丘成桐先生在香港中文大学演讲：我数学经历

MyPastExperienceinMathematicsShing-TungYauTheChineseUniversityofHongKongSep.19,20031Igrewupinthe(farming)countrysideofHongKong:YuenLongandShatin.Therewasnoelectricityandnopipewater.ItookbathintheriverwhenIwasveryyoung.Ihaveeightbrothersandsistersandfoodwasscarce.WhenIwasfiveyearsold,Itookanentranceexaminationtoagoodpublicschool.IfailedmathematicsbecauseImadethewrongconvention: Iwrote 57tobe75, 69tobe96.2SoIenteredaverysmallvillageschool.Thereweremanyroughkidsfromthefarm.Inamatterofhalfayear,Igotserioussickbecauseoftheintimidationoftheroughkidsandmistreatmentoftheteacher.Irestedathomeforhalfayear.Istartedtolearnhowtodealwithdifficultsituationwithclassmatesandteachers.BythetimewhenIenteredsixthgrade,Iwasaleaderofasmallgroupofstudentstowanderonthestreet.3Myfatherwasaprofessor.HetaughtmealotofChineseliteratureatthattime.

However,hedidnotrealizethatIdidnotattendschoolforaperiodoftime.

(PerhapsbecauseIdidwellathomeasIcanrecitemostoftheessaysthatheaskedmetodo.)4ThereasonthatIdidnotgotoschoolwasthattheteachersdidnotreallyteach.Igotboredinschool.ThenafterawhileIgotboredonthestreet,also.

Therewasajointexaminationforallprimaryschools.Ididpoorly.However,Iwasexactlyontheborderlinecase.

Thegovernmentallowedtheseborderlinekidstoapplyforprivateschoolandgavethemtuition.

IgotintoPuiChingMiddleSchool.5校歌何安東編李竹侯詞

培正培正何光榮，教育生涯慘淡營，

培後進兮其素志，正軌道兮樹風聲，

萬千氣象方蓬勃，鼓鑄群才備請纓，

愛我培正謨謀遠，永為真理之干城，永為真理之干城！青年向上歌 我要真誠，莫負人家信任深。我要潔淨，因為有人關心。我要剛強，人間痛苦才能當。我要膽壯，奮鬥才能得勝。我要膽壯，奮鬥才能得勝。 我要愛人，愛敵人也愛淪落人。我要施贈，心誠、義重、財輕。我要虛懷，不忘我身多弱點。我要向上，學主榜樣做人。我要向上，學主榜樣做人。6Thisisprobablyoneofthebestmiddleschool.

Nothingwasexcitinginthefirstyearofmiddleschool.Ididnotdotoowellthatyear.

However,Ilearntmuchmoreathome:Chineseliterature,Novels(ChineseandWestern),Philosophy,history.Allfrommyfatherandhisconversationswithhisstudents.

AlthoughIdidnotunderstandGreekPhilosophy.Itstartedtoimpressmeafterlisteningtomanyconversationofmyfatherwithhisstudents.7IstartedtoreadthefamousChinesehistorybooks:史記、左傳

Iamespeciallyfascinatedby史記.

Notonlybyitsbeautifulwritings,butalsobyitsoriginalandresponsiblewaytoreporttheancienthistory.

Uptopresentdays,Ireadthisbook.

Theglobalviewofhistoryfromagreatmasterresonantswiththethinkingofagreatscientist.8Thefollowingessaystrikesme:

Thoughindulgedinreading,Idonotpursuitprecisemeanings.Nonetheless,everytimeIhitonsomethingIwassooverwhelmedwithjoythatIforgotmymeals.晉‧陶淵明好讀書，不求甚解，每有會意，便欣然忘食。

