I am preparing a presentation discussing the structure of the cut locus on Riemannian and Finsler manifolds. Since my audience will consist mostly of applied mathematicians, I was hoping to include some nice applications to other (non-geometric) areas of mathematics which might appeal to the majority of the audience. While I'm vaguely aware that there are some applications to regularity theory, I am not quite sure I understand these results. I am therefore looking for a nice introduction to this topic in addition to any other applications which might appeal to a group of applied mathematicians.

I am preparing a presentation discussing the structure of the cut locus on Riemannian and Finsler manifolds. Since my audience will consist mostly of applied mathematicians, I was hoping to include some nice applications to other (non-geometric) areas of mathematics which might appeal to the majority of the audience. While I'm vaguely aware that there are some applications to regularity theory, I do not understand these results. I am therefore looking for a nice introduction to this topic in addition to any other applications which might appeal to a group of applied mathematicians.