New Spaces in Mathematics; Formal and Conceptual Reflections PDF
Preview New Spaces in Mathematics; Formal and Conceptual Reflections
NewSpacesinMathematics After the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This bookanditscompanionexplorevariousnewnotionsofspace,includingbothformal and conceptual points of view, as presented by leading experts at the New Spaces in MathematicsandPhysicsworkshopheldattheInstitutHenriPoincare´in2015. Thechaptersinthisvolumecoverabroadrangeoftopicsinmathematics,including diffeologies,syntheticdifferentialgeometry,microlocalanalysis,topostheory,infinity- groupoids, homotopy type theory, category-theoretic methods in geometry, stacks, derivedgeometry,andnoncommutativegeometry.Itisaddressedprimarilytomathe- maticiansandmathematicalphysicists,butalsotohistoriansandphilosophersofthese disciplines. mathieu anel isaVisitingAssistantProfessoratCarnegieMellonUniversity. His research interests include higher category theory, algebraic topology, and topos theory. gabriel catrenisPermanentResearcherinphilosophyofphysicsattheFrench National Centre for Scientific Research (CNRS). His research interests include the foundationsofclassicalandquantummechanics,andthefoundationsofgaugetheories. New Spaces in Mathematics Formal and Conceptual Reflections Editedby MATHIEU ANEL CarnegieMellonUniversity GABRIEL CATREN CNRS-Universite´ deParis UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108490634 DOI:10.1017/9781108854429 ©CambridgeUniversityPress2021 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2021 PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:Anel,Mathieu,editor.|Catren,Gabriel,1973–editor. Title:Newspacesinmathematics:formalandconceptualreflections/edited byMathieuAnel,CarnegieMellonUniversity,Pennsylvania,Gabriel Catren,CentreNationaldelaRechercheScientifique(CNRS),Paris. Description:NewYork:CambridgeUniversityPress,2020.| Includesbibliographicalreferencesandindex. Identifiers:LCCN2020006667|ISBN9781108490634(hardback)| ISBN9781108854399(epub) Subjects:LCSH:Mathematicalphysics–Research. Classification:LCCQC20.8.N4952020|DDC539.7/258–dc23 LCrecordavailableathttps://lccn.loc.gov/2020006667 ISBN-2VolumeSet978-1-108-85436-8Hardback ISBN-VolumeI978-1-108-49063-4Hardback ISBN-VolumeII978-1-108-49062-7Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents ContentsforNewSpacesinPhysics pagevii Introduction 1 Introduction 1 MathieuAnelandGabrielCatren PARTI DIFFERENTIALGEOMETRY 29 1 AnIntroductiontoDiffeology 31 PatrickIglesias-Zemmour 2 NewMethodsforOldSpaces:SyntheticDifferentialGeometry 83 AndersKock 3 MicrolocalAnalysisandBeyond 117 PierreSchapira PARTII TOPOLOGYANDALGEBRAICTOPOLOGY 153 4 Topo-logie 155 MathieuAnelandAndre´ Joyal 5 SpacesasInfinity-Groupoids 258 TimothyPorter 6 HomotopyTypeTheory:TheLogicofSpace 322 MichaelShulman PARTIII ALGEBRAICGEOMETRY 405 7 SheavesandFunctorsofPoints 407 MichelVaquie´ 8 Stacks 462 NicoleMestranoandCarlosSimpson v vi Contents 9 TheGeometryofAmbiguity:AnIntroductiontotheIdeas ofDerivedGeometry 505 MathieuAnel 10 Geometryindg-Categories 554 MaximKontsevich Contents for New Spaces in Physics ContentsforNewSpacesinMathematics pagevii Introduction 1 MathieuAnelandGabrielCatren PARTI NONCOMMUTATIVEAND SUPERCOMMUTATIVEGEOMETRIES 21 1 NoncommutativeGeometry,theSpectralStandpoint 23 AlainConnes 2 TheLogicofQuantumMechanics(Revisited) 85 KlaasLandsman 3 SupergeometryinMathematicsandPhysics 114 MikhailKapranov PARTII SYMPLECTICGEOMETRY 153 4 DerivedStacksinSymplecticGeometry 155 DamienCalaque 5 HigherPrequantumGeometry 202 UrsSchreiber PARTIII SPACETIME 279 6 StruggleswiththeContinuum 281 JohnC.Baez vii viii ContentsforNewSpacesinPhysics 7 TwistorTheory:AGeometricPerspectiveforDescribing thePhysicalWorld 327 RogerPenrose 8 QuantumGeometryofSpace 373 MuxinHan 9 StringyGeometryandEmergentSpace 407 MarcosMarin˜o Introduction MathieuAnelandGabrielCatren Contents 1 ABriefHistoryofSpace 1 2 ContemporaryMathematicalSpaces 3 3 SummariesoftheChapters 12 Acknowledgments 22 References 22 1 ABriefHistoryofSpace Spaceisacentralnotioninbothmathematicsandphysicsandhasalwaysbeen attheheartoftheirinteractions.FromGreekgeometrytoGalileoexperiences, mathematics and physics have been rooted in constructions performed in the same ambient physical space. But both mathematics and physics have eventually left the safe experience of this common ground for more abstract notionsofspace. The 17th century witnessed the development of projective geometry and the strange, yet effective, idea of points at infinity. This century witnessed also the advent of analytic geometry (with its use of coordinates) with Descartes and of differential calculus with Newton and Leibniz. Both have led to an approach to geometry fundamentally based on the manipulation of algebraic formulas. The capacity to manipulate spaces without relying on a spatial intuition has laid the foundations for one of the most important revolutions in geometry: the conception of spaces of arbitrary dimension. 1
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