On the full, flat space, Egidi-Veselic and Wang-Wang-Zhang-Zhang established that given a measurable set $\omega$, the following three properties are equivalent: (i) thickness: $\omega$ has equidistributed measure in every ball at some scale, (ii) a spectral inequality bounding the $L^2$ norm of functions with compact spectral support by their $L^2$ norm on $\omega$, and (iii) observability of the heat semigroup from $\omega$. The aim of the talk will be to discuss how these properties interact in the manifold setting and how to generalize the equivalence to non-compact manifolds, under relevant bounds on curvature. I will focus on ongoing work with Alix Deleporte and Jean Lagacé, in which we aim to prove that (i) implies (ii) on manifolds with bounded curvature, with a constant that depends only on curvature bounds rather than the metric itself. The proof crucially relies on elliptic estimates by Logunov and Malinnikova, combined with tools from geometric analysis.

We consider a class of dynamical systems that are described by time and space dependent partial differential equations. This class fits perfectly the port-Hamiltonian framework. We cover the wave equation, Maxwell's equations, the Kirchhoff-Love plate model, piezo-electromagnetic systems and many more. Our goal is to characterize boundary conditions that make the systems passive (the energy of solutions decays). This is done by constructing a boundary triple for the underlying differential operator. As a by-product we develop the theory of quasi Gelfand triples, which enables us to regard L2 boundary conditions even though the ``natural`` boundary spaces are neither included nor covering L2.