  The goal of this thesis is to study the singularities of the exponential map
of Riemannian and Finsler manifolds (a concept related to caustics and
catastrophes), and the object known as the cut locus (aka ridge, medial axis or
skeleton), to improve existing results about its structure, to look at it in
new ways, and to derive applications to the Ambrose conjecture and the
Hamilton-Jacobi equations.
