Bill Lionheart (UMIST)
Bill.Lionheart@umist.ac.uk
A typical inverse problem for a system of partial differential equations is to recover the coefficients of the system from over specified boundary data. For example to determine information about a Riemannian metric on a manifold with boundary from the Dirichlet to Neumann mapping for Laplace's equation. This data does not completely determine the metric, an ambiguity arises due to the action of diffeomorphisms which fix points on the boundary.
In practical situations, such as electrical imaging where the metric is determined by the conductivity, one might know additional geometric information, such as a conformal structure (CO(n)-structure). One then investigates the group of diffeomorphisms preserving this structure. As an example we prove that a conformal diffeomorphism which is the identity on the boundary is the identity.
We go on to discuss geometric aspects of other inverse problems for systems of PDEs in electromagnetics and elasticity.