Online Ph.D. course on translation surfaces, June-July 2021
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:
