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1、TheAtiyah-Patodi-SingerindextheoremRichardB.MelroseMassachusettsInstituteofTechnologycCopyrightR.B.Melrose,1992.Allrightsreserved.Version:5.0ERevised:7-12-1993Run:March14,1996Tomymother,IslaLouiseMelroseivPREFACEThistextisasomewhatexpandedversionoflecturenoteswrittendur-ing,
2、anddirectlyafter,acourseatMITintheSpringof1991.MostoftheparticipantshadattendedacoursetheprecedingFallonpseudodif-ferentialoperatorsoncompactmanifoldswithoutboundary,includingtheHodgetheoremandtheconstructionofthefundamentalsolutionforthewaveequation.Approximatelythislevelof
3、sophisticationisassumedofthereader.TheintentionofthesecondcoursewastoembedtheAtiyah-Patodi-Singerindextheoreminananalyticframeworkanalogoustothatprovidedbythetheoryofpseudodi erentialoperatorsfortheAtiyah-Singertheorem.Sincethistreatmentleadstoavarietyofcurrentresearchtopics
4、itispresentedhereinthehopethatitwillbeofusetoawideraudience.Therearemanypeopletothank.ForemostIamgratefultothemembersoftheaudienceofthecoursefortheirtoleranceandenthusiasm.IamespeciallygratefultoPaoloPiazzaforhiscommentsduringthecourseandalsoasacollaboratorinworkrelatedtothi
5、ssubject.OthersfromwhomIhavelearntinthiswayareXianzheDai,CharlieEpstein,DanFreed,RafeMazzeoandGerardoMendoza.TothelastoftheseIwouldliketotakethisopportunitytoapologizeformypart,whateverthatwas,inthesomewhatmysteriousnon-appearanceofthepaper[64]onwhichaconsiderablepartofChapt
6、er6,andindeedthegeneral`b-philosophy,'isbased.MoregenerallyIamhappytoacknowledgethein
uence,throughcon-versation,onmyapproachtothissubjectofMichaelAtiyah,EzraGetzler,LarsHormander,WernerMuller,BobSeeley,IzSingerandMichaelTay-lor.IamindebtedtoAnt^oniosaBarreto,XianzheDai,Cha
7、rlieEpsteinandMaciejZworskiwhotoleratedmyneglectofotherprojectsduringtheprocessofwriting,toTanyaChristiansen,XianzheDai,AndrewHassell,LarsHormander,GerdGrubb,RafeMazzeo,PaoloPiazzaandLorenzoRameroforcommentsonthemanuscriptandespeciallytoJillianMelroseforherforbearance.ToJudy
8、Romvosspecialthanksforturningmyrathercrudescrawlingsintotheoriginallecturen
