We are excited to announce "Curve counting: what's next?" to be held at the Hamilton Mathematics Institute, in Dublin, Ireland. A five-day workshop featuring invited talks by leading experts, selected talks by early-career researchers, and ample time for informal discussions and collaboration.
Dates: May 25–29, 2026
Venue: Hamilton Mathematics Institute (HMI), Trinity College Dublin
Organisers:
Accommodation:
Accommodation check-in is on Sunday, May 24, from 3pm, and check-out is on Friday, May 29, by 10am (unless you were specifically confirmed different dates).
For check-in, please go to the Accommodation Office (Trinity Summer Accommodation). Note that the office closes at 1am (see below for alternative).
https://maps.app.goo.gl/kKNw3jYa9NixMYti9
Accommodation Office is located in Front Square, House 10 to the left of the Chapel. (Except for those of you staying at a separate hotel.)
If the accommodation office is closed (after 1am), you can pick up your keys at the Front Gate:
https://maps.app.goo.gl/3K1783tSPiyDzczx5
Pre-cocktail
On Tuesday evening at 6:30pm, there will be a pre-cocktail gathering (platters/snacks) at JR Mahon's Public House and Brewery (1–2 Burgh Quay, Dublin 2):
https://share.google/cjK8Jj3dDF5U144RD
Talks
schedule.pdf—----------------------------------
Monday 25
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Time: 10-11
Speaker: Hugo Parlier
Title: The entropy spectrum of a hyperbolic surface
Abstract: The study of closed geodesics on hyperbolic surfaces sits at the crossroads of geometry, topology, and dynamics, and is also, quite conveniently, one of the main themes of the conference.
This talk will discuss new quantitative bounds for the growth of the number of closed geodesics on surfaces with geodesic boundary. The main theme is that coarse geometric data, such as boundary length, width, or diameter, already strongly constrain geodesic growth and entropy. The proofs use strip deformations inspired by W. Thurston’s work in Teichmüller theory.
The talk will also introduce the entropy spectrum of a hyperbolic surface, consisting of the entropies of its subsurfaces. This leads to the study of gaps in the entropy spectrum, rigidity phenomena, and discreteness results that reveal new geometric features of subsurface dynamics. This is joint work with one of the organizers, Ara Basmajian.
Time: 11.30-12.30
Speaker: Viveka Erlandsson
Title: Distribution of components of multi-curves
Abstract: Due to Mirzakhani’s work we know the asymptotic growth of the count of multi-curves of fixed topological type and bounded length, as the bound on the length goes to infinity. Being able to count these gives us a notion of a random multi-curve and we can ask what the expected properties of such a curve is. For example, how do the length of the individual components distribute? Mirzakhani answered this for random pants decompositions of a hyperbolic surface and this result has been generalized independently by Mingkun Liu and Francisco Arana-Herrera to other simple multi-curves. Using different methods we explain how to obtain these results and generalize them to any multi-curves (not necessarily simple) and a large class of metrics besides hyperbolic. This is joint work with Juan Souto.
Time: 14-15
Speaker: Juan Souto
Title: "Can you hear the shape of a hyperbolic surface? Now for real."
Abstract: We associate a musical instrument, a {\em hyperbolic marimba}, to every pair $(X,\Gamma)$ where $X$ is a hyperbolic surface and $\Gamma\subset X$ a simple multicurve labeled with musical keys. It works as follows: take a geodesic and every time it hits $\Gamma$, play the corresponding note. We investigate to which extent do the so-produced melodies characterize $(X,\Gamma)$ up to isometry. This is joint work with Ludovico Battista.
Time: 16-16.30
Speaker: Sophie Wright
Title: Random covers of hyperbolic surfaces
Abstract: Given a closed hyperbolic surface, I will discuss random covers whose fundamental group is isomorphic to the free group F_k. What do we expect such a cover to look like? I will show that, asymptotically, the covers distribute according to a probability measure on the moduli space of metric graphs. This allows us to determine expected properties of covers, with some nice explicit results for k=2.
Time: 16.30-17
Speaker: David Fisac
Title: Combinatorial curve counting for multi-curves on the once-punctured torus.
Abstract: I will present some work in progress joint with Mingkun Liu on the problem of counting the number of multicurves of given type and given word-length on the once-punctured torus.
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Tuesday 26
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Time: 10-11
Speaker: Dylan Thurston
Title: The intersection dual of geodesic currents
Abstract: Every geodesic current on a closed surface induces a dual function on
curves via the geometric intersection pairing. It is natural to ask
which curve functions are dual to geodesic currents, that is, which
arise as intersection functionals of a geodesic current.
In this paper we give an axiomatic, combinatorial characterization of
curve functionals dual to geodesic currents: a function on
multi-curves arises as the dual to a geodesic current if and only if
it is additive under disjoint union and satisfies the smoothing
property: it is non-increasing under surgery of essential crossings.
This yields a new definition of geodesic currents as curve functionals
or, equivalently, as functions on surface groups, without reference to
measures or flows.
As applications, we obtain new axiomatic characterizations of measured
laminations and hyperbolic length functions, and new descriptions of small
surface group actions on real trees, including a concise proof of a
classical theorem of Skora. We also provide a unified framework for
dual geodesic currents arising from metric structures and generalized
cross-ratios, including those associated with certain Anosov
representations. Our approach subsumes all previously
known constructions of dual geodesic currents and yields broad new
families of examples.
