Surfaces, Manifolds and Related Topics
Luxembourg, March 18-20, 2024


The workshop will take place on the Maison du Savoir (MSA), Belval Campus.

For the poster, please click here.

For the schedule, please click here.

For photos, please click here, here, here, and here.

Titles and Abstracts:

Federica Fanoni: Classification of mapping classes of infinite-type surfaces

Mapping classes of surfaces of finite type have been classified by Nielsen and Thurston. For surfaces of infinite type (e.g. surfaces of infinite genus), no such classification is known. I will talk about the difficulties that arise when trying to generalize the Nielsen-Thurston classification to infinite-type surfaces and present a first result in this direction, concerning maps which -- loosely speaking -- do not show any pseudo-Anosov behavior. Joint work with Mladen Bestvina and Jing Tao.

Niloufar Fuladi: Cross-cap drawings and signed reversal distance (slides)

A cross-cap drawing of a graph $G$ is a drawing on the sphere with $g$ distinct points, called cross-caps, such that the drawing is an embedding except at the cross-caps, where edges cross properly. A cross-cap drawing of a graph $G$ with $g$ cross-caps can be used to represent an embedding of $G$ on a non-orientable surface of genus $g$. Mohar conjectured that any triangulation of a non-orientable surface of genus $g$ admits a cross-cap drawing with $g$ cross-caps in which each edge of the triangulation enters each cross-cap at most once. Motivated by Mohar's conjecture, Schaefer and Stefankovic provided an algorithm that computes a cross-cap drawing with a minimal number of cross-caps for a graph $G$ such that each edge of the graph enters each cross-cap at most twice. In this talk, I will first outline a connection between cross-cap drawings and an algorithm coming from computational biology to compute the signed reversal distance between two permutations. This connection will then be leveraged to answer two computational problems on graphs embedded on surfaces.
First, I show how to compute a "short" canonical decomposition for a non-orientable surface with a graph embedded on it. Such canonical decompositions were known for orientable surfaces, but the techniques used to compute them do not generalize to non-orientable surfaces due to their more complex nature. Second, I explain how to build a counter example to a stronger version of Mohar's conjecture that is stated for pseudo-triangulations.
This is joint work with Alfredo Hubard and Arnaud de Mesmay.

Magali Jay: Tiling billiard in the wind-tree model (slides)

In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and Eaton lenses. The all three of them are motivated by physics. The wind-tree model was instoduced by Paul and Tatyana Ehrenfest to study a gaz: a particule is moving in a plane where obstacles are periodically placed, on which the particule 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. 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 indew of refraction.
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 traped in a strip, for almost every parameter. This behavior is similar to the one of Eaton lenses.

David Fisac: Exact curve counting with given word-length and self-intersection on the once-punctured torus (slides)

This talk will be on some work in progress on the problem of finding a closed formula for the exact number of curves with any given word-length and self-intersection on the once-punctured torus. We will first present the counting of the number of simple curves of given length by only combinatorial means using a well-known characterization of them. Followed by presenting an analogous characterization of all the curves with a single self-intersection, and hence their counting. This leads to a discussion on how to find a closed formula for any given self-intersection. Finally, we will also present the counting when there is no restriction on the intersection.

Javier Aramayona: Lego surfaces and their asymptotically rigid mapping class groups

A Lego surface is a non-compact surface obtained by gluing copies of a fixed compact surface in an inductive manner. To every Lego surface one may associate its asymptotically rigid mapping class group, whose elements are isotopy classes of homeomorphisms of the Lego surface which are “eventually trivial” in a suitable sense.
Through their action on the space of ends of the Lego surface, asymptotically rigid mapping class groups resemble some classical families of groups, namely Thompson groups and Houghton groups.
After introducing all these concepts, I will talk about the finiteness properties enjoyed by these groups, which are determined by the end space of the corresponding Lego surface.

Chris Leininger: Coarse geometry of surface bundles over Teichmüller curves

A Teichmüller curve is a totally geodesically embedded hyperbolic surface in the moduli space of Riemann surfaces, and the pull-back of the "universal curve" is a naturally associated surface bundle E over this hyperbolic surface. I'll start by concretely describing E and a geometric structure on it which is a singular "hyperbolic-by-Euclidean" structure, which is a four-dimensional analogue of the singular solv metric on a fibered hyperbolic 3-manifold. I'll then talk about how to use this structure to understand the coarse geometry of its fundamental group Gamma, proving that Gamma is hierarchically hyperbolic and satisfies a strong form of quasi-isometric rigidity. This is joint work with Spencer Dowdall, Matthew Durham, and Alessandro Sisto.

