This proposal brings together three researchers who will as a team use their diverse backgrounds and expertise to tackle a number of inter-connected inverse problems arising from differential geometry and partial differential equations, each having a natural physical interpretation. The specific problems include a dynamic version of the classical boundary rigidity problem for Riemannian metrics and its connection to the hyperbolic Dirichlet to Neumann map, an inverse problem for anisotropic electromagnetic bodies, and the detection of obstacles or cracks in an inhomogeneous conducting body from electrostatic measurements performed on the boundary.
The field of inverse problems addresses the problem of determining information about the interior of an object without destroying it, or cutting it open. One may send in waves: light waves, x-rays, or ultrasound, and analyze the transmitted waves which have been scattered by an unknown interior (scattering theory); or one may make boundary measurements of solutions to boundary value problems, the solutions of which depend on the material properties of the interior (inverse boundary value problems). Some of the problems discussed in this proposal also have applications to: electrical impedance tomography (application of a voltage potential to the boundary of a body and measurement of the induced current at the boundary yields information about the interior), medical imaging by ultrasound, non-destructive testing in industry (for cracks or wear in airplane wings, for instance), and seismology and oil exploration (the travel time of acoustic waves is used to reconstruct the earth's substructure).
