Inverse problems in partial differential equations and medical imaging

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When scientists and engineers seek to determine properties of the interior of an object (e.g. the human body, the nucleus of an atom, an airplane wing, the earth, or the sun) without penetrating or even accessing the object, the analyze its effect on acoustic, elastic, electromagnetic or quantum mechanical waves. Mathematically, wave propagation is modeled by Partial Differential Equations and the class of problems considered in the proposed research consists in determining the coefficients of such and equation (which encode the unknown medium properties), from available measurements of the solutions (the waves). Several breakthroughs in the analytical understanding of such problems have uncovered a beautiful underlying mathematical structure and have led to an explosion of activity in the field. There has been considerable progress, yet some fundamental and difficult problems remain open. My research will attempt to understand some of these analytic problems. I will also work in collaboration with researchers in engineering towards the development of next generation medical diagnostic instrumentation for imaging electric properties of tissue.