** Title: **

The geometry behind the asymptotic of the heat kernel for small time: from Riemannian manifolds to a generalization in optimal control theory

** Abstract: **

In this talk I will recall some deep properties of the asymptotic of the heat kernel for small time, which relate this analytic object with some geometric invariants of the manifold.
We will then generalize this study using optimal control theory and investigate if similar properties still hold.
In particular I will consider an optimal control problem in the n dimensional Euclidean space with a quadratic cost, where the dinamics is given by a drift vector field and k < n controlled vector fields, and I will introduce a related hypoelliptic operator. We will then find its fundamental solution in the linear problem and use this result to show the first terms of the asymptotic for a non-linear case.

This seminar is part of the AJS series of seminars.