The aim of this talk is to describe a new method, based on renormalization and Microlocal Analysis, to recover some celebrated results in the field of Teichmüller Theory of translation surfaces. In particular, we will show how these techniques give a new proof of effective unique ergodicity criterion for the linear flow on translation surfaces, firstly proved by Forni (2002). This is a joint project with Hamid Al-Saqban.
When the boundary is Anosov with simple length spectrum, the study of singularities in the trace of the wave operator reveals certain interesting spectral invariants via the Duistermaat-Guillemin trace formula. We will discuss how these invariants can be exploited and naturally combined with the injectivity of the geodesic X-ray transform to address the problem.
In this context, some recent positive results obtained in the class of conformal metrics will be presented and it will be explained how it is possible to hear the jet at the boundary of a conformal Steklov drum. Finally, we will briefly discuss some results obtained by similar techniques on the magnetic Steklov inverse problem, concerning the recovery of an electric potential and a magnetic field from the spectrum of the associated magnetic DN map.
