  We show show that the singularities of the Fresnel surface for Maxwell's
equation on an anisotrpic material can be accounted from purely topological
considerations. The importance of these singularities is that they explain the
phenomenon of conical refraction predicted by Hamilton. We show how to
de-singularise the Fresnel surface, which will allow us to use Morse theory to
find lower bounds for the number of critical wave velocities inside the
material under consideration. Finally, we propose a program to generalise the
results obtained to the general case of hyperbolic differential operators on
differentiable bundles.