Iamnotbitterforbeingpoorandobscure,noramIkeenonbeingrichandfamous.不戚戚於貧賤，不汲汲於富貴。9Infollowingyears,thishasbeentheguidingprincipleofmystudyofmanydifferentsubjects.Withmyfather’steaching,Istartedtosetthegoalofmylife.Animportantquotation:10Zuo'sBookofHistoryOnImmortality“Thefirstplaceistoreignbenevolently,thenexttogainvictory,andthelasttosayvaluablewords.Theseachievementswillstandforlongandnotbeabandoned.Thereaftertheyarecalledimmortal.”左傳叔孫豹論三不朽太上有立德，其次有立功，其次有立言，雖久不廢，此之謂不朽。Oneneedsbeinghumbleandsimpletoreachthesegoals.AformerstudentofminerecentlyrecitedthefollowinglinesfromaTangpoemduringaTVinterviewinChina:立德立功立言之道，必以謙讓質樸為主。「會當凌絕頂，一覽眾山小」，輕妄浮誇之言也。“Iwouldratherbeatthesummit,Soallmountainswillbecometinyunderme.”Ithinkhewasabittooarrogant.11SimaQian(司馬遷)onConfucius(孔子)Therearesoabundantlymanykingsandmenofvirtue!Widelyknownbytheircontemporaries,theirnamesfeelintooblivionsoonaftertheyperished.AndyetConfucius,amaninplaincloth,hasbeenheldingreatesteembyscholarsofmorethantengenerations.天下君王至於賢人，眾矣！當時則榮，沒則已焉，孔子布衣，傳十餘世，學者宗之。12Duringconversationsofmyfatherwithstudents,manyimportantpointsofhistoryofphilosophywerementioned.BasicprinciplesTherootoftheveryexistenceofmatter(basicaxioms,etc)GeneralphenomenaUnificationofallprinciples(unifiedfieldtheory)MethodsofunderstandingtruthbasedonlogicandreasoningHowtocombinedifferentknowledgeanddifferentphenomenaunderageneralprincipleThegreatphilosophersdidnotsimplyfollowothersindevelopingtheirviews,notevenfromtheirteachers.Theycreatedtheirownthoughts(basedonpreviousworks.)13Thegoalofwritinghistoryofphilosophy(求因)Theoriginofaphilosophicalthinkingmustcomefromdifferentsources.Itisourgoaltofindoutsuchsources.(明變)Therearemanycomplicatedphilosophicalthoughtsinthehistory.Itisimportanttofigureoutthetreadsoftheirthoughts.(評論)Acorticalcommentsoftheoccurrenceofallphilosophiesandtheirconsequences.14Ialsolearntthewaytodoresearchwithlastingimportance.WangGuowei(王國維)TheThreeLevelsToAchieveBreakthroughinResearch(borrowingsomelinesfromSongCiorlyricpoemsfromtheSungdynasty)

LevelOne...LastnightthewestwindwitheredthegreeneryofthetreesInloneliness,ImountedthetallbuildingCastingmyeyesightalongallroadstotheedgeoftheearth晏殊…昨夜西風凋碧樹，獨上高樓，望盡天涯路。15Leveltwo

...FormylooseningwaistbeltIfeelnoregret,Forheritisworthbeinghaggardandthin.柳永…衣帶漸寬終不悔，為伊消得人憔悴。LevelThreeLookingforherathousandtimesInacrowdAllofasuddenAsIturnedmyheadThereshewasStandingintheshadesoffadinglights辛棄疾…眾裏尋他千百度，驀然回首，那人卻在，燈火闌珊處。16InmysecondyearofPuiChing,Igotintoaproblemwithmyteacher.Theteacherwasaverydevotedheadmasterofmyclassandclearlymeantwellforme.Shewasshockedtofindoutthatmyfatherwasaprofessorandpoorlypaid.

Herpassionformyfuturechangedmybehaviorinclassroom.

Istudiedplanegeometryinsecondyearofhighschool.17Myclassmateswerenotusedtoreasonabstractly.ThemerefactthatIlistenedtomyfather’sphilo-sophicaldiscussionathomemademefeelathomewithaxiomaticapproach.

Infact,IfeltIcanunderstandmyfather’sconversationbetterafterIlearntgeometry.

Thecharmtoproveeleganttheoremsbasedonsimpleaxiomsexcitesme.18Withthepassionforgeometry,Istartedtodevelopmytasteformathematics,whichincludedalgebra.EverythingbecameeasyafterIfoundthesubjectexciting.

Ialsofoundthesubjectofhistoryinteresting.

IttaughtmeaglobalviewofeverythingthatIlearnt.

Howeventshappened?Whytheyhappened?Whatmayhappeninthefuture?19Atthistime,myfatherjustfinishedwritinghisbookonhistoryofWesternphilosophy.Hisconversationswithstudentstaughtmethewaythatweshouldseehistoryinaglobalmanner.

