2019-01513 - Post-Doctoral Research Visit F/M Delaunay triangulations of hyperbolic (2, 4, 6)-surfaces [s]
Le descriptif de l’offre ci-dessous est en Anglais

Type de contrat : CDD de la fonction publique

Niveau de diplôme exigé : Thèse ou équivalent

Fonction : Post-Doctorant

Contexte et atouts du poste

Team

Gamble, INRIA Nancy - Grand Est, Loria http://gamble.loria.fr

Contacts

Vincent Despré, Vincent.Despre@inria.fr
Monique Teillaud, Monique.Teillaud@inria.fr

http://gamble.loria.fr/members.html

 

Mission confiée

Context

Tesselations of space from repeating motifs have a long and involved history in mathemat- ics, art, engineering, and natural sciences. The mathematical literature has traditionally focused on patterns in Euclidean spaces but the role of hyperbolic geometry in the natural sciences is increasingly recognized. From a theoretical point of view, discrete differential geometry, a field that provides a basis for a theoretical analysis of discrete objects that recovers the rich theory of their smooth counterparts, treats discrete surfaces as triangular meshes. It is desirable to assume that the triangulation is Delaunay, as many things, such as the discrete Laplacian, become much simpler to compute and to interpret. As a result, Delaunay triangulations have become a central tool in the theoretical study of discrete surfaces.

Delaunay triangulations in Euclidean space have been extensively studied throughout the 20th century and are still a very active research topic. Their mathematical and algo- rithmic properties are by now well understood (see e.g., a survey book [AKL13]). Consid- ering their importance, it is surprising how little is known about geometric algorithms to robustly compute Delaunay triangulations for hyperbolic surfaces, even relatively simple ones. Recent progress has lead to an algorithm for one of the simplest hyperbolic surfaces, the Bolza surface, which is related to the triangle group (2, 3, 8) [BTV16, IT17]. However, the extension to general hyperbolic surfaces remains elusive.

The Computational Geometry Algorithms Library CGAL, developed within an Open Source Project, has facilitated the use of geometric algorithms in applications in an un- precedented way. The library offers several triangulation packages, including the recent package for the Bolza surface [IT19].

Principales activités

Objectives

Hyperbolic surfaces exhibiting the symmetries of the (2, 4, 6) triangle group are ubiquitous in nature, where they appear as triply-periodic minimal surfaces [ES15]. The aim of this project is to propose and analyze effective algorithms to compute triangulations of such surfaces. Fundamental questions must be addressed to establish the mathematical foundations used to describe triangulations in these spaces. To ensure that our results are as relevant and applicable as they can be, we will focus on recent and active research problems in the natural sciences as test cases for our theory. We plan to make the designed algorithms available for practical use as a new CGAL package.

References

[AKL13] Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and De- launay Triangulations. World Scientific, 2013.

[BTV16] Mikhail Bogdanov, Monique Teillaud, and Gert Vegter. Delaunay triangulations on orientable surfaces of low genus. In Proceedings of the Thirty-second Interna- tional Symposium on Computational Geometry, pages 20:1–20:17, 2016.

[ES15] Myfanwy E. Evans and Gerd E. Schro ̈der-Turk. In a material world: Hyperbolic geometry in biological materials. Asia Pacific Mathematics Newsletter, 5(2):21– 30, 2015.

[IT17] Iordan Iordanov and Monique Teillaud. Implementing Delaunay triangulations of the Bolza surface. In Proceedings of the Thirty-third International Symposium on Computational Geometry, pages 44:1–44:15, 2017.

[IT19] Iordan Iordanov and Monique Teillaud. 2D periodic hyperbolic triangulations. In CGAL User and Reference Manual. CGAL Editorial Board, 4.14 edition, 2019.

Compétences

Required qualifications

PhD thesis in Mathematics or Computer Science, and strong interest for the other discipline

Language

English is requested

Avantages

  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities
  • Access to vocational training
  • Social security coverage

Rémunération

Salary: 2653€ gross/month