This is joint work with Dídac Martínez-Granado and is available at
https://arxiv.org/abs/2605.04031 .
Time: 11.30-12
Speaker: Meenakshy Jyothis
Title: Title: Geodesic currents of coarse negative curvature
Abstract: To every geodesic current on a hyperbolic surface, one can associate a dual pseudometric on the hyperbolic plane, which is always Gromov hyperbolic. We show that a dense subset of geodesic currents satisfies the stronger property of strong hyperbolicity in the sense of Nica–Špakula. These dense families of currents are constructed using finite covers and transfer maps applied to linear combinations of Liouville currents. This allows us to obtain correlation counting results for a dense family of pairs of filling geodesic currents. This is joint work with Dídac Martínez-Granado available at: https://arxiv.org/abs/2605.14469
Time: 12-12.30
Speaker: Sayantika Mondal
Title: Designer Metrics and the Inf Spectrum of Curves
Abstract: To a filling curve on a surface, one may associate its inf-invariant, defined as the shortest possible length of the curve over all hyperbolic metrics on the surface. The collection of such values forms the inf spectrum, which naturally arises in questions related to counting and distinguishing curves.
In this talk, we discuss the relationship between the inf-invariant and geometric self-intersection number, as well as the geometry of the “designer” metrics realising the minimum length. We focus in particular on whether the inf invariant can distinguish mapping class group orbits of filling curves with the same self-intersection number. Finally, we describe recent and ongoing work on the growth and counting properties of the inf spectrum.
Time: 14-15
Speaker: Jing Tao
Title: Density of Penner-Thurston pseudo-Anosov mapping classes
Abstract: By Nielsen-Thurston Classification, every mapping class of a surface of finite type is one of three types: periodic, reducible or pseudo-Anosov. Pseudo-Anosov maps are precisely those with a representative preserving a pair of transverse measured foliations and they are shown to be generic in the mapping class group due to the work of Maher and Rivin. One of the main ways to construct explicit pseudo-Anosov mapping classes is via the Penner-Thurston construction, and fairly recently, it was shown by Shin-Strenner that not all pseudo-Anosov mapping classes arise from this construction. In this talk, I will discuss how dense/generic the Penner-Thurston pseudo-Anosovs are. This is joint with Justin Malestein and Joshua Pankau.
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Wednesday 27
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Time: 10-11
Speaker: Dragomir Saric
Title: Quadratic differentials and random walks on the dual graph of a pants decomposition
Abstract: Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. Using horizontal foliations of quadratic differentials, we prove that the geodesic flow on X is ergodic if and only if the random walk on the weighted dual graph of the pants decomposition of X is recurrent. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from ergodic to non-ergodic geodesic flows. In addition, we show that rough isometry of surfaces does not preserve the ergodicity of the geodesic flow, while rough isometry of their dual graphs does. This is a joint work with C. Bordenave and X. Dong.
Time: 11.30-12
Speaker: Marie Abadie
Title: Combinatorial approach to understand the Weil-Petersson geometry of Teichmüller space
Abstract: We give a brief overview of some combinatorial methods used to understand the geometry of the moduli space of hyperbolic metrics and its universal cover the Teichmüller space. Next we consider the hexagon graph model to obtain estimate on the Weil-Petersson distance on the Teichmüller space.
Time: 12-12.30
Speaker: Milo Banarse
Title: Counting essential surfaces in three-manifolds
Abstract: We give a short introduction to counting problems in three-manifolds, focussing on the question of counting isotopy classes of essential surfaces in a hyperbolic three-manifold M. Closely related to this question is the space ML(M) of measured laminations in M - an analogue to the space of measured geodesic laminations for a surface.
Mirroring the approach one dimension down, we explore how ML(M) may be used to count essential surfaces. We will discuss the broad strategy and highlight some key differences between dimensions two and three.
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Thursday 28
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Time: 10-11
Speaker: Marie Trin
Title: Large genus asymptotic for frequency of non-simple curves
Abstract: As an application of her counting results Mirzakhani proved that in genus 2 there is 48 times more non-separating than separating simple closed curves. This result has been extended to large genus genus by Delecroix-Goujard-Zograf-Zorich: they prove that in large genus almost every simple closed curves is non-separating. In this talk we will explain what happens if we get interested in curves with self-intersections. What is their frequency? How does it behave asymptotically? Which type of curves are more likely? This a joint work with M.Liu, K.Rafi and J.Souto
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Friday 29
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Time: 10-11
Speaker: Jean-Marc Schlenker
Title: Parameterizing quasifuchsian manifolds by measured laminations and foliations
Abstract:
Let $M$ be a quasifuchsian, or more generally convex co-compact hyperbolic 3-manifold.
We will recall why the ideal boundary of $M$ is equipped with a conformal structure $c$, but also with a measured foliation $f$. At the same time, the boundary of the convex core of $M$ is equipped with a hyperbolic metric $m$ and a measured bending lamination $l$.
We will then explain how $M$ is uniquely determined by $l$ (joint with Bruno Dular), and why, near the Fuchsian locus, $M$ is uniquely determined by $f$ (a recent result of Choudhury--Markovic).
We are grateful for the following funding:
