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. 

lun. 7 avril

1 Introduction to affine surfaces

Orateur: Bertrand Deroin

2 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

Toulouse-Jasmin.pdf

3 Introduction to the dynamics of interval exchange maps

Orateur: Corinna Ulcigrai

4 Zebra structures

We introduce a new class of geometric structures on surfaces, called zebra structures, which generalize translation and dilation structures yet still induce directional foliations for every slope. Our primary goal is to determine when a free homotopy class of loops (or a homotopy class of arcs with fixed endpoints) admits a canonical representative—or a canonical family of representatives—realized as closed leaves or as chains of leaves connecting singularities. These canonical representatives are analogs of geodesic representatives in translation surfaces.

Our main result shows that representatives always exist provided the surface admits a triangulation whose edges connect the singularities in the sense of the zebra structure. In the special case where the surface is closed, we further characterize several geometric conditions under which every homotopy class of closed curves has a canonical representative.

This is joint work with P. Hooper and B. Weiss. Reference: https://arxiv.org/abs/2301.03727

Orateur: Ferran Valdez

mar. 8 avril

5 Tiling billiard in the wind-tree model

In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and the Eaton lenses. The three of them are motivated by physics.
In the beginning of the 2000's, physicists have conceived metamaterials with negative index of refraction. Tilling billiards' trajectories consist of light rays moving in a arrangement of metamaterials with opposite index of refraction. The wind-tree model was introduced by Paul and Tatyana Ehrenfest to study a gaz: a particle is moving in a plane where obstacles are periodically placed, on which the particle bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens.
After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle. I show that these trajectories are trapped in a strip, for almost every parameter. This behavior is similar to the one of the Eaton lenses.

Orateur: Magali Jay

conf_toulouse.pdf

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

Toulouse talk Final-bitmap.pdf

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 Differentials on the sphere and representations of the braid groups

The fundamental group of a stratum of k-differentials naturally acts on the (co)-homology of the corresponding canonical cyclic covers via monodromy. In the genus zero case, these actions give rise to a series of representations of the pure braid groups. In this talk, I will report a result on the images of those representations. Specifically, I will discuss their Zariski closure and some sufficient conditions for those images to be arithmetic lattices. This is a joint work with G. Menet.

Orateur: Duc-Manh Nguyen

mer. 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 SL(2, R)-invariant measures on the moduli space of twisted holomorphic 1-forms and dilation surfaces

A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of SL(2, R) on the plane induces an action of SL(2, R) on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. As with translation surfaces, which can be identified with holomorphic 1-forms on Riemann surfaces, dilation surfaces can be thought of as “twisted” holomorphic 1-forms.

I will describe joint work with Nick Salter producing an SL(2, R)-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a K(pi,1) where pi is the framed mapping class group. I will describe further work with Bainbridge and Wang which shows that no such SL(2, R)-invariant measure can be finite. This will follow from showing that an open and dense set of surfaces in a stratum diverge under the action of the diagonal subgroup of SL(2, R) and hence the conclusion of Poincare recurrence fails.

Orateur: Paul Apisa

Dilation_Surfaces_Talk__Toulouse_Version_.pdf

jeu. 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 Dessins on Teichmüller curves in a special locus in genus 2

Origamis, also called square-tiled surfaces - are translation surfaces obtained by gluing finitely many copies of the unit square to each other along their boundaries. They are in particular closed Riemann surfaces. Their SL(2,Z)-orbit defines an algebraic curve in moduli space M_g, where g is the genus of the origami, which are special cases of Teichmüller curves.

The normalisation of a Teichmüller curve defined by an origami naturally comes with a Belyi morphism, i.e. a morphism to the sphere ramified over at most three points. This property equips the curve with a Grothendieck dessin d'enfants. We study these dessins in a special locus H(1,1), the stratum of translation surfaces of genus 2 with 2 singularities. For origamis in this locus, we can explicitly determine their Veech groups and leverage this information to describe the associated dessins d’enfants. This is joint work with Hannah Zeimetz.

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 Curvature gap and bounds on trajectories in flat and affine spheres

On a topological sphere endowed with a flat metric with conical singularities, the curvature gap quantifies the obstruction to realize a partition of the set of conical singularities into two sets of equal total angle defect. Unless its curvature gap is equal to zero, such a flat sphere cannot contain a simple closed geodesic. Drawing on the Delaunay decompositions of flat surfaces, we give a quantitative generalization of the latter statement. We prove that when the curvature gap is nonzero, there is an explicit upper bound on the lengths of simple geodesic trajectories in the flat sphere. If time allows, we will give extensions of this result to affine structures on punctured spheres. This is a work in collaboration with Kai Fu.

Orateur: Guillaume Tahar

ven. 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 Renormalization for foliations on surfaces

In this talk, I will discuss a well-known renormalization technique for the first return map of foliations on surfaces, known as Rauzy-Veech induction.
A key result in this setting is the exponential tail property of the renormalization process.
It was established by Avila-Gouezël-Yoccoz in the case of interval exchange maps (associated to orientable foliations on orientable surfaces) and later by Avila-Resende for linear involutions (emerging from non-orientable foliations).
These results have significant implications for the dynamics of the Teichmüller flow and were instrumental in proving a spectral gap property.

After providing an overview of these techniques, I will explain some other strong consequences and discuss the non-orientable cases as appearing for instance in tiling billiards.

Orateur: Charles Fougeron