Topology of the set of singularities of viscosity solutions of the Hamilton-Jacobi equation

We will mainly report on the progress done recently the connectedness properties of the set of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation. To make the lecture accessible to people with no previous knowledge in the subject, after stating the main result in its full generality, we will restrict to distance functions to closed subsets of Euclidean space, which contain all the relevant aspects of the problem.