Work to be done on this project is a continuation of mathematical analysis of inverse problems having roots in several applications to the physical world. The fundamental problem, known in the engineering literature as electrical impedance imaging or impedance computed tomography, is that of determining the unknown conductivity inside a body from knowledge of steady state direct current measurements at the boundary. Its successful resolution will provide a valuable diagnostic tool for both the medical and biological sciences. There are, however, serious mathematical as well as experimental difficulties. The fact that the boundary measurements contain sufficient information to distinguish any isotropic conductivity was only recently established for three dimensional bodies. In two- dimensions, the global uniqueness question - an essential first step - is still unresolved. There are related and more difficult questions still to be pursued in three dimensions. In particular, the problem of continuous dependence of solutions on the boundary values is only partially understood. To date, the bounds measuring continuity are weak (logarithmic) and require further study. Work will also be done on reconstruction of solutions and a characterization of the class of functions which can act as conductivity potentials. This research has natural relationships with other inverse problems such as multi-dimensional scattering and d-bar methods from several complex variables. Finally, it should be noted that the important anisotropic case will have to play a role in constructive approaches to the isotropic and will force researchers to understand more about this more general problem area.