Thiskindofpracticedeeplyinfluencemywayoflookingatmyresearchprojectsinthelatertime.20Around400A.D.Liu(whowrotethefirstbookoncomparativeliteratureonChinesewritingsuptothatperiod)文心雕龍諸子

Mybodymayperishalongwithtime.Mygoalandmyambitionwillextendwithmytheory.Myheartisinresonancewiththosegreatmeninancientdays.Myfeelingandmytheorywillgoforwardforthenextthousandyears.身與時桀，志共道申。標心於萬古之上，而送懷於千載之下。21WhenIwasfourteen,myfatherpassedaway.Thiswasperhapsthegreatestshocktomylife.Foralongtime,Icouldnotbelievethatmyfatherleftmeandthefamily.

Thefinancialsituationofthefamilywasreallybad.Itwasnotclearatallthatwecouldstillgotoschool.Thetremendouswillofmymotherandthehelpsofmyfather’sfriendsandstudentsmadeitpossible.22Thisdisastrouschangeoffamilysituationmadememuchmoremature.Theextremehard-shipshoweddifficultyofhumanrelationship,whatIlearntfrommyfatherbecamepractical.

ThepoemsandtheclassicalessaysthatIlearntbecamemuchmoremeaningful.IspenthalfayeartoreadclassicalliteratureandhistoryofChina.Itbecameawayformetorelaxduringverytensesituationinlife.Thebeautifulpoemsguidedmetoappreciatethebeautyofnature.23英國大詩人拜倫

〝希臘啊！你本是平和時代的愛嬌，你本是戰爭時代的天驕。撒芷波，歌聲高，女詩人，熱情好。更有那德羅士、菲波士榮光常照。此地是藝文舊壘，技術中潮。如今在否？算除卻太陽光線，萬般沒了。〞

〝馬拉頓前啊！山容縹緲。馬拉頓後啊！海門環繞。如此好山河，也應有自由回照。我向那波斯軍墓門憑眺。難道我為奴為隸，今生便了？不信我為奴為隸，今生便了。〞

梁啟超翻譯24FromByron’sDonJuanII--AvariationeditionbySteffanPrattCantoIII,861.TheislesofGreece,theislesofGreece!WhereburningSappholovedandsung,Wheregrewtheartsofwarandpeace,…WhereDelosrose,andPhoebussprung! Eternalsummergildsthenyet,Butall,excepttheirsun,isset.2.ThemountainslookonMarathon…AndMarathonlooksonthesea;Andmusingthereanhouralone,Idream’dthatGreecemightstillbefree;ForstandingonthePersian’sgrave,Icouldnotdeemmyselfaslave.

25Ireadalotofbooksinmathematics.Ithoughtabouttheproblemsinthosebooks.WhenIexhaustedallthoseproblems,IstartedtocreatemyownproblemsasIthoughtthattheymaybechallenging.Thepracticeofcreatingmyownproblemhadbeenthemostimportantpartofmyresearchinthefuture.Thetextbooksinschooldidnotsatisfyme.IwenttolibraryandIwenttobookstorestoreadbooks.IspenthoursandhoursinbookstoretoreadbooksthatIcouldnotaffordtobuy.26WhenIwasfifteen,Istartedtoteachlowergradelevelstudentstoearnmoney.Iwasproudthatbyteachinginanovelway,Iwasabletotransformsomeverypoorstudenttobecomethebeststudentinclass.

Itwasanexperiencetotrainyoungpeople.Ialsolearntthatitisbeneficialtomyselftoteach.27Myhighschoolteachersinmathematicswereexcellent.Westudiedratheradvancedtopicsinmathematics.Ihadnodifficultywiththem.

However,Iwasratherdisappointedthatmyphysicsteacherwasnotgoodenough.Thefundamentalintuitioninphysicswasnotestablishedduringmyhighschoolyear.Iregrettedituptonow.

IhadanexcellentteacherinChinese.Hewasmyfather’sfriend:hetaughtustothinkinanon-traditionalmanner.28Wewereaskedtothinkcreatively.Hesaidthatweshouldreadgoodbooksbutalsobadbooksaspossiblecomparisons.SoIreadeverything.Thisistrueevenformyscientificcareer.

