The Princeton Companion to Mathematics

1057 Pages·2008·8.02 MB·English

by  Timothy Gowers

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The Princeton Companion to Mathematics
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The Princeton Companion to Mathematics
editor
Timothy Gowers
University of Cambridge
associate editors
June Barrow-Green
The Open University
Imre Leader
University of Cambridge
PrincetonUniversityPress
PrincetonandOxford
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Copyright©2008byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,
6OxfordStreet,Woodstock,OxfordshireOX201TW
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
ThePrincetoncompaniontomathematics/TimothyGowers,editor;
JuneBarrow-Green,ImreLeader,associateeditors.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-691-11880-2(hardcover:alk.paper)
1.Mathematics—Studyandteaching(Higher)2.PrincetonUniversity.
I.Gowers,Timothy.II.Barrow-Green,June,date–III.Leader,Imre.
QA11.2.P7452008
510—dc22 2008020450
BritishLibraryCataloging-in-PublicationDataisavailable
Gratefulacknowledgmentismadeforpermission
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Contents
Preface ix III.15 Determinants 174
Contributors xvii III.16 DifferentialFormsandIntegration 175
III.17 Dimension 180
III.18 Distributions 184
III.19 Duality 187
Part I Introduction
III.20 DynamicalSystemsandChaos 190
III.21 EllipticCurves 190
I.1 WhatIsMathematicsAbout? 1
I.2 TheLanguageandGrammarofMathematics 8 III.22 The Euclidean Algorithm and
ContinuedFractions 191
I.3 SomeFundamentalMathematicalDefinitions 16
III.23 TheEulerandNavier–StokesEquations 193
I.4 TheGeneralGoalsofMathematicalResearch 47
III.24 Expanders 196
III.25 TheExponentialandLogarithmicFunctions 199
III.26 TheFastFourierTransform 202
Part II The Origins of Modern III.27 TheFourierTransform 204
Mathematics III.28 FuchsianGroups 208
III.29 FunctionSpaces 210
II.1 FromNumberstoNumberSystems 77 III.30 GaloisGroups 213
II.2 Geometry 83 III.31 TheGammaFunction 213
II.3 TheDevelopmentofAbstractAlgebra 95 III.32 GeneratingFunctions 214
II.4 Algorithms 106 III.33 Genus 215
II.5 The Development of Rigor in III.34 Graphs 215
MathematicalAnalysis 117 III.35 Hamiltonians 215
II.6 TheDevelopmentoftheIdeaofProof 129 III.36 TheHeatEquation 216
II.7 TheCrisisintheFoundationsofMathematics 142 III.37 HilbertSpaces 219
III.38 HomologyandCohomology 221
III.39 HomotopyGroups 221
Part III Mathematical Concepts III.40 TheIdealClassGroup 221
III.41 IrrationalandTranscendentalNumbers 222
III.1 TheAxiomofChoice 157 III.42 TheIsingModel 223
III.2 TheAxiomofDeterminacy 159 III.43 JordanNormalForm 223
III.3 BayesianAnalysis 159 III.44 KnotPolynomials 225
III.4 BraidGroups 160 III.45 K-Theory 227
III.5 Buildings 161 III.46 TheLeechLattice 227
III.6 Calabi–YauManifolds 163 III.47 L-Functions 228
III.7 Cardinals 165 III.48 LieTheory 229
III.8 Categories 165 III.49 LinearandNonlinearWavesandSolitons 234
III.9 CompactnessandCompactification 167 III.50 LinearOperatorsandTheirProperties 239
