Manchester Geometry Seminar 2001/2002

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 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 talk we will review the geometric formulation of the (scalar) anisotropic inverse conductivity problem. We will go on to consider generalizations of this anisotropic inverse boundary value problem to systems of Partial Differential Equations, including anisotropic, time harmonic Maxwell's equations and the result of Joshi and the speaker on the inverse boundary value problem for harmonic k-forms.

The talk will also include some examples of practical anisotropic inverse problems for Maxwell's equations including integral photoelasticity and electromagnetic imaging.