9322619 Uhlmann This award supports a continuation of mathematical research on problems arising in electrical impedance tomography. In particular, work will be done extending known results in the isotropic case to the physical case of non-smooth conductivities, to characterize, in two dimensions, all the possible outcomes of the experiments performed at the boundary and to consider the identifiability problem for anisotropic conductivities in dimensions three or greater. The project furthers work on other inverse boundary value problems arising in applications. They include the question of determining the Lame parameters of a medium by making measurements of displacements and stresses at the boundary and the problems of determining the magnetic and electric potentials of a body by making measurements at the boundary. Plans to consider two formally determined inverse problems which involve scattering observations will also be carried out. One is the inverse scattering problem at a fixed energy in two dimensions; the other is the inverse backscattering problem for singular potentials. Finally, work will be done on the admissibility problem in tomography, i.e. determining families of k-planes on which the k-plane transform is injective. It is planned to find relative inversion formulas (modulo smoothing) in such cases. This work falls within the general area of research on nonlinear partial differential equations and inverse problems. The questions often lie close to physical applications arising from diverse areas as tomography, conductivity and scattering. That is, to find characteristics of some unknown by observing the effect of some physical process which involves the unknown. ***