III.10 ComputationalComplexityClasses 169 III.51 LocalandGlobalinNumberTheory 241
III.11 CountableandUncountableSets 170 III.52 TheMandelbrotSet 244
III.12 C∗-Algebras 172 III.53 Manifolds 244
III.13 Curvature 172 III.54 Matroids 244
III.14 Designs 172 III.55 Measures 246
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vi Contents
III.56 MetricSpaces 247 IV.5 ArithmeticGeometry 372
III.57 ModelsofSetTheory 248 IV.6 AlgebraicTopology 383
III.58 ModularArithmetic 249 IV.7 DifferentialTopology 396
III.59 ModularForms 250 IV.8 ModuliSpaces 408
III.60 ModuliSpaces 252 IV.9 RepresentationTheory 419
III.61 TheMonsterGroup 252 IV.10 GeometricandCombinatorialGroupTheory 431
III.62 NormedSpacesandBanachSpaces 252 IV.11 HarmonicAnalysis 448
III.63 NumberFields 254 IV.12 PartialDifferentialEquations 455
III.64 OptimizationandLagrangeMultipliers 255 IV.13 GeneralRelativityandtheEinsteinEquations 483
III.65 Orbifolds 257 IV.14 Dynamics 493
III.66 Ordinals 258 IV.15 OperatorAlgebras 510
III.67 ThePeanoAxioms 258 IV.16 MirrorSymmetry 523
III.68 PermutationGroups 259 IV.17 VertexOperatorAlgebras 539
III.69 PhaseTransitions 261 IV.18 EnumerativeandAlgebraicCombinatorics 550
III.70 π 261 IV.19 ExtremalandProbabilisticCombinatorics 562
III.71 ProbabilityDistributions 263 IV.20 ComputationalComplexity 575
III.72 ProjectiveSpace 267 IV.21 NumericalAnalysis 604
III.73 QuadraticForms 267 IV.22 SetTheory 615
III.74 QuantumComputation 269 IV.23 LogicandModelTheory 635
III.75 QuantumGroups 272 IV.24 StochasticProcesses 647
III.76 Quaternions, Octonions, and Normed IV.25 ProbabilisticModelsofCriticalPhenomena 657
DivisionAlgebras 275 IV.26 High-Dimensional Geometry and Its
III.77 Representations 279 ProbabilisticAnalogues 670
III.78 RicciFlow 279
III.79 RiemannSurfaces 282
III.80 TheRiemannZetaFunction 283
Part V Theorems and Problems
III.81 Rings,Ideals,andModules 284
III.82 Schemes 285
V.1 TheABCConjecture 681
III.83 TheSchrödingerEquation 285
V.2 TheAtiyah–SingerIndexTheorem 681
III.84 TheSimplexAlgorithm 288
V.3 TheBanach–TarskiParadox 684
III.85 SpecialFunctions 290
V.4 TheBirch–Swinnerton-DyerConjecture 685
III.86 TheSpectrum 294
V.5 Carleson’sTheorem 686
III.87 SphericalHarmonics 295
V.6 TheCentralLimitTheorem 687
III.88 SymplecticManifolds 297
V.7 TheClassificationofFiniteSimpleGroups 687
III.89 TensorProducts 301
V.8 Dirichlet’sTheorem 689
III.90 TopologicalSpaces 301
V.9 ErgodicTheorems 689
III.91 Transforms 303
V.10 Fermat’sLastTheorem 691
III.92 TrigonometricFunctions 307
V.11 FixedPointTheorems 693
III.93 UniversalCovers 309
V.12 TheFour-ColorTheorem 696
III.94 VariationalMethods 310
V.13 TheFundamentalTheoremofAlgebra 698
III.95 Varieties 313
V.14 TheFundamentalTheoremofArithmetic 699
III.96 VectorBundles 313
V.15 Gödel’sTheorem 700
III.97 VonNeumannAlgebras 313
V.16 Gromov’sPolynomial-GrowthTheorem 702
III.98 Wavelets 313
V.17 Hilbert’sNullstellensatz 703
III.99 TheZermelo–FraenkelAxioms 314
V.18 The Independence of the