AtypicaltopicforouressayinourChinesewriting:thephilosophyofapig.

Sowestarttodreamaboutanythinginteresting.

Iwasnotthebestinmyhighschool.Ididnothavethehighestgradeinmathematics.ButIthinkdeeperthanmyclassmatesandIreadmuchmorebooks.29IenteredTheChineseUniversityofHongKongin1966.IchosemathematicsasmycareeralthoughIwasalsoveryinterestedinthesubjectofhistory.

Bythistime,IstartedtodigestthoseadvancedlevelmathematicsbooksthatIreadinhighschool.Ididnotquiteunderstandthosebooksatthebeginning.SuddenlyIunderstandthemandIwasmuchbetterthanthecontemporarystudents.30

崇基門前對聯

崇高惟博愛本天地立心無間東西溝通學術

基礎在育才當海山勝境有懷胞與陶鑄人群

31ChungChiCollegeAnthemMenfromfourseasfoundedChungChisothatheremightyouthHonourChrist,eternalteacher,whoHimselfistruth.ThroughthelongnightkeepthetorchbrightandtheworkbegunTillthelightsoffaithandknowledgeshowtheworldmadeone.China'sstillevolvingculture,grateful,weretain;EastandWest,throughfreelysharing,furtherstrengthobtain,BytheChurchupheldandnurtured,mindstodutydrawn,ChungChi,towardtheveryhighest,leaduson,andon!32Collegemathematicsopenedmyeyes.Thefactthatonecanderiveeverystatementinmathematicsfromsimpleaxiomsreallyexcitedme.AfterIunderstoodhowmathematicswasbuilt,IgotsoexcitedthatIwrotealettertomyprofessorshowingmygreatpleasure.Itwasacornerstoneformyappreciationofmathematics.

AnewPh.DcamefromBerkeleytoHongKong.HisnameisStephenSalaff.Hewassoimpressedbymyperformancethathewroteabookwithmetogether.Itstopicisonordinarydifferentialequations.33AnotherteacherDr.BrodycamefromPrinceton.Hehadaratheruniquewayofteaching.Hepickedanadvancedbookinmathematics.Heassignedachapterforthestudentstofindmistakesinthebookandcorrectedthemistakes.

Itisagoodmethodtotrainusnottodependontextbook.Atthesametime,Itrainedmyselftobecriticalabouttheestablishedtheoremsinthebook.SometimesIgeneralizedthetheorems.BrodywasverypleasedbymyperformancewhenIshowedwhatIcoulddoinclass.34Theimportanceofsuchtrainingsisthat:Ilearnthowtothinkindependently.Ifoundouthowimportantitistoexpressmathematicsinfrontofanaudience.Thesepointshavebeenimportantformeandformyteaching.35WiththehelpofDr.Salaff,IwasabletoenterthegraduateschoolofBerkeleydespitethatIdidnotfinishmycollegeinHongKong.

Ofcourse,Berkeleyhastheleadingmathematicsdepartmentintheworld.IarrivedinAugust.ImetProf.S.S.Chernwhobecomemythesisadvisorlater.36WhenIwasinHongKong,Iwastoomuchfascinatedbyveryabstractmathematics.(AlthoughIwastrainedquitesolidlyinanalysis.)Ithoughtmathematicscoveringaverygeneralareaisbestmathematics.IthoughtIwouldstudyfunctionalanalysis.Ilearntalotinthatsubject.IreadabigfatbookofDunford-Schwatzonfunctionalanalysis.Ialsoreadalotofbooksonoperatoralgebra.

WhenIarrivedinBerkeley,Imetsomebestmindsinmathematics.Ichangedmyview.37WhenImetthosefirstratedmathematicians,Iwasratherthirstyinlearningdifferentsubjectsfromthem.Iattendedmanyclassesfrom8amto5pm.(SometimesIatelunchinclass.)Thesearesubjectsrangingfromtopology,geometry,differentialequations,Liegroups,numbertheory,combinatorials,probabilitytheorytodynamicalsystem.Ididnotunderstandallofthem.ButIfocusmyeffortsonseveralofthem.38WhenIlearnttopology,itwassodifferentlyfromwhatIlearntbefore.Therewerefiftystudentsinclass.Allofthemseemtobesmartandfarbetterthanme.Theycouldperformandtalkednicely.

