Recherche
Publications
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with M. Ebbens F. Lazarus I. Yakovlev
"Algorithms for length spectra of combinatorial tori"
J. Comput. Geom. 15 (2024)
Journal zbMATHVoir résuméMasquer résumé
A combinatorial tori is a torus endowed with a metric induced by an embedded graph. We provide algorithms to compute problems related to the length spectra of such tori. Our approach relies on the equivalence between homotopy and homology for tori and the stable norm induced on homology by the metric. -
with M. Bonamy C. Legrand-Duchesne
"Kempe changes in degenerate graphs"
Eur. J. Comb. 119 (2024)
Journal arXiv zbMATHVoir résuméMasquer résumé
We consider Kempe changes on the k-colorings of a graph on n vertices. If the graph is (k-1)-degenerate, then all its k-colorings are equivalent up to Kempe changes. However, the sequence between two k-colorings that arises from the proof may be exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most k-1. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As a main result, we derive that given an n-vertex graph with maximum degree D, the D-colorings are all equivalent up to O(n^2) Kempe changes, unless D=3 and some connected component is a 3-prism. -
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with E. Goujard P. Zograf A. Zorich
"Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves"
Duke Math J. 12 (2021)
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with J. Schmitt J. van Zelm
"admcycles - a Sage package for calculations in the tautological ring of the moduli space of stable curves"
J. Softw. Algebra Geom. (2021)
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with A. Aggarwal E. Goujard P. Zograf A. Zorich
"Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants of strata of quadratic differentials"
Arnold Mathematical Journal 6 (2020) p. 149-161
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with E. Goujard P. Zograf A. Zorich
"Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes"
in Some aspects of the theory of dynamical systems: a tribute to Jean-Christophe Yoccoz. Volume I (S. Crovisier, R. Krikorian, C. Matheus, S. Senti, ed.), Astérisque 415, Société Mathématique de France (2020), p. 223-274
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with C. Matheus C. G. Moreira
"Approximation of the Lagrange and Markov spectra"
Math. Comput. (2020)
Journal arXiv zbMATHVoir résuméMasquer résumé
The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in Hausdorff distance. It also extends to approximate the Markov spectrum related to infimum of binary quadratic forms. -
with E. Goujard P. Zograf A. Zorich
"Enumeration of meanders and Masur-Veech volumes"
Forum Math., Pi (2020)
Journal arXiv zbMATHVoir résuméMasquer résumé
A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with crossings grows exponentially when grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics. -
with A. Avila
"Some monoids of Pisot matrices"
in New trends in one-dimensional dynamics (M. J. Pacifico, P. Guarino, ed.), Springer Proc. Math. Stat. 285, Springer (2019), p. 21-30
Journal arXiv zbMATHVoir résuméMasquer résumé
A matrix norm gives an upper bound on the spectral radius of a matrix. Knowledge on the location of the dominant eigenvector also leads to upper bound of the second eigenvalue. We show how this technique can be used to prove that certain semi-group of matrices arising from continued fractions have a Pisot spectrum: namely for all primitive matrices in this semi-group all eigenvalues except the dominant one is smaller than one in absolute value. -
with J.-F. Bertazzon
"Sommes de Birkhoff itérées sur des extensions finies d'odomètres. Construction de solutions auto-similaires à des équations différentielles avec délai"
Bull. SMF 146 (2018)
Journal arXiv zbMATHVoir résuméMasquer résumé
Nous étudions les sommes de Birkhoff itérées de fonctions sur certains systèmes dynamiques substitutifs. Les fonctions que nous regardons ont la propriété d'avoir toutes leurs sommes de Birkhoff itérées bornées (ce sont en particulier des cobords). Nous construisons une fonction continue comme limite de ces sommes de Birkhoff et montrons qu'elle vérifie une équation fonctionnelle. Cet article prolonge l'étude du premier auteur dans arXiv:1201.2502. -
![From a packing problem to quantitative recurrence in [0,1] and the Lagrange spectrum of interval exchanges](illustrations/l_infinity_delaunay.svg)
with M. Boshernitzan
"From a packing problem to quantitative recurrence in [0,1] and the Lagrange spectrum of interval exchanges"
Discrete Analysis 1749 (2017)
Journal arXiv zbMATHVoir résuméMasquer résumé
This article provides optimal constants for two quantitative recurrence problems. First of all for recurrence of maps of the interval [0,1] that preserve the Lebesgue measure. On the other hand, we study the bottom of the Lagrange spectrum of interval exchange transformations. Both results are based on a unconventional packing problem in the plane with respect to the "pseudo-norm" N(x,y) = sqrt(|xy|). -
