The Abel Prize: 2003–2007 The First Five Years

245 Pages·2010·2.79 MB·English

by  Arild Stubhaug (auth.)

The Abel Prize
NielsHenrikAbel1802–1829
TheonlycontemporaryportraitofAbel,paintedbyJohanGørbitzin1826
©Matematiskinstitutt,UniversitetetiOslo
Helge Holden (cid:2) Ragni Piene
Editors
The Abel Prize
2003–2007 The First Five Years
HelgeHolden RagniPiene
DepartmentofMathematicalSciences CentreofMathematicsforApplications
NorwegianUniversity andDepartmentofMathematics
ofScienceandTechnology UniversityofOslo
7491Trondheim 0316Oslo
Norway Norway
[email protected] [email protected]
Photosonthecoverofthebook:
PhotoofJean-PierreSerrebyArash.A.Nejad/Scanpix
PhotosofSirMichaelAtiyahandIsadoreSingerbyAnne-LiseFlavik
PhotoofPeterLaxbyKnutFalch/Scanpix
PhotoofLennartCarlesonbyKnutFalch/Scanpix
PhotoofS.R.SrinivasaVaradhanbyMortenF.Holm/Scanpix
AllphotosinconnectionwiththeAbelPrizeceremoniesaretakenbyScanpix
ISBN978-3-642-01372-0 e-ISBN978-3-642-01373-7
DOI10.1007/978-3-642-01373-7
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2009926160
MathematicsSubjectClassification(2000):00-02;00A15;01A70
©Springer-VerlagBerlinHeidelberg2010
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Preface
In 2002, the year marking the bicentennial of Abel’s birth, the Norwegian Par-
liament established the Niels Henrik Abel Memorial Fund with the objective
of creating an international prize for outstanding scientific work in the field of
mathematics—theAbelPrize.
InthisbookwewouldliketopresenttheAbelPrizeandtheAbelLaureatesof
thefirstfiveyears.ThebookresultsfromaninitiativeoftheMathematicssectionof
theNorwegianAcademyofScienceandLetters.Itisintendedasthefirstvolumein
aseries,eachvolumecomprisingfiveyears.
ThebookstartswiththehistoryoftheAbelPrize—astorythatgoesbackmore
thanahundredyears—writtenbyAbel’sbiographer,ArildStubhaug.Itisfollowed
by Nils A. Baas’ biographical sketch of Atle Selberg; at the opening ceremony of
the Abel Bicentennial Conference in Oslo in 2002, an Honorary Abel Prize was
presentedtoAtleSelberg.
Thereisonepartforeachoftheyears2003–2007.Eachpartstartswithanautobi-
ographicalpiecebythelaureate(s).Thenfollowsatextonthelaureate’swork:Pilar
BayerwritesontheworkofJean-PierreSerre,NigelHitchinonAtiyah–Singer’sIn-
dexTheorem,HelgeHoldenandPeterSarnakontheworkofPeterLax,TomKörner
onLennartCarleson,andTerryLyonsonSrinivasaVaradhan.Eachpartcontainsa
completebibliographyandacurriculumvitae,aswellasphotos—oldandnew.
TheDVDincludedinthebookcontainstheinterviewsthatMartinRaussenand
ChristianSkaumadewitheachlaureateinconnectionwiththePrizeceremoniesin
theyears2003–2007.Everyyearexceptthefirst,theinterviewswerebroadcaston
Norwegiannationaltelevision.Transcriptsofallinterviewshavebeenpublishedin
theEMSNewsletterandNoticesoftheAMS.
Wewouldliketoexpressourgratitudetothelaureatesforcollaboratingwithus
onthisproject,especiallyforprovidingtheautobiographicalpiecesandthephotos.
Wewouldliketothankthemathematicianswhoagreedtowriteaboutthelaureates,
andthusarehelpingusinmakingthelaureates’workknowntoabroaderaudience.
Thanks go to Martin Raussen and Christian Skau for letting us use the inter-
views, to David Pauksztello for his translations, to Marius Thaule for his LATEX
expertiseandthepreparationofthebibliographies,andtoAnne-MarieAstadofthe
v
vi Preface
Norwegian Academy for Science and Letters for her help with the interviews and
photos. HH gratefully acknowledgessupport from the Centre for AdvancedStudy
attheNorwegianAcademyofScienceandLettersinOsloduringtheacademicyear
2008–09.
The technical preparation of the manuscript was financed by the Niels Henrik
AbelMemorialFund.
Oslo HelgeHoldenandRagniPiene
July15,2009
Contents
TheHistoryoftheAbelPrize . . . . . . . . . . . . . . . . . . . . . . . . . 1
ArildStubhaug
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2002—anHonoraryAbelPrizetoAtleSelberg . . . . . . . . . . . . . . . 7
NilsA.Baas
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2003:Jean-PierreSerre
Jean-PierreSerre:Monpremierdemi-siècleauCollègedeFrance . . . . 13
MyFirstFiftyYearsattheCollègedeFrance . . . . . 13
MarcKirsch
Jean-PierreSerre:AnOverviewofHisWork . . . . . . . . . . . . . . . . 33
PilarBayer
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 TheBeginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.1 HomotopyGroupsofSpheres . . . . . . . . . . . . . . . . . . 35
1.2 Hochschild–SerreSpectralSequence . . . . . . . . . . . . . . 36
1.3 SheafCohomologyofComplexManifolds . . . . . . . . . . . 36
1.4 TheAmsterdamCongress . . . . . . . . . . . . . . . . . . . . 37
2 SheafCohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 FAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 CohomologyofAlgebraicVarieties . . . . . . . . . . . . . . . 38
3 LieGroupsandLieAlgebras . . . . . . . . . . . . . . . . . . . . . 40
4 LocalAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 ProjectiveModules . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 AlgebraicNumberFields . . . . . . . . . . . . . . . . . . . . . . . 42
7 ClassFieldTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.1 GeometricClassFieldTheory . . . . . . . . . . . . . . . . . . 43
