Program

Wednesday, November 5

Thursday, November 6

Friday, November 7

Titles and Abstracts

Bending parameterisation of hyperbolic structures on three-manifolds

Bruno Dular

By the simultaneous uniformization theorem, hyperbolic structures on a given three-manifold are parameterised by the induced conformal structure on their ideal boundary. At the heart of a hyperbolic three-manifold lies its convex core, whose boundary — homeomorphic to the ideal boundary — is a pleated surface: piecewise totally geodesic and bent along a geodesic lamination. This lamination and the bending angles define a measured geodesic lamination on the ideal boundary, called the bending lamination. In joint work with Jean-Marc Schlenker (2024), we proved that convex co-compact hyperbolic structures on a given three-manifold are uniquely determined by their bending lamination. I will describe the main ideas of the proof, which relies on a limiting argument. Then, I will explain how a similar strategy can be extended to geometrically finite and infinite structures.

Translation lengths in (fine) curve graphs

Federica Fanoni

Given a surface $S$, we consider two associated graphs: the by now classical curve graph, on which the mapping class group of S acts, and the recently defined fine curve graph, on which the homeomorphism group of S acts. In both cases, for a group element, having positive asymptotic translation length (i.e. displacing every element at linear speed, roughly speaking) corresponds to having interesting topological/dynamical properties. I will discuss joint work with Sebastian Hensel and Frédéric Le Roux where we establish relations and study differences between the asymptotic translation lengths of homeomorphisms and mapping classes.

Kissing number of punctured spheres

Jasmin Jörg

We consider $n$-punctured spheres equipped with a complete Riemannian metric of variable curvature. In joint work with Sebastian Baader, we determine the maximal number of systoles among all such spheres, known as the kissing number. We present a construction of $n$-punctured spheres with precisely $4n-11$ systoles, realizing the kissing number.

Exploring strong stable sets for the geodesic flow on hyperbolic surfaces

Françoise Dal-Bo

Let $S$ be a hyperbolic surface and $u\in T^1(S)$. My talk focuses on the strong stable set $\mathrm{Wss}(u)$ associated with the geodesic trajectory of $u$. Clearly $\mathrm{Wss}(u)$ contains the horocyclic trajectory of $u$, but do the two sets coincide? Joint work with Alexandre Bellis and Sergio Herreo-Vila.

Crossing number of dense graphs on surfaces

Arnaud de Mesmay

In this talk, we will first provide an introduction to the rich history of problems around crossing numbers of graphs, with an emphasis on complete graphs and geometric approaches. We will then present two recent results about crossing numbers of dense graphs on surfaces. First, we prove that if $G$ is a dense enough graph with $m$ edges and $\Sigma$ is a surface of genus $g$, then any drawing of $G$ on $\Sigma$ incurs at least \[ \Omega\left(\frac{m^2}{g} \log^2g\right) \] crossings. The polylogarithmic factor in this lower bound is new, even in the case of complete graphs and disproves a conjecture of Shahrokhi, Székely and Vrt'o from 1996. Then we prove a geometric converse to this lower bound: we provide an explicit family of hyperbolic surfaces such that for any graph $G$, sampling the vertices uniformly at random on this surface and connecting them with shortest paths yields \[ O\left(\frac{m^2}{g} \log^2g\right) \] crossings in expectation. Both results rely on hyperbolic geometry, and I will strive to explain why hyperbolic geometry comes up naturally in the study of these questions. Based on joint work with Alfredo Hubard and Hugo Parlier.

Bass notes of closed arithmetic hyperbolic surfaces

Bram Petri

The spectral gap (or bass note) of a closed hyperbolic surface is the smallest non-zero eigenvalue of its Laplacian. This invariant plays an important role in many parts of hyperbolic geometry. In this talk, I will speak about joint work with Will Hide on the question of which numbers can appear as spectral gaps of closed arithmetic hyperbolic surfaces.

Tame maps of surfaces of infinite type

Jing Tao

A cornerstone in low-dimensional topology is the Nielsen-Thurston Classification Theorem, which provides a blueprint for understanding homeomorphisms of compact surfaces up to homotopy. However, extending this theory to non-compact surfaces of infinite type remains an elusive goal. The complexity arises from the behavior of curves on surfaces with infinite type, which can become increasingly intricate with each iteration of a homeomorphism. To address some of the challenges, we introduce the notion of tame maps, a class of homeomorphisms that exhibit non-mixing dynamics. In this talk, I will present some recent progress on extending the classification theory to such maps. This is joint work with Mladen Bestvina and Federica Fanoni.