However,Ididmyhomeworkwellandinashorttime,IfoundoutthatIwasnotbad,afterall.Thekeyistoworkoutallthosetoughhomeworksandthinkaboutthemthoroughly.39IreadabookofJohnMilnorandwasfascinatedbythedescriptionoftheconceptofcurvature.JohnMilnorisanexcellenttopologist.

Istartedtothinkaboutproblemsthatisrelatedtoquestionsinthebook.Ispentalotoftimeinthelibrary.Therewasnoofficeforgraduatestudents.ThereweremanyfamousprofessorsinBerkeley.SoonIrealizethattheyarehumanbeingsafterall.Ireadmanyjournalsandbooksinthelibrary.40Istartedtobeabletoprovesomenontrivialtheoremsinthesecondquarter.TheyarerelatedtosometheoremsingrouptheorythatIlearntoversomecasualconversationwithateacherincollege.Iappliedittogeometry.Myprofessorsweresurprisedandpleasedbymyprogress.Oneoftheprofessorsstartedtoworkwithme.Wewrotetwopapers.ProfessorChernwasonsabbaticalleave.Whenhereturned,hewasverypleased.41IdidnotthinkwhatIdidwasgreat.IwasveryimpressedbyProf.Morreyonhisteachingofnonlinearpartialdifferentialequations.Hetaughtnonlineartechnique.Itwasnotfashionable.Thebookhewrotewasdifficulttoread.Ithoughtthosetechniquethathedevelopedareverydeepandmustbeimportantforthefutureofgeometry.Ilearntthosetechnology.DespiteofProf.Morrey’sbigname,veryfewstudentsorfacultiescaredabouthiscourse.Attheend,IwastheonlystudentinclassandProf.Morreytaughtmeinhisoffice.Thiscoursebuiltthefoundationofmymathematicscareer.42AfterIwroteseveralpapers,Prof.CherntoldmanypeoplehowbrilliantIwasalthoughIdidnotthinkheknewmyworkswell.Istartedtothinkmorethoroughlyaboutmathematicsandgeometryinparticular.Iworkedonotherpartsofgeometry.However,resultsdidnotcomeeasy.

MyfriendS.Y.ChengcamefromHongKongthatsummer.WesharedanapartmentrightnexttothecampusandIbecamemorerelax.43Inthatsummer,IaskedProf.Cherntobemyadvisor.Heagreedandafteronemonth,hesaidthatmypapersinthefirstyearshouldbeenoughformythesis.IwassurprisedbecauseIthoughttheyarenotgoodenoughandIwantedtolearnmore.

Inanycase,inthesecondyear,Ilearntmoreinthefieldofcomplexgeometryandtopology.Prof.Chernhadagreatexpectationonme.HesuggestedmetoworkonRiemanhypothesis.Unfortunatelyuptonow,Ineverthoughtaboutit.44InsteadIpursuedthegeneralunderstandingofcurvatureofspace.IdecidedakeytounderstandsuchaconceptwasaproposalmadebyCalabiinearlyfifties.NobodybelievedwhatCalabi’sthoughtistrue.Istartedtothinkaboutit.Itisnotthestandardthingthatageometerwoulddointhosedays.Itisclearlyahardquestionofanalysis.Nobodywouldtouchsuchproblem.45Istartedtodevelopmytasteintolearninghowtointroduceanalyticmethodsintogeometry.Previoustothis,therewereattemptstoapplynonlineartheorytosurfacesinthreespace.Iwantedtodealwithanabstractspaceinarbitrarydimension.