Yilin Wang: Two optimization problems of the Loewner energy

A Jordan curve on the Riemann sphere can be encoded by its conformal welding homeomorphism, which is a circle homeomorphism. The graph of the welding homeomorphism can be naturally viewed as a positive curve on the boundary of $\mathrm{AdS}^3$ space. For instance, Thurston's earthquake map associated with a circle homeomorphism has a geometric interpretation in $\mathrm{AdS}^3$.
The Loewner energy measures how far a Jordan curve is away from being a circle or, equivalently, how far its welding homeomorphism is away from being Mobius. I will discuss two optimizing problems for the Loewner energy, one under the constraint for the curve to pass through n given points on the Riemann sphere and the other under the constraint for the welding curve to pass through n given points in the boundary of $\mathrm{AdS}^3$. We show that these two problems exhibit mysterious symmetry.

Julien Boulanger: Algebraic intersection and regular polygons (slides)

Given a closed oriented Riemannian surface $X$ (possibly with singularities), one may wonder what is the maximal possible intersection of two closed curves of a given length on $X$. One way to approach this problem is to define the so-called "Interaction strength" of $X$, here denoted $\mathrm{KVol}(X)$. After a general introduction on the interaction strength, we will focus on the case of translation surfaces, which are instances of flat Riemannian surfaces with finitely many conical singularities. Namely, we explain how to compute $\mathrm{KVol}$ on the $\mathrm{SL}(2,\mathbb{R})$-orbit of several families of translation surfaces. This is based on joint work with E. Lanneau and D. Massart, as well as with I. Pasquinelli.

Anna Roig: On the length spectrum of random hyperbolic 3-manifolds (slides)

We are interested in studying the behaviour of geometric invariants of hyperbolic 3-manifolds as their complexity increases. A way to do so is through the study of random manifolds. In this talk, I will explain one of the principal probabilistic models of random manifolds for 3 dimensions, and I will present a result concerning the length spectrum -the set of lengths of all closed geodesics- of a 3-manifold constructed under this model. If time allows, I will also say a word about my work in progress regarding the systole.

Richard Webb: An equator theorem for the 2-sphere

We will focus on the group of Hamiltonian diffeomorphisms (and/or area-preserving homeomorphisms) of the 2-sphere. A tremendous amount of progress has been made in the study of these groups in the last few years, but many problems remain, including the Equator Conjecture. An equator on the 2-sphere is a simple closed curve whose complementary components have equal area. The Equator Conjecture predicts that for any positive K, there are pairs of equators such that any Hamiltonian diffeomorphism sending one equator to the other must have Hofer norm larger than K. We will prove an alternative conjecture, where we replace “Hofer norm” with “quantitative fragmentation norm”. To prove this, we construct new quasimorphisms defined on the group of area-preserving homeomorphisms of the 2-sphere, coming from methods inspired from mapping class groups and geometric group theory. Joint work with Yongsheng Jia.

Sayantika Mondal: Distinguishing filling curve types via special metrics (slides)

In this talk, we look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. In particular, explore the relations between the length infimum of curves and their self-intersection number. For any surface, we construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. I might also discuss some coarse bounds on the metrics associated to these minimum lengths.

Yuhao Xue: Number of simple and non-simple closed geodesics for random hyperbolic surfaces (slides)

On a closed hyperbolic surface $X$, the prime geodesic theorem says that the number of primitive closed geodesics of length $\leq L$ grows asymptotic as $e^L/L$. And Mirzakhani showed that the number of primitive simple closed geodesics of length $\leq L$ has polynomial growth rate. It is then a natural question to ask when will the most closed geodesics be non-simple? We show that for random hyperbolic surfaces of Weil-Petersson model, most closed geodesics of length $\ll\sqrt{g}$ are simple and non-separating, and most closed geodesics of length $\gg\sqrt{g}$ are non-simple. This is a joint work with Yunhui Wu.

Marie Trin: From counting problems to convergence of measures : the case of arcs in surfaces

We will explain how counting problems for curves on surfaces have been studied by Mirzakhani and Erlandsson-Souto by investigating convergence on sequences of counting measures. Afterwards, we will explain how this method can be used in the case of arcs rather than curves.

Anton Zorich: Random square-tiled surfaces and random multicurves in large genus (after joint works with V. Delecroix, E. Goujard and P. Zograf) (slides)

Moduli spaces of Riemann surfaces and related moduli spaces of quadratic differentials are parameterized by a genus g of the surface. Considering all associated hyperbolic (respectively flat) metrics at once, one observes more and more sophisticated diversity of geometric properties when genus grows. However, most of metrics, on the contrary, progressively share certain rules. Here the notion of “most of” has explicit quantitative meaning, for example, in terms of the Weil-Petersson measure. I will present some of these recently discovered large genus universality phenomena.
I will use count of metric ribbon graphs (after Kontsevich and Norbury) to express Masur-Veech volumes of moduli space of quadratic differentials through Witten-Kontsevich correlators. Then I will present Mirzakhani's count of simple closed geodesics on hyperbolic surfaces. We will proceed with description of random geodesic multicurves and of random square-tiled surfaces in large genus. I will conclude with a beautiful universal asymptotic formula for the Witten-Kontsevich correlators predicted by Delecroix, Goujard, Zograf and myself and recently proved by Amol Aggarwal.