Heat Kernel Asymptotics at the Cut Locus on Riemannian and Sub-Riemannian Manifolds

In this talk I will discuss the asymptotics of the heat kernel $p_t(x,y)$ on a Riemannian or sub-Riemannian manifold. We will consider the small time asymptotics, both off-diagonal and at the cut locus, showing how the asymptotic of $p_t(x,y)$ behave depending on whether (and how much) $y$ is conjugate to $x$. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context.
If time permits I will discuss how these techniques let us to identify the possible asymptotics for the heat kernel at the cut locus for a generic Riemannian manifolds (of dimension less or equal than $5$). This is a consequence of the fact that, among the stable singularities of Lagrangian maps appearing in the classification of Arnold, only two of them can appear as “optimal”, i.e. along minimizing geodesics.