# Mathematical Conversations

## Singularities of solutions of the Hamilton-Jacobi equation. A toy model: distance to a closed subset.

This is a joint work with Piermarco Cannarsa and Wei Cheng. Most of the lecture is about the distance function to a closed subset in Euclidean subset, at the level of a beginning graduate student. If $A$ is a closed subset of the Euclidean space $\mathbb R^k$, the Euclidean distance function $d_A: \mathbb R^k\to [0,+\infty[$ is defined by $d_A(x)=\min_{a\in A}\Vert x-a\Vert$. This function is Lipschitz, therefore differentiable almost everywhere. We will give topological properties of the set ${\rm Sing}(F)$ of points in $\mathbb R^k\setminus M$ where $F$ is not differentiable. For example it is locally connected. We will also discuss the homotopy type of ${\rm Sing}(F)$. We will sketch a proof. Although, we will concentrate on $d_A$, if time permits: 1) We will give applications in Riemannian geometry: Topology of the pairs of points that can be joined by at least 2 minimizing geodesics. 2) We will explain that it is a particular case of a more general result on the singularities of a viscosity solution $U:M\times ]0,+\infty[\to \mathbb R$ of the evolution Hamilton-Jacobi equation $$\partial_tU+H(x,\partial_xU)=0,$$ where $H:TM\to \mathbb R, (x,p)\mapsto H(x,p)$ is a C$^2$ Tonelli Hamiltonian, i.e. convex and superlinear in the momentum $p$, on the smooth manifold $M$.