BecauseofProf.MorreyandChern’sinterestsinminimalsurfaces,Ialsodevelopedinterestintothisfascinatingsubject.Inparticular,Iwasinterestingintoharmonicmaps.Ingeneral,IstudiedCalculusofvariation.46Iwasinterestedinallanalyticaspectsofgeometry.Thebasicideaistomergethesubjectofnonlineardifferentialequationandgeometry.Inordertounderstandnonlinearequation,itisfundamentaltounderstandlinearequations.HenceIestablishthefirstmajortheoremforharmonicfunctionsonmanifolds.IgotmyfriendS.Y.Chengtolookintoeigenvalueandeigenfunctionproblems.Togetherwewroteseveralimportantpapersonthesubject.Theyarestillfundamentalformodernresearch.47WhenIgraduated,Igotseveraloffers.MyteacherChernsuggestedmetogotoInstituteforAdvancedStudy.ThesalarywaslessthanhalfofwhatIcouldhavegottenfromHarvard.ButIwenttotheInstituteforAdvancedStudy.Imetsomeothergroupofdistinguishedmathematicians.Idevelopedsometasteintotopology,especiallythetheoryofsymmetriesofspace.IdidsolvesomeimportantproblemsinthissubjectbasedonanalyticalideasIdeveloped(groupactionsonmanifolds).48Becauseofproblemofvisa,IwenttoNewYorkStateUniversityofStonyBrook.Atthatpoint,itwassupposedtobethecenterofmetricgeometry.Itwasindeedagoodplace,fullofenergeticyounggeometers.Ilearnttheirtechnique.ButIdidnotthinkthatwastherightdirectionforgeometry.

