Offre de thèse
DISCRETE HYPERBOLIC LAPLACIAN
Date limite de candidature
10-07-2026
Date de début de contrat
01-10-2026
Directeur de thèse
POUGET Marc
Encadrement
This Phd is offered within the framework of the associated team Confor- lux, which aims to develop new tools for studying low-dimensional conformal structures from a discrete perspective. As an interdisciplinary collaboration between mathematics and computer science, we regard mathematical and algorithmic results as equally valuable contributions. The PhD will be primarily based in Nancy (at least for the first 3 years), at LORIA, with regular visits to the Department of Mathematics at the University of Luxembourg. These visits will allow the candidate to interact with researchers from both institutions and to benefit from a stimulating cross-disciplinary environment. The Conforlux project will fund the first 3 years via the University of Lorraine (ISITE - IRP) and the 4th year of the PhD via the Luxembourg partner. This PhD project will be a “co-tutelle” between the two universities. Jean-Marc Schlenker will be the advisor and Wayne Lam the co-advisor on the Luxembourg site and Vincent Despr´e will be the advisor and Marc Pouget the co-advisor on the Lorraine site.
Type de contrat
école doctorale
équipe
GAMBLEcontexte
The goal of this PhD is to study geometric graph Laplacians in hyperbolic and, more generally, non-Euclidean geometries. Such discrete Laplacians are expected to serve as effective approximations of the continuous Laplace- Beltrami operator on hyperbolic surfaces and to shed light on the interplay between discrete and smooth geometry.spécialité
Informatiquelaboratoire
LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Mots clés
surface hyperbolique, géométrie algorithmique
Détail de l'offre
A Riemann surface is a complex manifold of dimension 1. In other words,
it is a manifold that admits an atlas whose transition maps are biholomor-
phisms. In the simply connected case, the Riemann uniformization theorem
tells us that there are only three distinct surfaces up to conformal deforma-
tion: the sphere, the Euclidean plane, and the hyperbolic plane. The case
that we focus on here is the case of hyperbolic surfaces that can be seen as
quotient of the hyperbolic plane [Bus92].
The Laplace operator on a Riemannian surface is a fundamental analytic
tool that encodes rich geometric and spectral information about the surface.
Inspired by electrical networks, Laplacians on graphs are defined using edge
weights that represent conductances. When the edge weights are uniform,
one obtains the combinatorial Laplacian, which captures deep combinatorial
properties of the graph.
For graphs embedded on surfaces, it is natural to define a geometric Lapla-
cian whose edge weights reflect the underlying geometry [IL25]. In the Eu-
clidean setting, the celebrated cotangent formula expresses the edge weights
of a geodesic triangulation in terms of the local metric, providing discrete
analogues of classical differential-geometric structures. This discrete Lapla-
cian has proven to be a powerful tool, closely related to problems such as the
deformation theory of circle patterns and discrete harmonic maps [Bob16].
The goal of this PhD is to study geometric graph Laplacians in hyperbolic
and, more generally, non-Euclidean geometries. Such discrete Laplacians
are expected to serve as effective approximations of the continuous Laplace-
Beltrami operator on hyperbolic surfaces and to shed light on the interplay
between discrete and smooth geometry.
Keywords
hyperbolic surface, computational geometry
Subject details
A Riemann surface is a complex manifold of dimension 1. In other words, it is a manifold that admits an atlas whose transition maps are biholomor- phisms. In the simply connected case, the Riemann uniformization theorem tells us that there are only three distinct surfaces up to conformal deforma- tion: the sphere, the Euclidean plane, and the hyperbolic plane. The case that we focus on here is the case of hyperbolic surfaces that can be seen as quotient of the hyperbolic plane [Bus92]. The Laplace operator on a Riemannian surface is a fundamental analytic tool that encodes rich geometric and spectral information about the surface. Inspired by electrical networks, Laplacians on graphs are defined using edge weights that represent conductances. When the edge weights are uniform, one obtains the combinatorial Laplacian, which captures deep combinatorial properties of the graph. For graphs embedded on surfaces, it is natural to define a geometric Lapla- cian whose edge weights reflect the underlying geometry [IL25]. In the Eu- clidean setting, the celebrated cotangent formula expresses the edge weights of a geodesic triangulation in terms of the local metric, providing discrete analogues of classical differential-geometric structures. This discrete Lapla- cian has proven to be a powerful tool, closely related to problems such as the deformation theory of circle patterns and discrete harmonic maps [Bob16]. The goal of this PhD is to study geometric graph Laplacians in hyperbolic and, more generally, non-Euclidean geometries. Such discrete Laplacians are expected to serve as effective approximations of the continuous Laplace- Beltrami operator on hyperbolic surfaces and to shed light on the interplay between discrete and smooth geometry.
Profil du candidat
The candidate should have an academic background at the Master's level
(M2) in mathematics or theoretical computer science. A solid foundation
in geometry, complex analysis, or differential geometry is highly desirable
along with an interest in bridging theoretical mathematics with algorithmic
and computational approaches.
Candidate profile
The candidate should have an academic background at the Master's level
(M2) in mathematics or theoretical computer science. A solid foundation
in geometry, complex analysis, or differential geometry is highly desirable
along with an interest in bridging theoretical mathematics with algorithmic
and computational approaches.
Référence biblio
[Bob16] Alexander I Bobenko. Advances in discrete differential geometry. Springer Nature,
2016.
[Bus92] Peter Buser. Geometry and Spectra of Compact Riemann Surfaces. Birkh¨auser,
Boston, 1992.
[IL25] Ivan Izmestiev and Wai Yeung Lam. Discrete laplacians—spherical and hyper-
bolic. Journal of the London Mathematical Society, 112(1):e70235, 2025.