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PhD student-Triangulations of hyperbolic manifolds, France


Established in 1967, Inria is the only public research body fully dedicated to computational sciences.
Combining computer sciences with mathematics, Inria’s 3,500 researchers strive to invent the digital technologies of the future. Educated at leading international universities, they creatively integrate basic research with applied research and dedicate themselves to solving real problems, collaborating with the main players in public and private research in France and abroad and transferring the fruits of their work to innovative companies.
The researchers at Inria published over 4,500 articles in 2013. They are behind over 300 active patents and 120 start-ups. The 172 project teams are distributed in eight research centers located throughout France.

Scientific Context :

Computational geometry emerged in the 70’s and is now a well established field. While research in computational geometry was essentially theoretical for about 20 years, it has progressivey gained  a wide impact both in academia and in industry through CGAL, the Computational Geometry Algorithms Library.

Delaunay triangulations in the Euclidean space Rd have been extensively studied. Their mathematical properties are well understood, many algorithms to construct them have been proposed and analyzed. CGAL contains packages to efficiently compute them.

Needs for non-Euclidean geometries are arising, e.g., in geometric modeling, neuromathematics, physics…

An algorithm was proposed for closed flat manifolds, i.e. compact quotient spaces of the Euclidean space for a discrete group of isometries. This direction was initially motivated by needs for simulations or modeling of periodic structures. A CGAL package was developed for the special case of the 3D flat torus with cubic domain, which has already found users in various fields including particle physics and material engineering. This work led to a fruitful collaboration with astrophysicists.

It was proved that Delaunay complexes in the hyperbolic space Hd  can be deduced from their Euclidean counterparts [2].

Little is known about Delaunay triangulations of  closed orientable hyperbolic surfaces, i.e. quotient spaces  of H2 by some group of hyperbolic isometries. This raises new challenges, in particular since hyperbolic isometries do not commute. First attempts have been made to extend the above-mentioned studies to the simplest possible case of the Bolza surface, homeomorphic to a torus having two handles [3].

 The mathematical study of Delaunay triangulations in hyperbolic manifolds will be pursued. Algorithms to construct them will be designed. The algorithms will be analyzed both in theory and in practice after prototype implementations. Implementations will be improved to target longer-term integrations into CGAL, whenever prototypes show that the approach is promising.

Profil recherché

Skills and profile :

                Required qualification: Master in mathematics or computer science

                Required skills and knowledge:

    Algorithmics, hyperbolic geometry, computational geometry
    C++ (knowledge of CGAL will be a plus)
    Good communication and writing skills
    Good English (French is welcome but not required)


Help and benefits :

-          Possibility of free French courses

-          Help for finding housing

-          Help for the resident card procedure and for husband/wife visa

-          Lunch cost at Inria is 2,72 €
Informations complémentaires

- Bibliography :

[1] Manuel Caroli and Monique Teillaud, Delaunay Triangulations of Point Sets in Closed Euclidean d-manifolds, Proceedings 27th Annual Symposium on Computational Geometry, 2011, pp 274-282,

[2] Mikhail Bogdanov and Olivier Devillers and Monique Teillaud, Hyperbolic Delaunay complexes and Voronoi diagrams made practical, Proceedings 29th Annual Symposium on Computational Geometry, 2013, pp 67-76,

[3] Mikhail Bogdanov and Monique Teillaud, Delaunay triangulations and cycles on closed hyperbolic surfaces, Research Report INRIA No 8434, 2013,

More references are available at

-  Additional information :

Supervision and contact : Monique Teillaud

Additional links: CGAL, the Computational Geometry Algorithms Library,

- Duration : 3 years

- Starting date : between Oct. 1st 2015 and Jan. 1st 2016
- Salary: 1 958 euros gross monthly (about 1 580 euros net) during the first and the second years. 2 059 euros the last year (about 1 661 euros net). Medical insurance is included.

The required documents for applying are the following :

- CV;

- a motivation letter;

- your degree certificates and transcripts for Bachelor and Master (or the last 5 years if not applicable).

- Master thesis (or equivalent) if it is already completed, or a description of the work in progress, otherwise;

- all your publications, if any (it is not expected that you have any).

- At least one recommendation letter from the person who supervises(d) your Master thesis (or research project or internship); you can also send at most two other recommendation letters.

The recommendation letter(s) should be sent directly by their author to the prospective PhD advisor.

All the documents should be sent in at most 2 pdf files; one file should contain the publications, if any, the other file should contain all the other documents. These two files should be sent to your prospective PhD advisor (in addition to the application on the web).

Further Information

Application Deadline : 5 April 2015

PhD student-Triangulations of hyperbolic manifolds : PDF

Contact Email:

Posted on 2015-03-28 01:53:50

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