The theory of surfaces of finite or infinite types, with their geometric structures, with moduli spaces of geometric structures and with some related dynamical systems on these moduli spaces, constitutes some of the most important aspects of low-dimensional topology, geometry and dynamics. In this school, we propose a set of coordinated courses that will concentrate on several aspects of this field and which will give the students the opportunity, at the same time, to learn the basic aspects of these topics, and to have access to important research problems.

The topics studied in these 8 × 6 hours of courses will include hyperbolic geometry and constructions of hyperbolic structures on surfaces, Counting simple closed geodesics on surfaces: Mirzakhani’s work, recent results and open problems; Funk and Hilbert geometry in Euclidean and non-Euclidean geometry, Funk geometries and Teichmüller theory: the Weil-Petersson-Funk metric; The metric theory of Teichmüller spaces of surfaces, The Thurston metric on the Teichmüller space of hyperbolic surfaces and of Euclidean surfaces, The Finsler structure of the Thurston metric; The space of measured laminations and the space of unmeasured laminations on a hyperbolic surface; Combinatorial group theory in surface theory, the braid group and the mapping class group; Moduli spaces of linkages; Differential geometry and combinatorics of curves on surfaces; Finsler geometry from the distance point of view, The Busemann-Mayers-Kobayashi theorem for Finsler manifolds.
A special stress will be on pedagogical aspect, and each of the teachers will conduct an additional session of exercises for the students