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

• Martin Deraux
• Louis Esperet
• Francis Lazarus
• Matthijs Ebbens
• Greg McShane
• Anne Parreau
• Andrea Seppi
• Matej Stehlìk
• Yibo Zhang

Scientific Context

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.

The project includes two new participants funded by ToFu

• Yibo Zhang, now PhD student under the supervision of Greg McShane and Louis Funar
• Matthijs Ebbens, now post-doc in ToFu

ToFu is funding the following events

• The winter school : Billiards at L'escandille, Autrans, 17 - 21 January 2022
• The conference Complex Hyperbolic Geometry and Related Topics, CIRM, 4 - 8 July, 2022
• the Special Session on Combinatorial and Computational Aspects in Topology at the AMS-EMS-SMF joint international meeting, Grenoble UGA, 18 - 22 July, 2022

meetings

Working group on combinatorial maps and dessins d'enfants animated by Francis Lazarus.

1. 4 October, 2021, 15h30-17h, introduction to the category of combinatorial maps and their monodromy groups. In this formalism, one can state a Riemann-Hurwitz formula with a simple combinatorial proof. Description of covering and quotient maps. A combinatorial map has a natural topological realization.
2. 19 October, 2021, 15h30-17h, proof of the Hurwitz bound on the number of automorphisms of a combinatorial map. Other formalisms for combinatorial maps: constellations and hypermaps. This last formalism happens to be slightly more convenient for studying the dessins d'enfants. The topological realization actually comes with a branched covering over the sphere with three ramification values. This realization and its branched covering can be made in the realm of Riemann surfaces.
3. 9 November, 2021, 14h-16h30, correspondence between Riemann surfaces and algebraic curves. Theorem of Belyi: a Riemann surface is defined over \bar{Q} if and only if it admits a Belyi function, i.e. a branched covering over the Riemann sphere with three ramification values. This is the case for the realization of hypermaps. This allows us to define an action of the absolute Galois group over hypermaps. This action is faithfull and the hope is (was?) to understand/approach the absolute Galois group more easily.