Research

Global analysis and PDEs on manifolds

Invariant geometric PDEs are a great tool for probing geometry of manifolds. The most known example of this interaction is the Atiyah-Singer index theorem, which allows to relate topological invariants of the manifold to the Fredholm index of an elliptic differential operator. But one can also see the volume and dimension of a manifold in its spectrum or spot the presence or absence of closed geodesics using dispersive estimates.

Recently I have been interested in the study of the Laplace operator on almost-Riemannian manifold, which constitute a certain type of singular Riemannian manifolds. Those manifolds have very interesting properties. For example, they are well defined metric spaces with smooth geodesics, but with infinite volume and unbounded curvature. For this reason the natural quantum-classical correspondence from Riemannian geometry breaks down. In particular, often the Laplace-Beltrami operator on a domain bounded by the singularity is essentially self-adjoint and quantum particles or the heat can not cross the singular set, while the geodesics can. This phenomena is known as quantum confinement.

In the work I studied minimal domains of the Laplace-Beltrami operator on generic 2D almost-Riemannian manifolds including the most difficult case of structures with tangency points, which cause a lot of headache to specialists in sub-Riemannian geometry. In a joint work with Ugo Boscain and Eugenio Pozzoli we showed that presence of quantum confinement depends on the quantization chosen (a phenomena absent in the case of Riemannian manifolds). On a different note, together we Ugo Boscain and Mario Sigalotti in a joint article we studied controllability properties of systems under classical-quantum correspondence on Riemannian manifolds. We found several geometric obstructions to small controllability of the time-dependent controlled Schroedinger equation.