Table Of ContentCAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 204
EditorialBoard
J. BERTOIN, B. BOLLOBÁS, W. FULTON, B. KRA, I. MOERDIJK,
C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO
GEOMETRICINVERSEPROBLEMS
This up-to-date treatment of recent developments in geometric inverse problems
introduces graduate students and researchers to an exciting area of research. With
an emphasis on the two-dimensional case, topics covered include geodesic X-ray
transforms,boundaryrigidity,tensortomography,attenuatedX-raytransforms,andthe
Calderónproblem.
Thepresentationisself-containedandbeginswiththeRadontransformandradial
soundspeedsasmotivatingexamples.Therequiredgeometricbackgroundisdeveloped
in detail in the context of simple manifolds with boundary. An in-depth analysis of
various geodesic X-ray transforms is carried out together with related uniqueness,
stability,reconstruction,andrangecharacterizationresults.Highlightsincludeaproof
ofboundaryrigidityforsimplesurfacesaswellasscatteringrigidityforconnections.
The concluding chapter discusses current open problems and related topics. The
numerousexercisesandexamplesmakethisbookanexcellentself-studyresourceor
textforaone-semestercourseorseminar.
Gabriel P. Paternain is Professor of Mathematics at the Department of Pure Math-
ematics and Mathematical Statistics at the University of Cambridge and a Fellow of
TrinityCollege.Hisresearchhascoveredanamplemathematicallandscape,including
Hamiltonian dynamics, symplectic geometry, and geometric inverse problems. He is
theauthorofthemonographGeodesicFlows(1999),andwasawardedthePilkington
Teaching Prize at Cambridge for his ability to explain analysis and geometry with a
claritythathaswonhimtheadmirationandrespectofhisstudents.
Mikko Salo is Professor of Mathematics at the University of Jyväskylä, Finland.
Hehasreceivedseveralawardsforhisworkoninverseproblemsinpartialdifferential
equations and geometry, including the Calderón prize, the Väisälä prize, an ERC
StartingGrant,andanERCConsolidatorGrant.
GuntherUhlmannistheWalkerFamilyEndowedProfessorattheUniversityofWash-
ington and the Si Yuan Professor at the Hong Kong University of Science and
Technology. He has worked on microlocal analysis and a broad spectrum of inverse
problems. He was awarded the AMS Bocher Prize, the Kleinman Prize from SIAM,
theSolomonLefschetzMedalfromtheMathematicalCounciloftheAmericas,andthe
BirkhoffPrize,awardedjointlybySIAMandtheAMS.
Published online by Cambridge University Press
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
EditorialBoard
J.Bertoin,B.Bollobás,W.Fulton,B.Kra,I.Moerdijk,C.Praeger,P.Sarnak,B.Simon,B.Totaro
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165 P.Gille&T.SzamuelyCentralSimpleAlgebrasandGaloisCohomology(2ndEdition)
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167 D.Li&H.QueffelecIntroductiontoBanachSpaces,II
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Published online by Cambridge University Press
Geometric Inverse Problems
With Emphasis on Two Dimensions
GABRIEL P. PATERNAIN
UniversityofCambridge
MIKKO SALO
UniversityofJyväskylä,Finland
GUNTHER UHLMANN
UniversityofWashington
Published online by Cambridge University Press
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316510872
DOI:10.1017/9781009039901
©GabrielP.Paternain,MikkoSaloandGuntherUhlmann2023
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress&Assessment.
Firstpublished2023
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Paternain,GabrielP.(GabrielPedro),1964–author.|
Salo,Mikko,author.|Uhlmann,Gunther,1952–author.
Title:Geometricinverseproblems:withemphasisontwodimensions/
GabrielP.Paternain,MikkoSalo,GuntherUhlmann.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2023.|
Series:Cambridgestudiesinadvancedmathematics,0950-6330;204|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2022030681(print)|LCCN2022030682(ebook)|
ISBN9781316510872(hardback)|ISBN9781009039901(epub)
Subjects:LCSH:Inverseproblems(Differentialequations)|
Inversions(Geometry)|Geometry,Differential.
Classification:LCCQA378.5.P382023(print)|LCCQA378.5(ebook)|
DDC515/.357–dc23/eng20221013
LCrecordavailableathttps://lccn.loc.gov/2022030681
LCebookrecordavailableathttps://lccn.loc.gov/2022030682
ISBN978-1-316-51087-2Hardback
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Contents
ForewordbyAndrásVasy pagexi
Preface xv
Acknowledgements xxiii
1 TheRadonTransforminthePlane 1
1.1 UniquenessandStability 1
1.2 RangeandSupportTheorems 5
1.3 TheNormalOperatorandSingularities 8
1.4 TheFunkTransform 19
2 RadialSoundSpeeds 25
2.1 GeodesicsofaRadialSoundSpeed 25
2.2 TravelTimeTomography 31
2.3 GeodesicsofaRotationallySymmetricMetric 36
2.4 GeodesicX-rayTransform 39
2.5 ExamplesandCounterexamples 45
3 GeometricPreliminaries 52
3.1 Non-trappingandStrictConvexity 52
3.2 RegularityoftheExitTime 58
3.3 TheGeodesicFlowandtheScatteringRelation 63
3.4 ComplexStructure 65
3.5 TheUnitCircleBundleofaSurface 75
3.6 TheUnitSphereBundleinHigherDimensions 81
3.7 ConjugatePointsandMorseTheory 87
3.8 SimpleManifolds 99
4 TheGeodesicX-rayTransform 107
4.1 TheGeodesicX-rayTransform 107
4.2 TransportEquations 110
vii
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viii Contents
4.3 PestovIdentity 112
4.4 InjectivityoftheGeodesicX-rayTransform 114
4.5 StabilityEstimateinNon-positiveCurvature 117
4.6 StabilityEstimateintheSimpleCase 123
4.7 TheHigherDimensionalCase 126
5 RegularityResultsfortheTransportEquation 130
5.1 SmoothFirstIntegrals 130
5.2 FoldsandtheScatteringRelation 132
5.3 AGeneralRegularityResult 135
∗
5.4 TheAdjointI 138
A
6 VerticalFourierAnalysis 142
6.1 VerticalFourierExpansions 142
6.2 TheFibrewiseHilbertTransform 149
6.3 SymmetricTensorsasFunctionsonSM 151
6.4 TheX-rayTransformonTensors 157
6.5 Guillemin–KazhdanIdentity 162
6.6 TheHigherDimensionalCase 166
7 TheX-rayTransforminNon-positiveCurvature 171
7.1 TensorTomography 171
7.2 StabilityforFunctions 173
7.3 StabilityforTensors 179
7.4 CarlemanEstimates 182
7.5 TheHigherDimensionalCase 186
∗
8 MicrolocalAspects,SurjectivityofI 189
0
8.1 TheNormalOperator 189
∗
8.2 SurjectivityofI 196
0
8.3 StabilityEstimatesBasedontheNormalOperator 201
8.4 TheNormalOperatorwithaMatrixWeight 203
9 InversionFormulasandRange 208
9.1 Motivation 208
9.2 PropertiesofSolutionsoftheJacobiEquation 211
9.3 TheSmoothingOperatorW 213
9.4 FredholmInversionFormulas 218
9.5 RevisitingtheEuclideanCase 224
9.6 Range 227
9.7 NumericalImplementation 229
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