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1、GiovanniLandiAnIntroductiontoNoncommutativeSpacesandtheirGeometriesWith27FiguresAugust11,1997Springer-VerlagBerlinHeidelbergNewYorkLondonParisTokyoHongKongBarcelonaBudapestadAnnaeJacopoperilloroamoreelaloropazienzaVIPrefaceThesenotesarosefromaseriesofintro
2、ductoryseminarsonnoncommutativegeometryIgaveattheUniversityofTriesteinSeptember1995duringtheXWorkshoponDi erentialGeometricMethodsinClassicalMechanics.ItwasBeppeMarmo'ssuggestionthatIwrotenotesforthelectures.ThenotesaremainlyanintroductiontoConnes'noncommu
3、tativege-ometry.Theycouldserveasa` rstaidkit'beforeoneventuresintothebeautifulbutbewilderinglandscapeofConnes'theory.Themaindi erencefromotheravailableintroductionstoConnes'work,notablyKastler'spapers[86]andalsotheGracia-BondaandVarillypaper[130],istheem
4、phasisonnoncommutativespacesseenasconcretespaces.Importantexamplesofnoncommutativespacesareprovidedbynoncom-mutativelattices.ThelatterarethesubjectofintenseworkIamdoingincollaborationwithA.P.Balachandran,GiuseppeBimonte,ElisaErcolessi,FedeleLizzi,GianniSpa
5、ranoandPauloTeotonio-Sobrinho.Thesenotesarealsomeanttobeanintroductiontothisresearch.Thereisstillalotofworkinprogressandbynomeanscanthesenotesbeconsideredasareviewofev-erythingwehaveachievedsofar.Rather,Ihopetheywillshowtherelevanceandpotentialityforphysic
6、altheoriesofnoncommutativelattices.Acknowledgement.Iamindebtedtoseveralpeopleforhelpandsuggestionsofdi erentkindsatvariousstagesofthisproject:A.P.Balachandran,G.Bimonte,U.Bruzzo,T.Brzezinski,M.Carfora,R.Catenacci,A.Connes,L.Dabrowski,G.F.Dell'Antonio,M.Dub
7、ois-Violette,B.Dubrovin,E.Elizalde,E.Ercolessi,J.M.Gracia-Bonda,P.Hajac,D.Kastler,A.Kempf,F.Lizzi,J.Madore,G.Marmo,A.Napoli,C.Reina,C.Ro-velli,G.Sewell,P.Siniscalco,G.Sparano,P.Teotonio-Sobrinho,G.Thompson,J.C.Varilly,R.Zapatrin.VIIContents1.Introductio
8、n::::::::::::::::::::::::::::::::::::::::::::::12.NoncommutativeSpacesandAlgebrasofFunctions:::::::72.1Algebras::::::::::::::::::::::::::::::::::::::::::::::72.2CommutativeSpaces::::::::::::::::::::::::::::::