ContinuumHypothesis 703
V.19 Inequalities 703
Part IV Branches of Mathematics V.20 TheInsolubilityoftheHaltingProblem 706
V.21 TheInsolubilityoftheQuintic 708
IV.1 AlgebraicNumbers 315 V.22 Liouville’sTheoremandRoth’sTheorem 710
IV.2 AnalyticNumberTheory 332 V.23 Mostow’sStrongRigidityTheorem 711
IV.3 ComputationalNumberTheory 348 V.24 ThePversusNPProblem 713
IV.4 AlgebraicGeometry 363 V.25 ThePoincaréConjecture 714
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Contents vii
V.26 The Prime Number Theorem and the VI.34 JánosBolyai(1802–1860) 762
RiemannHypothesis 714 VI.35 CarlGustavJacobJacobi(1804–1851) 762
V.27 Problems and Results in VI.36 PeterGustavLejeuneDirichlet(1805–1859) 764
AdditiveNumberTheory 715 VI.37 WilliamRowanHamilton(1805–1865) 765
V.28 From Quadratic Reciprocity to VI.38 AugustusDeMorgan(1806–1871) 765
ClassFieldTheory 718 VI.39 JosephLiouville(1809–1882) 766
V.29 Rational Points on Curves and VI.40 ErnstEduardKummer(1810–1893) 767
theMordellConjecture 720
VI.41 ÉvaristeGalois(1811–1832) 767
V.30 TheResolutionofSingularities 722
VI.42 JamesJosephSylvester(1814–1897) 768
V.31 TheRiemann–RochTheorem 723
VI.43 GeorgeBoole(1815–1864) 769
V.32 TheRobertson–SeymourTheorem 725
VI.44 KarlWeierstrass(1815–1897) 770
V.33 TheThree-BodyProblem 726
VI.45 PafnutyChebyshev(1821–1894) 771
V.34 TheUniformizationTheorem 728
VI.46 ArthurCayley(1821–1895) 772
V.35 TheWeilConjectures 729
VI.47 CharlesHermite(1822–1901) 773
VI.48 LeopoldKronecker(1823–1891) 773
VI.49 Georg Friedrich Bernhard Riemann
Part VI Mathematicians (1826–1866) 774
VI.50 JuliusWilhelmRichardDedekind(1831–1916) 776
VI.1 Pythagoras(ca.569b.c.e.–ca.494b.c.e.) 733 VI.51 ÉmileLéonardMathieu(1835–1890) 776
VI.2 Euclid(ca.325b.c.e.–ca.265b.c.e.) 734 VI.52 CamilleJordan(1838–1922) 777
VI.3 Archimedes(ca.287b.c.e.–212b.c.e.) 734 VI.53 SophusLie(1842–1899) 777
VI.4 Apollonius(ca.262b.c.e.–ca.190b.c.e.) 735 VI.54 GeorgCantor(1845–1918) 778
VI.5 AbuJa’farMuhammadibnMu¯s¯aal-Khw¯arizm¯ı VI.55 WilliamKingdonClifford(1845–1879) 780
(800–847) 736 VI.56 GottlobFrege(1848–1925) 780
VI.6 LeonardoofPisa(knownasFibonacci) VI.57 ChristianFelixKlein(1849–1925) 782
(ca.1170–ca.1250) 737 VI.58 FerdinandGeorgFrobenius(1849–1917) 783
VI.7 GirolamoCardano(1501–1576) 737 VI.59 Sofya(Sonya)Kovalevskaya(1850–1891) 784
VI.8 RafaelBombelli(1526–after1572) 737 VI.60 WilliamBurnside(1852–1927) 785
VI.9 FrançoisViète(1540–1603) 737 VI.61 JulesHenriPoincaré(1854–1912) 785
VI.10 SimonStevin(1548–1620) 738 VI.62 GiuseppePeano(1858–1932) 787
VI.11 RenéDescartes(1596–1650) 739 VI.63 DavidHilbert(1862–1943) 788
VI.12 PierreFermat(160?–1665) 740 VI.64 HermannMinkowski(1864–1909) 789
VI.13 BlaisePascal(1623–1662) 741 VI.65 JacquesHadamard(1865–1963) 790
VI.14 IsaacNewton(1642–1727) 742 VI.66 IvarFredholm(1866–1927) 791
VI.15 GottfriedWilhelmLeibniz(1646–1716) 743 VI.67 Charles-JeandelaValléePoussin(1866–1962) 792
VI.16 BrookTaylor(1685–1731) 745 VI.68 FelixHausdorff(1868–1942) 792
VI.17 ChristianGoldbach(1690–1764) 745 VI.69 ÉlieJosephCartan(1869–1951) 794
