Inverse Boundary Problems

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The proposer will address basic research questions arising in the study of Inverse Problems. The fundamental task of science is to probe the world around us. The most powerful method to do this is to send waves to an object and measure the response. Inverse Problems consist in determining properties of the object being probed from these observations. These problems arise in all fields of science and applications where causes for a desired or observed effect are to be determined. The proposed research considers fundamental questions in the mathematical theory of inverse problems which are motivated by applications to medical imaging, geophysics, and remote sensing among others. Development of mathematical techniques to address these difficult questions will undoubtedly have real world applications in the future that will benefit mankind. The mathematics of X-ray tomography has, for instance, contributed to the development of CT scans and other medical imaging techniques that help save lives everyday. The proposed research will also involve graduate students as part of their course work and PhD theses, and also postdoctoral fellows.Specifically, the proposer plans four major topics of research. The first one is Electric Impedance Tomography. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at the boundary. This inverse method has been proposed as a valuable diagnostic tool in medicine for the early detection of breast cancer. The second is on the subject on how to "cloak" objects, that is how to make them invisible to electromagnetic waves and other type of waves, a subject that has received a lot of attention in recent years. The third is Travel Time Tomography that arises in determining the inner structure of the Earth by measuring the travel times of seismic waves produced by earthquakes. The final topic is inverse scattering that involves far field measurements and arises, for instance, in remote sensing, sonar and radar.