Spectral characterizations of Besse and Zoll Reeb flows

In this talk, I will address a geometric inverse problem from contact geometry: is it possible to recognize whether all orbits of a given Reeb flow are closed from the knowledge of the action spectrum? Borrowing the terminology from Riemannian geometry, Reeb flows all of whose orbits are closed are sometimes called Besse, and Besse Reeb flows all of whose orbits have the same minimal period are sometimes called Zoll. In the talk I will summarize recent results on this inverse problem in a few settings: geodesic flows (joint work with Stefan Suhr), closed contact 3-manifolds (joint work with Daniel Cristofaro-Gardiner), convex contact spheres and, more generally, restricted contact type hypersurfaces of symplectic vector spaces (joint work with Viktor Ginzburg and Basak Gürel). I will also mention a few conjectures and open problems.