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:
- Introduction, motivation, brief survey of Teichmüller’s theorems;
- Basic definitions, examples, Masur’s criterion, Sketch of Kerkhoff, Masur and Smillie;
- Veech’s dichotomy, Examples of Veech surfaces (without proofs);
- Lyapunov exponents, motivations: Zorich’s asymptotic flag. Forni’s proof of Kontsevich-Zorich conjecture for genus 2;
- Detour, an explicit counterexample: the Eierlegende Wollmilchsau;
- Sketch of the exponential mixing of the Teichmüller flow (following Avila, Gouëzel and Yoccoz);
- An introduction to Transfer Operator Theory;
- Ruelle resonances for Pseudo-Anosov transformations.
Practical info: The course will last 5 weeks, for a total of approximately 30 hours. The first lecture will be on the 4th of June at 15,30 (Rome Time). From the 7th of June to the 10th of July the schedule is:
Monday 11,30 – 13,30;
Wednesday 15,30 – 17,30;
Friday 15,30 – 17,30
BEWARE: Due to the Italian tradition of “quarto d’ora accademico” lectures begin 15 minutes after the scheduled time.
The course will be held virtually on Teams, hosted by the University of Pisa. Some of the lectures will be streamed and recorded (for the teachers’ use) via Zoom. Accordingly, the links will be provided both by mailing list and the Teams channel.