheir costumes. But he cares.onlyabaut the sound .oftheir voices,andhe says that the acoustics are betterwhere he sits than anywhereelse in thetheatre. Perelman viewsthe mathemat-icscommunity~nd much ofthe largerworld-from a similar remove.Befare we arrived in St. Petersburg,an June 23rd, wehad sent severalmes-

sagesto his e:-mailaddress,atthe SteklovInstitute, haping ta arrange a meeting,but hehad nat replied.We taak ataxitohis apartment building and, reluctant taintrude on his privacy, left a baok-acollectian ofJohn Nash's papers-in hismailbox, alang with a card saying thatwe wauld be sit4ng ona bench in anearby playgraund the fallawing after-naan. The next day, after Perelmanfailedto appear,we lefta box afpearl teaand a note describingsameof the ques-tians we hoped to discusswith him.We'repeated this ritual a'thirdtime. Finally,believingthat Perelmanwasaut of town,we pressed the buzzerfar hisapartment,hoping atleastto speakwith ,hismather.A woman answered and let us inside.Perelman met us in the dimlylit hallway.ofthe apartment. I~ turned out that hehad nat checked his Steklove-mail ad-dress far manths, and had nat loaked inhis mailbax all week. He had no ideawha wewere.We arranged ta meet at ten the fal-

lawing morning on Nevsky Praspekt.From there,Perelman, dressedin aspartscaat and loafers,taok us an a faur-haurwalking taur of the city,cammenting oneverybuilding and vista. After that, weallwent to avacal campetitian at the St.Petersburg Canservatary, which lasted

, , '

for fivehaurs. Perelman repeatedly saidthat he had retired from the mathemat-icscommunity andna langer cansideredhimself a professional mathematician.He mentianed adispute that he had hadyearsearlierwith acollaboratoroverhowto credit the authar .ofa particular proaf,andsaidthat hewasdismayedbythe dis-cipline'slax ethics. "It is not peaple whobreakethical standardswho areregardedas aliens,"he said. "It is people like mewho areisolated."We askedhimwhetherhe had read Cao and Zhu's paper. "It isnat clear ta me what new contributiandid they make," he said. "Apparently,Zhudid nat quite understand the argu-ment and rewarked it." As for Yau, ,Perelman said, "I can't sayfm .outraged.Other peaple do :warse.Of course, therearemany ,mathematicianswho aremore.orlesshanest. But almast allof them areconfarmists. They are more or lesshon-est, but they talerate thase who are nothanest."The prospect .ofbeing awarded aFields Medal had forced him ta make a

complete breakwith his prafessian. "Aslang as I was nat canspicuaus, I had achaice," Perelmanexplained. "Either tamakesomeuglything"~ fussaboutthemath community's lack of integrity-"or,if I didn't da thiskind .ofthing, to betreated asa pet. Now, when I become averyconspicuousperson, I cannat stayapet and say nathing. That iswhy I hadta quit."We asked Perelman whether,

, by refusing the Fields and withdrawingfrom his prafessian, he was eliminatingany possibility of infl.uencingthe disci-pline. "I am nat a politician!"he replied,angrily.Perelmanwauld nat saywhetherhis objectian to awards extended to theClayInstitute's million-:dallarprize."Pmnot going ta decide whether ta acceptthe prize until it is offered,"he said.Mikhail Gromav, the Russiangeam-eter, said that he 1.illderstaadPerelman'slogic: "Ta do great work, yau have tohave a pure mind. You can think .onlyabaut the mathematics. Everything elseis human weakness. Accepting prizes isshawing weakness."Others might viewPerelman's refusal ta accept a Fields asarrogant, Gromov said, but his princi-ples are admirable. "Theideal scientistdoes science and cares abaut nothing,else,"he said."He wants ta livethis ideal.Naw, I dan't think he reallyliveson thiside~ plane. But hewants ta." .THE NEW YOI\KEI\, AUGUST 28, 2006 57

manifold destiny

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