Perspectives on Geometric Analysis---丘成桐）

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1、PERSPECTIVESONGEOMETRICANALYSISSHING-TUNGYAUThisessaygrewfromatalkIgaveontheoccasionoftheseventiethanniversaryoftheChineseMathematicalSociety.IdedicatethelecturetothememoryofmyteacherS.S.Chernwhohadpassedawayhalfayearbefore(December2004).Duringmygrad

2、uatestudies,Iwasratherfreeinpickingresearchtopics.I[724]workedonfundamentalgroupsofmanifoldswithnon-positivecurvature.Butinthesecondyearofmystudies,Istartedtolookintodifferentialequationsonmanifolds.However,atthattime,Chernwasverymuchinterestedinthewo

3、rkofBottonholomorphicvectorfields.AlsohetoldmethatIshouldworkonRiemannhypothesis.(Weilhadtoldhimthatitwastimeforthehypothesistobesettled.)WhileCherndidnotexpresshisopinionsaboutmyresearchongeometricanalysis,hestartedtoappreciateitafewyearslater.Infact

4、,afterCherngaveacourseonCalabi’sworksonaffinegeometryin1972atBerkeley,S.Y.Chengtoldmeabouttheseinspiringlectures.By1973,ChengandIstartedtoworkonsomeproblemsmentionedinChern’slectures.WedidnotrealizethatthegreatgeometersPogorelov,CalabiandNirenbergwerea

5、lsoworkingonthem.WewereexcitedthatwesolvedsomeoftheconjecturesofCalabionimproperaffinespheres.ButsoonafterwefoundoutthatPogorelov[556]publishedhisresultsrightbeforeusbydifferentarguments.Neverthelessourideasareusefulinhandlingotherproblemsinaffinegeometry

6、,andmyknowledgeaboutMonge-Amp`ereequationsstartedtobroadenintheseyears.Chernwasverypleasedbymywork,especiallyafterI[729]solvedtheproblemofCalabionK¨ahlerEinsteinmetricin1976.IhadbeenatStanford,andChernproposedtotheBerkeleyMathDepartmentthattheyhireme

7、.IvisitedBerkeleyin1977forayearandgaveacourseongeometricanalysiswithemphasisonisometricembedding.ChernnominatedmetogiveaplenarytalkattheInternationalCongressinHelsinki.Thetalk[730]wentwell,butmydecisionnottostayatBerkeleydidnotquitepleasehim.Neverthe

8、lessherecommendedmeforapositiononthefacultyattheInstituteforAdvancedStudy.BeforeIacceptedafacultypositionattheInstitute,Iorganizedaspecialyearongeometryin1979attheInstituteattheinvitationofBorel.Thatwasanexcitingyearbecausemostpeopleingeometricanalys