从物理到数论——非交换几何（孔涅）

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1、FromPhysicstoNumberTheoryviaNoncommutativeGeometryAlainConnes1andMatildeMarcolli21Coll`egedeFrance,3,rueUlm,F-75005Paris,FranceI.H.E.S.35routedeChartresF-91440Bures-sur-Yvette,FranceDepartmentofMathematics,VanderbiltUniversity,TN-37240,USAconnes@ihes.fr,alain@connes.org2Max–PlanckInstitutf¨urMa

2、thematik,Vivatsgasse7,D-53111Bonn,Germany,marcolli@mpim-bonn.mpg.deIntroduction...........................................................270References.....................................................278QuantumStatisticalMechanicsofQ-Lattices..................2801Introduction...............

3、...............................2802QuantumStatisticalMechanics...........................2833QabandKMSstates.......................................2884FurtherDevelopments....................................2925FabulousStates...........................................2946ThesubalgebraAQandEisens

4、teinSeries..................2967TheDeterminantpartoftheGL2-System.................3058CommensurabilityofQ-LatticesinCandthefullGL2-System...............................................3119ThesubalgebraAQandtheModularField................32710Thenoncommutativeboundaryofmodularcurves........33911

5、TheBCalgebraandopticalcoherence....................343References.....................................................347270AlainConnesandMatildeMarcolliIntroductionevoltanostrapoppanelmattino,de’remifacemmoalialfollevolo——Dante,Inf.XXVI124-125Severalrecentresultsrevealasurprisingconnectionbetwe

6、enmodularformsandnoncommutativegeometry.Thefirstoccurrencecamefromtheclassificationofnoncommutativethreespheres,[C–DuboisViolette-I][C–DuboisViolette-II].HardcomputationswiththenoncommutativeanalogoftheJacobianinvolvingtheninthpoweroftheDedekindetafunctionwerenecessaryinordertoanalyzetherelationb

7、etweensuchspheresandnoncommutativenilmanifolds.AnotheroccurrencecanbeseeninthecomputationoftheexplicitcycliccohomologyCherncharacterofaspectraltripleonSUq(2)[C–02].AnothersurprisecamerecentlyfromaremarkableactionoftheHopfalgebraof