Geometric Inverse Problems With Emphasis on Two Dimensions

Contents
Foreword -- András Vasy
Preface
Acknowledgements
1. The Radon Transform in the Plane
	1.1 Uniqueness and Stability
	1.2 Range and Support Theorems
	1.3 The Normal Operator and Singularities
	1.4 The Funk Transform
2. Radial Sound Speeds
	2.1 Geodesics of a Radial Sound Speed
	2.2 Travel Time Tomography
	2.3 Geodesics of a Rotationally Symmetric Metric
	2.4 Geodesic X-ray Transform
	2.5 Examples and Counterexamples
3. Geometric Preliminaries
	3.1 Non-trapping and Strict Convexity
	3.2 Regularity of the Exit Time
	3.3 The Geodesic Flow and the Scattering Relation
	3.4 Complex Structure
	3.5 The Unit Circle Bundle of a Surface
	3.6 The Unit Sphere Bundle in Higher Dimensions
	3.7 Conjugate Points and Morse Theory
	3.8 Simple Manifolds
4. The Geodesic X-ray Transform
	4.1 The Geodesic X-ray Transform
	4.2 Transport Equations
	4.3 Pestov Identity
	4.4 Injectivity of the Geodesic X-ray Transform
	4.5 Stability Estimate in Non-positive Curvature
	4.6 Stability Estimate in the Simple Case
	4.7 The Higher Dimensional Case
5. Regularity Results for the Transport Equation
	5.1 Smooth First Integrals
	5.2 Folds and the Scattering Relation
	5.3 A General Regularity Result
	5.4 The Adjoint I*_A
6. Vertical Fourier Analysis
	6.1 Vertical Fourier Expansions
	6.2 The Fibrewise Hilbert Transform
	6.3 Symmetric Tensors as Functions on SM
	6.4 The X-ray Transform on Tensors
	6.5 Guillemin–Kazhdan Identity
	6.6 The Higher Dimensional Case
7. The X-ray Transform in Non-positive Curvature
	7.1 Tensor Tomography
	7.2 Stability for Functions
	7.3 Stability for Tensors
	7.4 Carleman Estimates
	7.5 The Higher Dimensional Case
8. Microlocal Aspects, Surjectivity of I~*_0
	8.1 The Normal Operator
	8.2 Surjectivity of I*_0
	8.3 Stability Estimates Based on the Normal Operator
	8.4 The Normal Operator with a Matrix Weight
9. Inversion Formulas and Range
	9.1 Motivation
	9.2 Properties of Solutions of the Jacobi Equation
	9.3 The Smoothing Operator W
	9.4 Fredholm Inversion Formulas
	9.5 Revisiting the Euclidean Case
	9.6 Range
	9.7 Numerical Implementation
10. Tensor Tomography
	10.1 Holomorphic Integrating Factors
	10.2 Tensor Tomography
	10.3 Range for Tensors
11. Boundary Rigidity
	11.1 The Boundary Rigidity Problem
	11.2 Boundary Determination
	11.3 Determining the Lens Data and Volume
	11.4 Rigidity in a Given Conformal Class
	11.5 Determining the Dirichlet-to-Neumann Map
	11.6 Calderón Problem
	11.7 Boundary Rigidity for Simple Surfaces
12. The Attenuated Geodesic X-ray Transform
	12.1 The Attenuated X-ray Transform in the Plane
	12.2 Injectivity Results for Scalar Attenuations
	12.3 Surjectivity of I*
	12.4 Discussion on General Weights
13. Non-Abelian X-ray Transforms
	13.1 Scattering Data
	13.2 Pseudo-linearization Identity
	13.3 Elementary Background on Connections
	13.4 Structure Equations Including a Connection
	13.5 Scattering Rigidity and Injectivity for Connections
	13.6 An Alternative Proof of Tensor Tomography
	13.7 General Skew-Hermitian Attenuations
	13.8 Injectivity for Connections and Higgs Fields
	13.9 Scattering Rigidity for Connections and Higgs Fields
	13.10 Matrix Holomorphic Integrating Factors
	13.11 Stability Estimate
14. Non-Abelian X-ray Transforms II
	14.1 Scattering Rigidity and Injectivity Results for gl(n,C)
	14.2 A Factorization Theorem from Loop Groups
	14.3 Proof of Theorems 14.1.1 and 14.1.2
	14.4 General Lie Groups
	14.5 Range of I_{A,0} and I_{A,⊥}
	14.6 Surjectivity of I*_{A,0} and I*_{A,⊥}
	14.7 Adding a Matrix Field
15. Open Problems and Related Topics
	15.1 Open Problems
	15.2 Related Topics
References
A B
C D
E F G
H
I J K
L M
N
O P
Q R S
T U V W Z
Index