A discrete isosystolic inequality for flat tori

Supervision and Contact: Vincent Despré and Monique Teillaud
Location: Gamble group, INRIA Nancy - Grand Est, LORIA

The work may extend to a PhD on a related topic.

Context

A flat torus is a genus-1 surface endowed with a metric that is everywhere flat, where flat means that the Gaussian curvature is null. The systole of a surface is its smallest non-contractible curve. For a given fixed area, the optimal surface, i.e., the surface with the longest systole, is of great interest [Gro]. In the case of flat tori of fixed area, the optimal surface is the one defined by a regular hexagon with opposite sides identified.
We are interested in the discrete setting and we consider triangulated tori. In this setting, the systole is the smallest (in terms of its number of edges) non-contractible closed path in the triangulation. The problem becomes as follows: Which triangulation of the flat torus with n triangles has the longest systole? Ultimately, it would be very interesting to know if the only limit metric corresponding to a sequence of optimal triangulations , when n tends to infinity, is the continuous metric of the optimal flat torus.

Goal of the internship

The student will have to compute the optimal triangulations for small values of n. The idea is to use a computer approach: enumerate all possible triangulations and compute the systole of each one. To this aim, (s)he will need to understand the algorithms to compute all the triangulations of a torus with a fixed number of vertices [DGL] and to compute discrete systoles [EHP]. A formal proof may be imagined but being able to avoid a computation, at least to get a correct intuition, seems really unlikely.

Required skills

  • Good understanding of topology and geometry.
  • Understanding of algorithmic problems.
  • Basic programming skills.

References

[DGL] Vincent Despré, Daniel Gonçalves and Benjamin Lévèque. Encoding Toroidal Triangulations. Discrete & Computational Geometry, 2017.
[EHP] Jeff Erickson and Sariel Har-Peled. Optimally cutting a surface into a disk. Discrete & Computational Geometry, 2004.
[GRO] Mikhael Gromov. Filling Riemannian Manifolds. Journal of Differential Geometry, 1983.