SoS is co-funded by ANR (ANR-17-CE40-0033) and FNR as a PRCI (Projet de Recherche Collaborative Internationale).
Starting date: April 1st, 2018.
Duration: 4 years.
The central theme of this project is the study of geometric and combinatorial structures related to surfaces and their moduli. Even though they work on common themes, there is a real gap between communities working in geometric topology and computational geometry and SoS aims to create a long lasting bridge between them. Beyond a common interest, techniques from both ends are relevant and the potential gain in perspective from long-term collaborations is truly thrilling.
In particular, SoS aims to extend the scope of computational geometry, a field at the interface between mathematics and computer science that develops algorithms for geometric problems, to a variety of unexplored contexts. During the last two decades, research in computational geometry has gained wide impact through CGAL, the Computational Geometry Algorithms Library. In parallel, the needs for non-Euclidean geometries are arising, e.g., in geometric modeling, neuromathematics, or physics. Our goal is to develop computational geometry for some of these non-Euclidean spaces and make these developments readily available for users in academy and industry.
To reach this aim, SoS will follow an interdisciplinary approach, gathering researchers whose expertise cover a large range of mathematics, algorithms and software. A mathematical study of the objects considered will be performed, together with the design of algorithms when applicable. Algorithms will be analyzed both in theory and in practice after prototype implementations, which will be improved whenever it makes sense to target longer-term integration into CGAL.
Our main objects of study will be Delaunay triangulations and circle patterns on surfaces, polyhedral geometry, and systems of disjoint curves and graphs on surfaces.
Concerning the closely related notions of Delaunay triangulations and circle packings, we intend to study the cases of non-compact surfaces and general hyperbolic surfaces. We will explore the possibility to unify their study and to design algorithms for surfaces equipped with a complex projective structure. We will study isometric embeddings into Euclidean spaces of a cell complex endowed with a compatible metric structure.
In the area of combinatorial structures on surfaces and moduli spaces, we intend to develop efficient algorithms in coordination with a deeper understanding of the core objects. For example, we will consider shortest graphs with given topological properties on a given surface and shortest paths between triangulations. Moreover we also intend to improve the mathematical understanding of the relations between combinatorial structures such as curve, pants and flips graphs and related continuous objects such as Teichmüller and moduli spaces. We have ambitious mathematical goals (such as proving expander type properties for moduli spaces and their combinatorial analogues) as well as plans for software which will allow us to discover new directions of research.