Translation, Dilation, Affine and other Structures on Surfaces

7 avr. 2025, 08:00 → 11 avr. 2025, 17:00 Europe/Paris

Amphithéatre Schwartz ( Institut de Mathématiques de Toulouse)

Selim Ghazouani (University College London), Slavyana Geninska (Institut de mathématiques de Toulouse)

This conference will centre around the theme of affine structures (translation, dilation, complex affine, ...) on surfaces, in particular on the interplay between the geometry of their moduli space and the dynamical properties of their geodesic foliations. 

Invited speakers: 
Paul Apisa
Bertrand Deroin
Charles Fougeron
Pat Hooper
Magali Jay
Carlos Matheus
Martin Mion-Mouton
Duc-Manh Nguyen
Jasmin Raissy
Anja Randecker
Guillaume Tahar
Ferran Valdez
Jane Wang
Gabriela Weitze-Schmithüsen

Practical information: 

Please note that room and board for supported participants will be taken care directly by the organization. The organization will also cover the expenses for lunches (from Tuesday to Friday) and for the social dinner for all participants. Non-supported participants are expected to arrange for their own travels and accommodations in Toulouse. 

lundi 7 avril ¶

1 Holomorphic dynamics in dimension 2 and geometry of surfaces¶

In this introductory lecture I will present the connections between the dynamics of germs of biholomorphisms of ${\mathbb{C}}^2$ tangent to the identity at a fixed point, the real-time dynamics of homogeneous vector fields in ${\mathbb{C}}^2$ and the dynamics of the geodesic flow on affine surfaces, focusing on open problems.

Orateur: Jasmin Raissy

2 Introduction to ... TBA¶

3 Introduction to ... TBA¶

4 TBA¶

Orateur: Ferran Valdez

mardi 8 avril ¶

5 TBA¶

Orateur: Magali Jay

6 Symmetric surfaces from dynamics on triangles¶

Half-dilation surfaces are fun to build; you can snap together triangles like Magna-tiles®. I will describe a construction of half-dilation surfaces built from triangles produced by a (ℤ/2ℤ)(ℤ/2ℤ)(ℤ/2ℤ) action on homothety-equivalence classes of triangles in the plane. The advantage of this construction is that it produces surfaces with non-elementary Veech groups. Some of the surfaces that arise have infinite type, some others are already well-known: the Bouw-Möller lattice surfaces. This talk is about joint work with Seth Foster and Zhi Heng Liu.

Orateur: Pat Hooper

7 The topology of the moduli spaces of dilation surfaces¶

Dilation surfaces, which can be thought of as polygons whose sides are identified by translation and dilation, are a natural generalization of translation surfaces. While translation surfaces are well-studied, much less is known about dynamics on dilation surfaces and their moduli spaces. In this talk, we will survey recent progress in understanding the topology of moduli spaces of dilation surfaces, in particular the homotopy groups of these spaces. We will do this by understanding the action of the mapping class group on the moduli spaces of dilation surfaces. This talk is based on joint work with Paul Apisa and Matt Bainbridge.

Orateur: Jane Wang

8 TBA¶

Orateur: Duc-Manh Nguyen

mercredi 9 avril ¶

9 Foliated affine and projective structures¶

In this talk, I will report on some joint work with Adolfo Guillot on the existence of affine or projective structures along the leaves of one dimensional algebraic foliations. We will also discuss the problem of uniformizability of such structures.

Orateur: Bertrand Deroin

10 TBA¶

Orateur: Paul Apisa

jeudi 10 avril ¶

11 Lorentzian metrics with conical singularities and bi-foliations of the torus¶

The constant curvature Lorentzian metrics having a finite number of conical singularities offer new examples of geometric structures on the torus, naturally generalizing the analogous Riemannian case. In the latter, works of Troyanov show that the data of the conformal structure and of the angles at the singularities entirely classify the metrics with conical singularities. In this talk, we will introduce the Lorentzian metrics with conical singularities, construct some examples, and present a rigidity phenomenon: de-Sitter tori with a singularity of fixed angle are determined by the topological equivalence class of their lightlike bifoliation. Contrarily to the Riemannian case, we will see that in the Lorentzian case this rigidity is intimately linked to one-dimensional dynamics phenomena.

Orateur: Martin Mion -Mouton

12 TBA¶

Orateur: Gabriela Weitze-Schmithüsen

13 Sous-variétés bi-algébriques non-linéaires dans les espaces de modules de différentielles abéliennes¶

Les strates des espaces de modules de différentielles abéliennes sont des espaces non-homogènes portant des structures bi-algébriques naturelles. Klingler et Lerer ont montré qu'une courbe bi-algébrique dans un strate d'un espace de modules de différentielles abéliennes est linéaire pourvu qu'une certaine condition (*) soit satisfaite. Dans cet exposé, on discutera un travail en collaboration avec Bertrand Deroin sur l'existence de sous-variétés bi-algébriques non-linéaires dans les espaces de modules de différentielles abéliennes de genres 7 et 10.

Orateur: Carlos Matheus

14 TBA¶

Orateur: Guillaume Tahar

vendredi 11 avril ¶

15 Lengths of saddle connections for large genus¶

Inspired by the well-studied case of hyperbolic surfaces, we can ask about the expected value of geometric properties of translation surfaces for large genus.
In this talk, we consider the number of saddle connections in a given length range as a random variable on a stratum and show that for genus going to infinity, this converges in distribution to a Poisson distributed random variable.
This is based on joint work with Howard Masur and Kasra Rafi.

Orateur: Anja Randecker

16 TBA¶

Orateur: Charles Fougeron