# News

This term I am on secondment at the CNRS. Second term 2019-2020 i will be in Paris Saclay in the TROPICAL team of INRIA.

# Position

I am presently senior associate professor (Maître de conférence hors classe) at the IMAG of the Université de Montpellier.

If you want to join me, see my address and office phone number above.
You can browse my résumé and my list of publications.

# Research

My main research interest is in Riemannian Geometry.

My present research focus on the following themes

**Hilbert's Geometries**Hilbert geometries are a generalisation of the Hyperbolic space's projective model (also known as Klein's model). They posses a Finsler structure (less smooth than in the original definition) and as such can be seen as length metric spaces. I am particularly interested in those which are hyperbolic in the sens of Gromov, for I think of them as being the Finslerian analogue of the Riemannian spaces of constant non-positive curvature. This point of view is based on my previous research related to the bottom of the spectrum, the volume of ideal triangles and the volume entropy

Concerning the bottom of the spectrum, having obtained a characterisation of its nullity and a uniform upper bound, I would like to prove that this upper bound is only achieved by the Hyperbolic geometry. But I would also like to relate the regularity of the boundary of the convex set with the nullity of the bottom of its spectrum, for instance, by proving or disproving my hypothese that the bottom of the spectrum is zero if and only if there is a polygone in the closure of the orbit of the convex set by the action of the group of homographies.

The volume entropy of these geometry is also a subject I am currently working on. Recently with G. Berck and A. Bernigwe obtained an equivalent of the volume of spheres of large radius for a familly of Hilbert geometries for which the entropy is n-1. The general case is still open for that question. Since the summer of 2015, thanks to N. Tholozan we now know that the volume entropy is bounded from above by $n-1$. In another direction I proved that in dimension 2 and 3 the volume entropy was actually equal to the approximability of the convex set. With Cormac Walsh we ate currently working on a generalisation of that result in higher dimension.

**Spectrum of balls and spheres in non-positive curvature**Mixing a recent result by A. Savo et P. Guerini and an older by Cheng, we find out that the spectrum of large balls in the hyperbolic space converges towards the bottom of the spectrum of the Laplacian of the whole space with a speed inversely proportional to the square of the radius of the ball. This is the same phenomenon I observed during my work on the spectrum of nilmanifolds. Hence I am naturally trying to find an equivalent of the difference between the bottom of the spectrum of the whole space and the spectrum of balls of large radii.

Regarding the case of spheres, they happen to be euclidean spheres in the constant curvature case. Up to a dilation involving the entropy, they all have the same spectrum. Now let us have a look at the universal covering of a negatively curved compact manifold. There the spheres of large radii look almost like the boundary at infinity. Hence I am trying to find an equivalent of the spectrum of large spheres involving the entropy.

To solve these two problems, I am trying to mix a kind of two-scale convergence "à la" G. Allaire and G. Nguetseng with the work of G. Knieper.

**Asymptotic volume of nilmanifolds**In my work on the macroscopic spectrum of nilmanifolds I proved an inequality involving the asymptotic volume of balls of larger radii on the universal covering which characterizes some metrics. However, in the light of the results obtained by D. Burago and S. Ivanov on the asymptotic volume of tori (which I re-proved in the two-dimensional case), I don't expect my inequality to be the best one. I strongly believe that we can get a better inequality which may help characterize the left invariant metrics.

I really believe that this problem is of Sub-Riemannian nature. That is the reason why I am focusing on Sub-Riemannian metrics and not on Riemannian ones.

**Optimal transport**Like many other riemannian geometers I had to go into optimal mass transportation to understand the notions of spaces with Ricci curvature bounded from below introduced by C. Villani, J. Lott and K.T. Sturm and following the work of Y. Brenier, R.J. McCann, D. Cordero-Erausquin, M. Schmuckenschläger. For instance I am trying to see if these definitions apply to the Hilbert Geometries (At the moment I believe that they don't :) ).

# Teaching

- 2019-2020

I am on secondment in CNRS and INRIA year, so no teaching.

# Tex/Latex

During my Phd, if I would sometimes relax playing with xbill, I also initiated myself to the wondrous pleasures of computer typesetting with TeX/LaTeX . You will find here some of my latex macros (in french, alas, did not find the time to translate it, yet).