Geometric Inverse Problems
With Emphasis on Two Dimensions

Cutting-edge mathematical tools are used in this treatment of recent developments in geometric inverse problems.

Gabriel P. Paternain (Author), Mikko Salo (Author), Gunther Uhlmann (Author)

9781316510872, Cambridge University Press

Hardback, published 5 January 2023

350 pages
23.5 x 15.8 x 2.8 cm, 0.71 kg

'This monograph gives a beautiful introduction to Geometric inverse problems, largely in dimension two, by three of the most prominent contributors to the field. The Geometric problems are interesting as pure mathematics, but they also arise from applications to tomography, such as the Calderon problem of determining (M, g) from its Dirichlet-to-Neumann map. Roughly speaking, the underlying physics problem is to determine electrical properties of a medium by making voltage and current measurements on the boundary. Techniques of microlocal analysis relate such PDE boundary inverse problems to geometric inverse problems. These inverse problems furnish problems of great interest in PDE and in geometry in a rather concrete setting, and are masterfully conveyed by the authors. The level is appropriate for a graduate class in mathematics but is also an excellent entrée into the field for research mathematicians.' Steve Zelditch, Northwestern University

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, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The presentation is self-contained and begins with the Radon transform and radial sound speeds as motivating examples. The required geometric background is developed 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 and range characterization results. Highlights include a proof of boundary rigidity for simple surfaces as well as scattering rigidity for connections. The concluding chapter discusses current open problems and related topics. The numerous exercises and examples make this book an excellent self-study resource or text for a one-semester course or seminar.

Foreword András Vasy
Preface
1. The Radon transform in the plane
2. Radial sound speeds
3. Geometric preliminaries
4. The geodesic X-ray transform
5. Regularity results for the transport equation
6. Vertical Fourier analysis
7. The X-ray transform in non-positive curvature
8. Microlocal aspects, surjectivity of $I^{*}_{0}$
9. Inversion formulas and range
10. Tensor tomography
11. Boundary rigidity
12. The attenuated geodesic X-ray transform
13. Non-Abelian X-ray transforms
14. Non-Abelian X-ray transforms II
15. Open problems and related topics
References
Index.

Subject Areas: Mensuration & systems of measurement [PDDM], Geometry [PBM], Numerical analysis [PBKS], Tomography [MMPJ]