Translation, Dilation, Affine and other Structures on Surfaces

Contribution List

2. Introduction to affine surfaces

Bertrand Deroin

4/7/25, 9:30 AM

1. Holomorphic dynamics in dimension 2 and geometry of surfaces

Jasmin Raissy

4/7/25, 11:00 AM

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.

3. Introduction to the dynamics of interval exchange maps

Corinna Ulcigrai

4/7/25, 2:30 PM

4. Zebra structures

Ferran Valdez

4/7/25, 4:00 PM

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...

5. Tiling billiard in the wind-tree model

Magali Jay

4/8/25, 9:30 AM

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...

6. Symmetric surfaces from dynamics on triangles

Pat Hooper

4/8/25, 11:00 AM

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...

10. The topology of the moduli spaces of dilation surfaces

Jane Wang

4/8/25, 2:30 PM

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...

8. Differentials on the sphere and representations of the braid groups

Duc-Manh Nguyen

4/8/25, 4:00 PM

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...

9. Foliated affine and projective structures

Bertrand Deroin

4/9/25, 9:30 AM

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.

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

Paul Apisa

4/9/25, 11:00 AM

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...

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

Martin Mion -Mouton

4/10/25, 9:30 AM

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...

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

Gabriela Weitze-Schmithüsen

4/10/25, 11:00 AM

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...

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

Carlos Matheus

4/10/25, 2:30 PM

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...

14. Curvature gap and bounds on trajectories in flat and affine spheres

Guillaume Tahar

4/10/25, 4:00 PM

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...

15. Lengths of saddle connections for large genus

Anja Randecker

4/11/25, 9:30 AM

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...

16. Renormalization for foliations on surfaces

Charles Fougeron

4/11/25, 11:00 AM

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...