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Inverse Problems in Partial Differential Equations and Geometry

Final Report Summary - INVPROBGEOMPDE (Inverse Problems in Partial Differential Equations and Geometry)

Inverse problems research concentrates on the mathematical theory and practical interpretation of indirect measurements. Applications are found in virtually every research field involving scientific, medical, or industrial imaging and mathematical modelling. Familiar examples include X-ray Computed Tomography (CT) and ultrasound imaging. Inverse problems research forms a vibrant research field in the intersection of pure and applied mathematics, drawing techniques from several different areas and generating new research questions.

The main focus of this ERC research has been on fundamental mathematical questions in the theory of inverse problems. A major topic is the famous inverse conductivity problem due to Calderón forming the basis of Electrical Impedance Tomography (EIT), a modality in seismic, medical and industrial imaging. The project has made considerable progress in the theory of EIT and related methods in materials where anisotropy occurs in transversal directions, where the measurements are taken only on a part of the boundary, or where the current obeys a power law leading to nonlinear models. Another important topic is geometric inverse problems that arise in Travel Time Tomography in seismic imaging and in differential geometry. We have solved related tensor tomography conjectures on manifolds both with and without boundary, with potential applications in the study of the interior structure of the Earth.