Geometric Inverse Problems: With Emphasis on Two Dimensions

ISBN-13: 9781316510872 / Angielski / Twarda / 2023 / 350 str.

Gunther (University of Washington) Uhlmann

Opis

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

Recenzje

'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

Spis treści

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 $I^{*}_$; 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.