A_walk_in_the_noncommutative_garden.pdf

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1、AWALKINTHENONCOMMUTATIVEGARDENALAINCONNESANDMATILDEMARCOLLIContents1.Introduction22.Handlingnoncommutativespacesinthewild:basictools23.Phasespacesofmicroscopicsystems64.Noncommutativequotients95.Spacesofleavesoffoliations106.Thenoncommutativetori177.Dualsofdiscretegroups228.Brilloui

2、nzoneandthequantumHalleffect249.Tilings2910.Noncommutativespacesfromdynamicalsystems3211.Noncommutativespacesfromstringtheory3612.Groupoidsandtheindextheorem3813.Riemannianmanifolds,conicalsingularities3914.Cantorsetsandfractals4515.SpacesofdimensionzandDimReg4816.Localalgebrasinsupe

3、rsymmetricQFT5017.Spacetimeandthestandardmodelofelementaryparticles5118.Isospectraldeformations5719.Algebraicdeformations5920.Quantumgroups6221.Sphericalmanifolds66arXiv:math.QA/0601054v13Jan200622.Q-lattices6923.ModularHeckealgebras7424.Noncommutativemodulispaces,Shimuravarieties81

4、25.Theadelesclassspaceandthespectralrealization8426.ThermodynamicsofendomotivesandtheTehranprogram90References10012CONNESANDMARCOLLI1.IntroductionIfyoucleavethehearthofonedropofwaterahundredpureoceansemergefromit.(MahmudShabistari,Gulshan-i-raz)Wehavedecidedtocontributetothevolumeof

5、theIPMlecturesonnoncommutativege-ometryatextthatcollectsalistofexamplesofnoncommutativespaces.AsthequoteoftheSufipoethereabovesuggests,itisoftenbettertoapproachanewsubjectbyanalyzingspecificexamplesratherthanpresentingthegeneraltheory.Wehopethatthediversityofexamplesthereaderswillenco

6、unterinthistextwillsufficetoconvincethemofthefactthatnoncommutativegeometryisaveryrichfieldinrapidevolution,fullofinterestingandyetunexploredlandscapes.ManyoftheexamplescollectedherehavenotyetbeenfullyexploredfromthepointofviewofthegeneralguidelinesweproposeinSection2andthemainpointoft

7、histextistoprovideagreatnumberofopenquestions.Thereadershouldinterpretthissurveyasasuggestionofpossibleinterestingproblemstoinvestigate,bothinthesettingsdescribedhere,aswellasinotherexamplesthatareavailablebutdidnotfitinthislist,andinthemanymorethatstillawaittobediscovered.Besidesthe

8、existingbooksonNCGs