Square-tiled surfaces & quasimodular forms

After having introduced the Hürwitz problem on coutings of ramified coverings of a Riemann compact surface with prescribed degree, we explain a result due to Dijkgraaf and Kaneko & Zagier on coverings of the torus. Then, we introduce the notion of quasimodular forms and give a proof of a Theorem first proved by Eskin, Masur & Schmoll on coutings of ramified coverings of the torus with prescribed ramification type. This talk has been given after joint work with Samuel Lelièvre.

School and Conference on Analytic Number Theory

The Abdus Salam International Center for Theoretical Physics, Trieste, Italy