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1、TheAtiyah-Patodi-SingerindextheoremRichardB.MelroseMassachusettsInstituteofTechnologycCopyrightR.B.Melrose,1992.Allrightsreserved.Version:5.0ERevised:7-12-1993Run:March14,1996Tomymother,IslaLouiseMelroseivPREFACEThistextisasomewhatexpandedversionoflecturenoteswritte
2、ndur-ing,anddirectlyafter,acourseatMITintheSpringof1991.MostoftheparticipantshadattendedacoursetheprecedingFallonpseudodif-ferentialoperatorsoncompactmanifoldswithoutboundary,includingtheHodgetheoremandtheconstructionofthefundamentalsolutionforthewaveequation.Approx
3、imatelythislevelofsophisticationisassumedofthereader.TheintentionofthesecondcoursewastoembedtheAtiyah-Patodi-SingerindextheoreminananalyticframeworkanalogoustothatprovidedbythetheoryofpseudodierentialoperatorsfortheAtiyah-Singertheorem.Sincethistreatmentleadstoavar
4、ietyofcurrentresearchtopicsitispresentedhereinthehopethatitwillbeofusetoawideraudience.Therearemanypeopletothank.ForemostIamgratefultothemembersoftheaudienceofthecoursefortheirtoleranceandenthusiasm.IamespeciallygratefultoPaoloPiazzaforhiscommentsduringthecourseanda
5、lsoasacollaboratorinworkrelatedtothissubject.OthersfromwhomIhavelearntinthiswayareXianzheDai,CharlieEpstein,DanFreed,RafeMazzeoandGerardoMendoza.TothelastoftheseIwouldliketotakethisopportunitytoapologizeformypart,whateverthatwas,inthesomewhatmysteriousnon-appearance
6、ofthepaper[64]onwhichaconsiderablepartofChapter6,andindeedthegeneral`b-philosophy,'isbased.MoregenerallyIamhappytoacknowledgethein
uence,throughcon-versation,onmyapproachtothissubjectofMichaelAtiyah,EzraGetzler,LarsHormander,WernerMuller,BobSeeley,IzSingerandMichael
7、Tay-lor.IamindebtedtoAnt^oniosaBarreto,XianzheDai,CharlieEpsteinandMaciejZworskiwhotoleratedmyneglectofotherprojectsduringtheprocessofwriting,toTanyaChristiansen,XianzheDai,AndrewHassell,LarsHormander,GerdGrubb,RafeMazzeo,PaoloPiazzaandLorenzoRameroforcommentsonthe
8、manuscriptandespeciallytoJillianMelroseforherforbearance.ToJudyRomvosspecialthanksforturningmyrathercrudescrawlingsintotheoriginallecturen