Compactness and contradiction Tao.pdf

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1、CompactnessandcontradictionTerenceTaoDepartmentofMathematics,UCLA,LosAngeles,CA90095E-mailaddress:tao@math.ucla.eduToGarthGaudry,whosetmeontheroad;Tomyfamily,fortheirconstantsupport;Andtothereadersofmyblog,fortheirfeedbackandcontributions.ContentsPrefacexiAremarkonnotationxiAcknowledgmentsxiiCh

2、apter1.Logicandfoundations1x1.1.Materialimplication1x1.2.Errorsinmathematicalproofs2x1.3.Mathematicalstrength4x1.4.Stableimplications6x1.5.Notationalconventions8x1.6.Abstraction9x1.7.Circulararguments11x1.8.Theclassicalnumbersystems12x1.9.Roundnumbers15x1.10.Theoself-defeatingobject"argument,

3、revisited16x1.11.Theoself-defeatingobject"argument,andthevaguenessparadox28x1.12.Acomputationalperspectiveonsettheory35Chapter2.Grouptheory51x2.1.Torsors51x2.2.Activeandpassivetransformations54x2.3.Cayleygraphsandthegeometryofgroups56x2.4.Groupextensions62viiviiiContentsx2.5.AproofofGromov'st

4、heorem69Chapter3.Analysis79x3.1.Ordersofmagnitude,andtropicalgeometry79x3.2.Descriptivesettheoryvs.Lebesguesettheory81x3.3.Complexanalysisvs.realanalysis82x3.4.Sharpinequalities85x3.5.Impliedconstantsandasymptoticnotation87x3.6.Browniansnow akes88x3.7.TheEuler-Maclaurinformula,Bernoullinumbers,

5、thezetafunction,andreal-variableanalyticcontinuation88x3.8.Finitaryconsequencesoftheinvariantsubspaceproblem104x3.9.TheGuth-KatzresultontheErd}osdistanceproblem110x3.10.TheBourgain-Guthmethodforprovingrestrictiontheorems123Chapter4.Nonstandardanalysis133x4.1.Realnumbers,nonstandardrealnumbers,a

6、ndniteprecisionarithmetic133x4.2.Nonstandardanalysisasalgebraicanalysis135x4.3.Compactnessandcontradiction:thecorrespondenceprincipleinergodictheory137x4.4.Nonstandardanalysisasacompletionofstandardanalysis150x4.5.Concentrationcompactnessvianonstandardanalysis167Chapter5.Partialdierentialequa

7、tions181x5.1.Quasilinearwell-posedness181x5.2.Atypediagramforfunctionspaces189x5.3.Amplitude-frequencydynamicsforsemilineardispersiveequations194x5.4.TheEuler-Arnoldequation203Chapter6.Miscellaneous217x6.1.Multiplicityofperspectiv