ConforLux

Overview

ConforLux (Computational Conformal Geometry) is a collaborative research initiative uniting the GAMBLE project-team (Inria, France) and the research group of Jean-Marc Schlenker (University of Luxembourg).

The project operates at the frontier of theoretical mathematics and computer science, focusing on Geometric Topology and Algorithmic Geometry. The collaboration aims to leverage complementary expertise: the mathematical depth of the Luxembourg group regarding complex structures on surfaces, and the algorithmic expertise of the GAMBLE team in designing data structures and implementations for the CGAL library.

Scientific Context

This project requires expertise both in mathematics and computer sciences. The primary objects of study are:

• Circle packings
• Delaunay triangulations
• Projective complex structures on surfaces

While circle packings are well-understood on surfaces with constant curvature (Euclidean or Hyperbolic), this project aims to generalize these concepts to surfaces equipped with a Complex Projective Structure (\(\mathbb{C}P^{1}\)-structure). A \(\mathbb{C}P^{1}\)-structure allows for a well-defined notion of circles and angles, invariant under Möbius transformations.

From a computer science perspective, the goal is to design generic algorithms and data structures capable of handling these geometric objects.

Objectives & Work Program

The collaboration aims to produce theoretical results as well as practical software tools.

• Conjecture Exploration: We intend to computationally explore the conjecture by Kojima et al. regarding the realization of triangulations as incidence graphs of circle packings on complex projective structures.
• Proper Delaunay Decompositions: Determine sufficient conditions under which a \(\mathbb{C}P^{1}\)-surface and a set of points define a "proper" Delaunay decomposition (where empty disks are compact).
• Algorithm Extension: Extend the existing "flip algorithm" (analyzed in [DST20] for Fuchsian structures) to arbitrary \(\mathbb{C}P^{1}\)-structures.
• CGAL Integration: Algorithms will be implemented and integrated into the CGAL (Computational Geometry Algorithms Library).

The Team

History & Funding

The partnership between these groups dates back to 2016 with the Antonishing Associate Team, followed by the ANR SOS project in 2018. Previous joint successes include analyzing flip algorithms for hyperbolic surfaces and computing Dirichlet domains.

Funding Support:

• LUE (Lorraine Université d'Excellence): Funds a PhD position dedicated to this collaboration.
• GRACE Project: Funded by FNR (Luxembourg National Research Fund), supporting the training of the cohort in Luxembourg.