The University of Pisa offers an online Ph.D. course taught by Mauro Artigiani (Universidad del Rosario, Colombia) and Paolo Giulietti (Università di Pisa) titled “Translation Surfaces: From Geometry to Spectral Theory”.
Description: Translation surfaces are a generalization of flat tori to higher genuses, with rich and interesting geometrical and dynamical properties. In fact, they can be seen both from a complex geometry point of view, stemming from the works of Teichmüller, Ahlfors and Bers, and also from a Euclidean geometry point of view, connecting to the works of the Russian school on low dimensional dynamics. These two different point of views have been fruitfully exploited in the last 50 years to obtain many deep and beautiful results.
In this course, we will introduce translation surfaces motivating their interest. Then, we will survey some of the classical results about them, focusing on the dynamical point of view: the geodesic flow on a translation surface is an important example of a parabolic system, in which nearby points diverge slowly from each other. We will stress the general philosophy of renormalization, which connects the study of the geodesic flow on a surface with the study of the geodesic flow on the moduli space of translation surfaces, which has a chaotic behavior that can be exploited to obtain many information on our initial flow.
Prerequisites are measure theory, some familiarity with complex analysis and functional analysis.
A sketch of the course is as follows:
