Inverse boundary problems are a class of problems in which one seeks to determinethe internal properties of a medium by performing measurements along the boundaryof the medium. These inverse problems arise in many important physical situations,ranging from geophysics to medical imaging to the non-destructive evaluation andmaterials.
The appropriate mathematical model of the physical situation is usually given by a partial differential equation (or a system of such equations) inside the medium. The boundary measurements are then encoded in a certain boundary map. The inverseboundary problem is to determine the coefficients of the partial differential equationinside the medium from knowledge of the boundary map.
The prototypical example of an inverse boundary problem is the inverse conductivityproblem, also called electrical impedance tomography, first proposed by A. P. Calderon.In this case the boundary map is the voltage to current map; that is, the map assignsto a voltage potential on the boundary of a medium the corresponding induced currentflux at the boundary of the medium. The inverse problem is to recover the electricalconductivity of the medium from the boundary map.
This problem can be recast in geometric terms as determining a Riemannian manifoldfrom the DN map associated to the Laplace-Geltrami operator. We will discuss recent resultsconcerning the determination of the manifold itself from this information.