VI.18 TheBernoullis(fl.18thcentury) 745 VI.70 EmileBorel(1871–1956) 795
VI.19 LeonhardEuler(1707–1783) 747 VI.71 BertrandArthurWilliamRussell(1872–1970) 795
VI.20 JeanLeRondd’Alembert(1717–1783) 749 VI.72 HenriLebesgue(1875–1941) 796
VI.21 EdwardWaring(ca.1735–1798) 750 VI.73 GodfreyHaroldHardy(1877–1947) 797
VI.22 JosephLouisLagrange(1736–1813) 751 VI.74 Frigyes(Frédéric)Riesz(1880–1956) 798
VI.23 Pierre-SimonLaplace(1749–1827) 752 VI.75 LuitzenEgbertusJanBrouwer(1881–1966) 799
VI.24 Adrien-MarieLegendre(1752–1833) 754 VI.76 EmmyNoether(1882–1935) 800
VI.25 Jean-BaptisteJosephFourier(1768–1830) 755 VI.77 WacławSierpin´ski(1882–1969) 801
VI.26 CarlFriedrichGauss(1777–1855) 755 VI.78 GeorgeBirkhoff(1884–1944) 802
VI.27 Siméon-DenisPoisson(1781–1840) 757 VI.79 JohnEdensorLittlewood(1885–1977) 803
VI.28 BernardBolzano(1781–1848) 757 VI.80 HermannWeyl(1885–1955) 805
VI.29 Augustin-LouisCauchy(1789–1857) 758 VI.81 ThoralfSkolem(1887–1963) 806
VI.30 AugustFerdinandMöbius(1790–1868) 759 VI.82 SrinivasaRamanujan(1887–1920) 807
VI.31 NicolaiIvanovichLobachevskii(1792–1856) 759 VI.83 RichardCourant(1888–1972) 808
VI.32 GeorgeGreen(1793–1841) 760 VI.84 StefanBanach(1892–1945) 809
VI.33 NielsHenrikAbel(1802–1829) 760 VI.85 NorbertWiener(1894–1964) 811
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viii Contents
VI.86 EmilArtin(1898–1962) 812 VII.8 MathematicsandEconomicReasoning 895
VI.87 AlfredTarski(1901–1983) 813 VII.9 TheMathematicsofMoney 910
VI.88 AndreiNikolaevichKolmogorov(1903–1987) 814 VII.10 MathematicalStatistics 916
VI.89 AlonzoChurch(1903–1995) 816 VII.11 MathematicsandMedicalStatistics 921
VI.90 WilliamVallanceDouglasHodge(1903–1975) 816 VII.12 Analysis,MathematicalandPhilosophical 928
VI.91 JohnvonNeumann(1903–1957) 817 VII.13 MathematicsandMusic 935
VI.92 KurtGödel(1906–1978) 819 VII.14 MathematicsandArt 944
VI.93 AndréWeil(1906–1998) 819
VI.94 AlanTuring(1912–1954) 821
VI.95 AbrahamRobinson(1918–1974) 822
Part VIII Final Perspectives
VI.96 NicolasBourbaki(1935–) 823
VIII.1 TheArtofProblemSolving 955
VIII.2 “WhyMathematics?”YouMightAsk 966
Part VII The Influence of Mathematics VIII.3 TheUbiquityofMathematics 977
VIII.4 Numeracy 983
VII.1 MathematicsandChemistry 827 VIII.5 Mathematics:AnExperimentalScience 991
VII.2 MathematicalBiology 837 VIII.6 AdvicetoaYoungMathematician 1000
VII.3 WaveletsandApplications 848 VIII.7 AChronologyofMathematicalEvents 1010
VII.4 TheMathematicsofTrafficinNetworks 862
VII.5 TheMathematicsofAlgorithmDesign 871
VII.6 ReliableTransmissionofInformation 878
VII.7 MathematicsandCryptography 887 Index 1015
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Preface
1 WhatIsTheCompanion? togiveapreciseanswertothequestion,“Whatmakesa
mathematicalstatementinteresting?”itsimplyaimsto
Bertrand Russell, in his book The Principles of Mathe-
presentforthereaderalargeandrepresentativesam-
matics, proposes the following as a definition of pure
pleoftheideasthatmathematiciansaregrapplingwith
mathematics.