vii
viii Contents
7.2 LocalClassFieldTheory . . . . . . . . . . . . . . . . . . . . 45
7.3 ALocalMassFormula . . . . . . . . . . . . . . . . . . . . . . 46
8 p-adicAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9 GroupCohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9.1 CohomologyofProfiniteGroupsandp-adicLieGroups . . . . 47
9.2 GaloisCohomology . . . . . . . . . . . . . . . . . . . . . . . 48
9.3 GaloisCohomologyofLinearAlgebraicGroups . . . . . . . . 48
9.4 Self-dualNormalBasis . . . . . . . . . . . . . . . . . . . . . 50
9.5 EssentialDimension . . . . . . . . . . . . . . . . . . . . . . . 50
10 DiscreteSubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 51
10.1 CongruenceSubgroups . . . . . . . . . . . . . . . . . . . . . 52
10.2 CohomologyofArithmeticGroups . . . . . . . . . . . . . . . 53
11 ArithmeticofAlgebraicVarieties . . . . . . . . . . . . . . . . . . . 54
11.1 ModularCurves . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.2 VarietiesOverFiniteFields . . . . . . . . . . . . . . . . . . . 56
11.3 NumberofPointsofCurvesOverFiniteFields . . . . . . . . . 56
11.4 DiophantineProblems . . . . . . . . . . . . . . . . . . . . . . 58
12 FieldTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
13 GaloisRepresentations. . . . . . . . . . . . . . . . . . . . . . . . . 62
13.1 Hodge–TateModules . . . . . . . . . . . . . . . . . . . . . . 62
13.2 EllipticCurvesand(cid:2)-adicRepresentations . . . . . . . . . . . 62
13.3 ModularFormsand(cid:2)-adicRepresentations . . . . . . . . . . . 65
13.4 AbelianVarietiesand(cid:2)-adicRepresentations . . . . . . . . . . 69
13.5 Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
14 GroupTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.1 RepresentationTheory . . . . . . . . . . . . . . . . . . . . . . 73
14.2 AlgebraicGroups . . . . . . . . . . . . . . . . . . . . . . . . 76
14.3 FiniteSubgroupsofLieGroupsandofAlgebraicGroups . . . 77
15 MiscellaneousWritings . . . . . . . . . . . . . . . . . . . . . . . . 78
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
ListofPublicationsforJean-PierreSerre . . . . . . . . . . . . . . . . . . 81
CurriculumVitaeforJean-PierreSerre . . . . . . . . . . . . . . . . . . . 95
2004:SirMichaelAtiyahandIsadoreM.Singer
Autobiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
SirMichaelAtiyah
Autobiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
IsadoreM.Singer
TheAtiyah–SingerIndexTheorem . . . . . . . . . . . . . . . . . . . . . . 115
NigelHitchin
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Contents ix
2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.1 TheIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.2 Riemann–Roch . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.3 TheBeginning . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.4 TheSignature . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.5 Hirzebruch–Riemann–Roch . . . . . . . . . . . . . . . . . . . 121
2.6 TheDiracOperator . . . . . . . . . . . . . . . . . . . . . . . 123
3 TheIntegerIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.1 FormulationoftheTheorem . . . . . . . . . . . . . . . . . . . 126
3.2 IntegralityTheorems . . . . . . . . . . . . . . . . . . . . . . . 128
3.3 PositiveScalarCurvature . . . . . . . . . . . . . . . . . . . . 129
3.4 Gauge-TheoreticModuliSpaces. . . . . . . . . . . . . . . . . 130
4 TheEquivariantIndex . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.1 K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2 FixedPointTheorems . . . . . . . . . . . . . . . . . . . . . . 134
4.3 RigidityTheorems . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Themod2Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1 RealK-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 ThetaCharacteristics . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 PositiveScalarCurvature . . . . . . . . . . . . . . . . . . . . 140
6 TheIndexforFamilies . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1 FredholmOperators . . . . . . . . . . . . . . . . . . . . . . . 141
6.2 JumpingofDimension . . . . . . . . . . . . . . . . . . . . . . 142
7 TheLocalIndexTheorem . . . . . . . . . . . . . . . . . . . . . . . 144
7.1 TheHeatKernel . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 TheEtaInvariant . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 QuantumFieldTheory . . . . . . . . . . . . . . . . . . . . . . 147
7.4 TheSupersymmetricProof . . . . . . . . . . . . . . . . . . . 148
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
ListofPublicationsforSirMichaelAtiyah . . . . . . . . . . . . . . . . . 151
ListofPublicationsforIsadoreM.Singer . . . . . . . . . . . . . . . . . . 165
CurriculumVitaeforSirMichaelFrancisAtiyah,OM,FRS,FRSE . . . 173
CurriculumVitaeforIsadoreManualSinger . . . . . . . . . . . . . . . . 177
2005:PeterD.Lax
Autobiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
PeterD.Lax
ASurveyofPeterD.Lax’sContributionstoMathematics . . . . . . . . . 189
HelgeHoldenandPeterSarnak
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189