Afteroneyear,IwenttoStanfordwheretherewasnogeometer.Itisaverypeacefulenvironmentandisverygoodinnonlinearpartialdifferentialequations.ImetmyverygoodfriendLeonSimonandmyformerstudentRichardSchoen.Togetherwedevelopedthesubjectofnonlinearanalysisingeometry.49TaoYuan-Ming(晉‧陶淵明)LongIlivedcheckedbythebarsofacage.NowIhaveturnedagaintoNatureandFreedom. 久在樊籠裡，復得返自然。50WhenIarrivedatStanford,therewasabigconferenceingeometry.Aphysicistwasinvolvedtogiveatalkongeneralrelativity.AlthoughIdidnotunderstandPhysicswell,Iimmediatelyfellinloveofthegeometryproblemassociatedtogeneralrelativity.Itisfascinatingtogivephysicalmeaningofspacethatwesawandviceversa.51Theproblemwastoodifficulttosolveatthattime.ButIkeptthatinmind.Duringtheconference,IthoughtIfoundawaytodisprovetheproposalofProf.Calabi.Iwasaskedtogiveapresentationofmythoughts.Itallsoundedgood.Nobodyobjected.Soeverypeoplewashappythatthegeneralexpectionwastrue:theCalabiConjectureiswrongafterall.52Aftertwomonths,Prof.Calabiwrotetomeforclarificationofmythoughts.Ifoundaseriousgaponmyreasoning.Itwasthemostpainfulperiodofmyresearchlife.Icouldnotsleep.53Forabouttwoweeks,IcouldnotsleepasIconsideredthatmyreputationwasbadlydamagedbynotabletoreproducewhatIclaimed.(AlthoughIneverwroteanyannouncementofit.)However,thepainofgoingthroughofeachsingledetailsoftheproblemconvincedmethattheoppositedirectionshouldberight.TheargumenttogivecounterexampletotheCalabiConjecturewasthatifitweretrue,somethingmusthappen.Henceafewyearslater,whenIsettledthisproblem,Iknewmanynaturalconsequencesofit.54AfterIdecidedthatitmustbetrue,Iworkedtowardstherightdirection.Manypreparatoryworksweredonetoprepareforthefinalproof.IworkedwithChengonunderstandingmyquestionsrelatedtoMonge-Ampereequations,affinegeometry,maximalsurfacesandmanyrelatedproblems.RichardSchoenworkedwithmeonharmonicmaps.Schoen,SimonandIworkedonminimalsurfaces.Inamatteroftwoyears,weunderstoodagreatdealofnonlinearanalysisrelatedtogeometry.Itwasanexcitingperiodoftimeingeometry.55QuYuan(屈原)ThisiswhatIamafter.Iwillnotturnbackevenninedeathsareahead.固余心之所善兮，雖九死而猶未悔。56RightafterIgotmarried,IgottherightideatofinishtheproofoftheCalabiConjecture.IfeltIfinallyunderstoodcurvatureofKahlergeometry.Manyimportantapplicationswerefoundtosolvesomeoldproblemsinalgebricgeometry.57“Oneproblemisthatstringtheoryrequires10dimensionsofspace-time,whereasweappeartoliveinfour.Dr.StromingerrememberedbeingexcitedwhenhefoundapaperbythemathematicianDr.Shing-TungYau,nowofHarvardandtheChineseUniversityofHongKong.ItprovedaconjecturebyDr.EugenioCalabi,nowretiredfromtheUniversityofPennsylvania,thattheextradimensionscouldbecurledupinmicroscopicallyinvisiblewaysliketheloopsinacarpet.”OneCosmicQuestion,TooManyAnswersSeptember2,200358WhenIfinishedtheproofoftheCalabiConjecture,IfeltIhavesetupaframeworkformathematicianstomergetwoimportantfieldstogether:nonlinearpartialdifferentialequationandgeometry.In1976,IwasinUCLAandImetmyfriendMeekswhowasmyclassmateingraduateschool.Hewasnotingoodshape.Buthewasaveryoriginalmathematician.SoIproposedtoworkwithhimonrelatingideasfromminimalsurfacesandtopologyofthreedimensionalmanifoldtogether.59Wehadagreatsuccess.Wesolvedtwoclassicalproblemsinbothfields:Iftheboundaryofasoapfilmisconvex,thesoapfilmcannotcrossitself.TogetherwithworkswithThurston,thefamousproblemofSmithConjectureissolved.Oncethedirectionwassetintherightway,manyclassicalquestionscanbeanswered.60Nextyear,IvisitedBerkeleywhereIgaveseminarsonnonlinearproblemsingeometry.BothRichardSchoenandS.Y.Chengwerethere.WithSchoen,wefinallysolvedtheproblemthatIwasexcitedaboutingeneralrelativity.Theproblemiscalledpositivemassconjectureandisfundamentalforgeneralrelativity.(Onlyifthemassispositivethatspacetimecanbestable.)61In1978,IreturnedtoStanford,Iappliedideasofminimalsurfacestosolveafamousproblem(FrenkelConjecture)incomplexgeometrywithY.T.Siu.Ialsointroducedideasofharmonicmaptostudyingdiscretegroupsymmetries.Theseideasarestillusefuluptonow.SchoenandIdevelopedthestructuretheoryofmanifoldswithpositivescalarcurvaturebasedonourworksongeneralrelativity.62In1979,wehadaspecialyearindifferentialgeometryintheInstituteforAdvancedStudy.Practicallyallgeometerscame.Wesetagooddirectionforgeometry.Iproposedonehundredinterestingopenproblemsforthefieldofgeometry.Someofthemweresolvedandsomeofthemwerenot.The1970’shavebeenoneofthemostfruitfuleraofgeometry.63Bylateseventies,Iwaswell-recognizedbymycolleagues.ThereweremanynewscoverageonproblemsthatIsolved.However,itwillbemisleadingtothinkthatmygoalistoearnmedalsandgainrecognition.Ithasneverbeenthepriorityofmystudy.IaminterestedinMathematicsbecauseitissoexcitingtoseehowhumanthoughtcanbeextendedtounderstandnature.Thebeautyofnaturefromthepointofviewofgeometryiseverlasting.64TogetherwithmyfriendsSchoen,Simon,Cheng,Meeks,Uhlenbeck,Hamilton,andlaterbyDonaldson,TaubesandHuisken,andothers,NonlinearAnalysisinGeometryhasbeenestablishedasarichsubject.Itsimportanceinunderstandingbeautyofnaturecanneverbeunderestimated.ThemostrecentdevelopmentsshowtheirimportanceinPhysicsandinAppliedScience.65Onceanaturalmergeofseveralgreatsubjects:Geometry,Non-linearAnalysis,AlgebraicGeometry,MathematicalPhysics,isdone,classicalanddifficultproblemswerenaturallysolved.Problemsolvingcanbeconsideredaslampstandsonourroadtounderstandnature.66Confucius(孔子)

Studyingwithoutthinkinggoesnowhere;thinkingwithoutstudyingleadstobewilderment.學而不思則罔，思而不學則殆。67Han(韓愈600AD)ontheprocessoflearning:Whengeneralpeoplepraisemywriting,Iamnotpleased.Whengeneralpeopleslightmywriting,Iamnotdepressed.譽之則不以為喜，毀之則不以為憂。68C.F.Gauss(1817)Iambecomingmoreandmoreconvincingthatthenecessityofourgeometrycannotbeproved,atlea