This project is financed by the Labex Persyval. It gathers 8 people from Institut Fourier and G-SCOP.

- Martin Deraux
- Louis Esperet
- Francis Lazarus
- Greg McShane
- Anne Parreau
- Joanny Perret
- Andrea Seppi
- Matej Stehlìk

Problems combining geometrical and analytical aspects frequently lead to rich exchanges between those taking theoretical and numerical viewpoints. The study of Riemann surfaces requires many tools coming from dierent branches of mathematics - algebra, topology and dierential geometry but also combinatorial and algorithmic analysis. An important topic is understanding the dieomorphisms of a surface up to isotopy. This is a group which one studies via its actions on spaces constructed from the combinatorics of curves on the surface. In particular it acts on Teichmüller space, Thurston's measured lamination space and Harvey's curve complex. The latter is an innite graph associated to a surface : a vertex is a homotopy class of simple curves and vertices are joined by an edge if the corresponding curves can be chosen to be disjoint. There are many interesting questions concerning the metric geometry/combinatorics of this graph and several algorithms related to them : the BestvinaHandel algorithm, BellWebb algorithm, Leasure and Shackleton algorithms.

The goal of this project is to combine the knowledge of theoretical mathematicians and computer scientists in order to study questions in geometry which have a combinatorial and algorithmic nature. More precisely, the comprehension of the lengths of closed simple geodesics and the mapping class group of a surface requires many analytic tools coming from hyperbolic geometry and Teichmüller theory but has also an algorithmic approach. On the analytic side we hope to extend the inequalities between entropy of dieomorphisms and hyperbolic invariants [8] to other problems and relate this to combinatorial problems of curve graphs etc. On the eective side we will develop and use programs to explore the combinatorics of the various graphs that one can associate to a surface.