Integral geometry and unique continuation principles

TL;DR

This thesis explores advanced inverse problems in medical and seismic imaging, focusing on integral geometry, fractional PDEs, and unique continuation principles to improve data reconstruction techniques.

Contribution

It introduces new methods and theoretical insights into inverse problems involving fractional operators, partial data, and geometric analysis on Riemannian and Finsler manifolds.

Findings

Enhanced understanding of fractional Calderón problems

New uniqueness results for travel time tomography

Development of integral geometry techniques for partial data

Abstract

This is the introductory part of my PhD thesis on inverse problems arising in medical and seismic imaging. The topics include X-ray tomography of scalar and vector fields with partial data, higher order fractional Calder\'on problems, travel time tomography on Riemannian and Finsler manifolds, and unique continuation of fractional Laplacians. The thesis additionally includes these articles: arXiv:1909.05585, arXiv:2001.06210, arXiv:2006.05790, arXiv:2008.10227, arXiv:2009.01043, arXiv:2010.11484, arXiv:2103.14385.