Geometric methods for anisotopic inverse boundary value problems.

Abstract

Electromagnetic fields have a natural representation as differential forms. Typically the measurement of a field involves an integral over a submanifold of the domain. Differential forms arise as the natural objects to integrate over submanifolds of each dimension. We will see that the (possibly anisotropic) material response to a field can be naturally associated with a Hodge star operator. This geometric point of view is now well established in computational electromagnetism, particularly by Kotiuga, and by Bossavit and and others. The essential point is that Maxwell’s equations can be formulated in a context independent of the ambient Euclidean metric. This approach has theoretical elegance and leads to simplicity of computation. In this paper we will review the geometric formulation of the (scalar) anisotropic inverse conductivity problem, amplifying some of the geometric points made in Uhlmann’s paper in this volume. We will go on to consider generalizations of this anisotropic inverse boundary value problem to systems of Partial Differential Equation, including the result of Joshi and the author on the inverse boundary value problem for harmonic k-forms