  This is the text from a talk at the Arbeitstagung 2011, which can serve as an
introduction to arxiv:1009.0736 and arXiv:1007.0907. I first discuss how a
global field is determined by a certain dynamical system, and how this relates
to abelian L-series determining those fields. I then discuss an analog in
Riemannian geometry, and how it leads to a metric in the space of closed
Riemannian manifolds.
