ToFu

It consists of 8 people from the Fourier Institute and G-SCOP:

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

Scientific Background

Problems that combine geometric and analytical aspects often give rise to fruitful exchanges between those who adopt theoretical and numerical perspectives. The study of Riemann surfaces requires numerous tools from various branches of mathematics—algebra, topology, and differential geometry, as well as combinatorial and algorithmic analysis. An important topic is the understanding of diffeomorphisms of a surface up to isotropy. This is a group that is studied 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 innate graph associated with a surface: a vertex is a homotopy class of simple curves, and vertices are connected by an edge if the corresponding curves can be chosen to be disjoint. There are many interesting questions regarding the metric/combinatorial geometry of this graph and several algorithms related to it: the Bestvina-Handel algorithm, the Bell-Webb algorithm, and the Leasure and Shackleton algorithms.

The goal of this project is to combine the expertise of theoretical mathematicians and computer scientists to study geometric problems that have combinatorial and algorithmic aspects.
More specifically, understanding the lengths of simple closed geodesics and the mapping class group of a surface requires numerous analytical tools from hyperbolic geometry and Teichmüller theory, but also involves an algorithmic approach. On the analytical side, we hope to extend the inequalities between the entropy of diomorphisms and hyperbolic invariants to other problems, and relate them to combinatorial problems involving graphs of curves, etc. On the computational side, we will develop and use programs to explore the combinatorics of the various graphs that can be associated with a surface.

The project includes two new participants funded by ToFu:

• Yibo Zhang, currently a Ph.D. student under the supervision of Greg McShane and Louis Funar
• Matthijs Ebbens, now a postdoctoral researcher at ToFu

ToFu funds the following events:

• Winter School: Billiards at L'Escandille, Autrans, January 17–21, 2022
• The Conference on Complex Hyperbolic Geometry and Related Topics, CIRM, July 4–8, 2022
• The special session on combinatorial and computational aspects of topology at the joint AMS-EMS-SMF international meeting, Grenoble UGA, July 18–22, 2022

Meetings

Working group on combinatorial maps and children's drawings, led by Francis Lazarus.

1. October 4, 2021, 3:30–5:00 p.m., Introduction to the category of combinatorial maps and their monodromy groups. Within this framework, we 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. October 19, 2021, 3:30–5:00 p.m., proof of the Hurwitz bound on the number of automorphisms of a combinatorial map. Other formalisms for combinatorial maps: constellations and hypermaps. The latter formalism happens to be somewhat more practical for studying children’s drawings. The topological realization is in fact accompanied by a ramified covering on the sphere with three ramification values. This realization and its ramified covering can be constructed within the field of Riemann surfaces.
3. 9 novembre 2021, 14h-16h30, correspondance entre les surfaces de Riemann et les courbes algébriques. Théorème de Belyi : une surface de Riemann est définie sur \bar{Q} si et seulement si elle admet une fonction de Belyi, c'est-à-dire un recouvrement ramifié sur la sphère de Riemann avec trois valeurs de ramification. C'est le cas pour la réalisation des hypermaps. Cela nous permet de définir une action du groupe de Galois absolu sur les hypermaps. Cette action est fidèle et l'espoir est (était ?) de comprendre/approcher le groupe de Galois absolu plus facilement.