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1、ANINTRODUCTIONTONONCOMMUTATIVEGEOMETRYJosephC.V´arillyDepartamentodeMatem´aticas,UniversidaddeCostaRica,2060SanJos´e,CostaRicaIntroductionThesearelecturenotesforacoursegivenattheSummerSchoolonNoncommuta-tiveGeometryandApplications,sponsoredbytheEuropeanMathematicalSociety,atMo
2、nsaraz,PortugalandatLisboa,fromthe1sttothe10thofSeptember,1997.Noncommutativegeometry,whichalreadyoccupiesanextensiveandwide-rangingareaofmathematics,hascomeunderincreasingscrutinyfromphysicistsinterestedinwhatithastosayaboutfundamentalproblemsofNature.Thiscoursesoughttoaddres
3、samixedaudienceofstudentsandyoungresearchers,bothmathematiciansandphysicists,andtoprovideagatewaytosomeofitsmorerecentdevelopments.Manyapproachescanbetakentointroducingnoncommutativegeometry.IdecidedtofocusonthegeometryofRiemannianspinmanifoldsandtheirnoncommutativecousins,whi
4、charegeometriesdeterminedbyasuitablegeneralizationoftheDiracoperator.ThesegeometriesunderlietheNCGapproachtophenomenologicalparticlemodelsandrecentattemptstoplacegravityandmatterfieldsonthesamegeometricalfooting.Thefirsttwolecturesaredevotedtocommutativegeometry;wesetupthegenera
5、lframeworkandthencomputeasimpleexample,thetwo-sphere,innoncommutativeterms.Thegeneraldefinitionofageometryisthenlaidoutandexemplifiedwiththenoncom-mutativetorus.EnoughdetailsaregivensothatonecanseeclearlythatNCGisjustordinarygeometry,extendedbydiscardingthecommutativityassumptio
6、nonthecoordi-natealgebra.Classificationuptoequivalenceisdealtwithbrieflyinlecture7.Otherlecturesexploresomeofthetoolsofthetrade:thenoncommutativeintegral,therˆoleofquantization,andthespectralactionfunctional.Physicalmodelsarenottreateddirectly,sincethesewerethesubjectofotherlect
7、uresattheSummerSchool,butmostofthemathematicalissuesneededfortheirunderstandingaredealtwithhere.Iwishtothankseveralpeoplewhocontributedinnosmallwaytoassemblingtheselectures.Jos´eM.Gracia-Bond´ıagavedecisivehelpatmanypoints;heandAlejandroRiveroprovidedconstructivecriticismthrou
8、ghout.IthankDanielKastler,BrunoIochum,arXiv:physics/9709045v1