atthebeginningofthetwenty-firstcentury,andtodo
soinasattractiveandaccessibleawayaspossible.
Pure Mathematics is the class of all propositions of
the form “p implies q,” where p and q are proposi-
tions containing one or more variables, the same in 2 TheScopeoftheBook
thetwopropositions,andneitherpnorqcontainsany
constants except logical constants. And logical con-
The central focus of this book is modern, pure math-
stants are all notions definable in terms of the fol-
ematics, a decision about which something needs to
lowing: Implication, the relation of a term to a class
of which it is a member, the notion ofsuch that, the be said. “Modern” simply means that, as mentioned
notionofrelation,andsuchfurthernotionsasmaybe above, the book aims to give an idea of what math-
involved in the general notion of propositions of the ematicians are now doing: for example, an area that
aboveform.Inadditiontothese,mathematicsuses a developedrapidlyinthemiddleofthelastcenturybut
notionwhichisnotaconstituentofthepropositions
thathasnowreachedasettledformislikelytobedis-
whichitconsiders,namelythenotionoftruth.
cussed less than one that is still developing rapidly.
ThePrincetonCompaniontoMathematicscouldbesaid However, mathematics carries its history with it: in
tobeabouteverythingthatRussell’sdefinitionleaves ordertounderstandapieceofpresent-daymathemat-
out. ics,onewillusuallyneedtoknowaboutmanyideasand
Russell’s book was published in 1903, and many resultsthatwerediscoveredalongtimeago.Moreover,
mathematiciansatthattimewerepreoccupiedwiththe ifonewishestohaveaproperperspectiveontoday’s
logicalfoundationsofthesubject.Now,justoveracen- mathematics,itisessentialtohavesomeideaofhowit
turylater,itisnolongeranewideathatmathematics cametobeasitis.Sothereisplentyofhistoryinthe
can be regarded as a formal system of the kind that book,evenifthemainreasonforourincludingitisto
Russell describes, and today’s mathematician is more illuminatethemathematicsoftoday.
likely to have other concerns. In particular, in an era Theword“pure”ismoretroublesome.Asmanyhave
where so much mathematics is being published that commented, there is no clear dividing line between
noindividualcanunderstandmorethanatinyfraction pure and applied mathematics, and, just as a proper
ofit,itisusefultoknownotjustwhicharrangements appreciation of modern mathematics requires some
of symbols form grammatically correct mathematical knowledge of its history, so a proper appreciation of
statements,butalsowhichofthesestatementsdeserve puremathematicsrequiressomeknowledgeofapplied
ourattention. mathematics and theoretical physics. Indeed, these
Of course, one cannot hope to give a fully objec- areas have provided pure mathematicians with many
tive answer to such a question, and different math- fundamental ideas, which have given rise to some of
ematicians can legitimately disagree about what they the most interesting, important, and currently active
find interesting. For that reason, this book is far less branches of pure mathematics. This book is certainly
formal than Russell’s and it has many authors with notblindtotheimpactonpuremathematicsofthese
manydifferentpointsofview. Andratherthantrying other disciplines, nor does it ignore the practical and