with V. Berthé F. Dolce D. Perrin C. Reutenauer G. Rindone
"Return words of linear involutions and fundamental groups"
Erg. Th. and Dyn. Sys. 37 , n° 3 (2017) p. 693-715
Journal arXiv zbMATHVoir résuméMasquer résumé
We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet A. The set of return words with respect to a subgroup of finite index G of the free group on A is also proved to be a symmetric basis of G -
with A. Avila
"Weak-mixing directions in non-arithmetic Veech surfaces"
J. of AMS 29 (2016) p. 1167-1208
Journal arXiv zbMATHVoir résuméMasquer résumé
In this paper we prove the genericity of weak mixing in non-arithmetic Veech surfaces (arithmetic is synonym for square tiled). We know since the work A. Avila and G. Forni (Ann. of Math. 165 (2007), see also arXiv:math/0406326) that weak-mixing is prevalent in the space of translation surfaces. Nevertheless, there was no known example of surfaces for which the weak-mixing is prevalent in almost every direction. Our result applies in particular to billiard in regular polygons studied by Veech (Inv. Math. 97 (1989)) and billiard in L-shaped tables introduced by C. McMullen (J. Amer. Math. Soc. 16 (2003)). -
with C. Matheus
"Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich"
Math Res. Lett. 22 , n° 6 (2015) p. 1667-1678
Journal arXiv zbMATHVoir résuméMasquer résumé
Forni proved that a certain geometric quantity gives a lower bound for the number of positive Lyapunov exponents of the Kontsevich-Zorich cocycle (J. Mod. Dyn. 5, No. 2 (2011), see also arXiv:1009.4655). In this short note, we exhibit an example for which the geometric criterion is not satisfied but for which all Lyapunov exponents are positive. In order to prove positivity, we use a result of C. Matheus, M. Moeller and J.-C. Yoccoz (arXiv:1305.2033) -
with C. Ulcigrai
"Diagonal changes in hyperelliptic components. A natural extension to Ferenczi-Zamboni induction"
Geom. Ded. 176 , n° 1 (2015) p. 117-174
Journal arXiv zbMATHVoir résuméMasquer résumé
In this article, we introduce an induction scheme for translation surfaces in hyperelliptic strata. It can be considered as a geometric counterpart to Ferenczi-Zamboni construction (J. Analyse Math. 112 (2010)) -
with V. Berthé
"Beyond substitutive dynamical systems: S-adic expansions"
RIMS Kôkyûroku Bessatsu B46 (2014) p. 81-123
arXiv zbMATHVoir résuméMasquer résumé
Self-similar dynamical systems are example of highly structured dynamical systems. But there are much more systems of low complexity. One way to consider all of them is to slightly weaken the notion of self-similarity: we allow to see different patterns at different scales but all of them belong to a fixed family. The combinatorial counterpart of this construction are the so-called S-adic systems. -
with P. Hubert S. Lelièvre
"Diffusion for the periodic wind-tree model"
Ann. Sc. ENS 47 , n° 6 (2014) p. 1085-1110
Journal arXiv zbMATHVoir résuméMasquer résumé
The wind-tree model is a billiard in the plane where scatterers are rectangles randomly displaced. We study a periodic version and prove that the diffusion rate is 2/3. More precisely the maximum distance reached by a particule before time T is around T^(2/3). It makes a large difference with random walks in the plane for which that quantity equals T^(1/2). -
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"Cardinality of Rauzy classes"
Ann. Inst. Fourier 63 , n° 5 (2013) p. 1651-1715
Journal arXiv zbMATHVoir résuméMasquer résumé
Rauzy classes are set of permutations that appear in a renormalization scheme of interval exchange transformations introduced by Rauzy and further studied by Veech. In this article we provide a formula for the cardinalities of Rauzy classes and make a conjecture about their asymptotics.
Prépublications
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with O. Fontaine A. de Mesmay
"On the size of k-irreducible triangulations"
arXivVoir résuméMasquer résumé
A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of a surface of genus g has O(k^2 g) triangles, which is optimal. -
with M. Bell V. Gadre Rodolfo Gutiérrez-Romo S. Schleimer
"Coding Teichmüller flow using veering triangulations"
arXivVoir résuméMasquer résumé
We develop the theory of veering triangulations on oriented surfaces adapted to moduli spaces of half-translation surfaces. We use veering triangulations to give a coding of the Teichmüller flow on connected components of strata of quadratic differentials. We prove that this coding, given by a countable shift, has an approximate product structure and a roof function with exponential tails. This makes it conducive to the study of the dynamics of Teichmüller flow. -
"Asymptotics of lieanders with fixed composition sizes"
arXivVoir résuméMasquer résumé
Lieanders are special cases of meanders. We derive a polynomial asymptotics via the equidistribution and uncorrelation results from Delecroix-Goujard-Zograf-Zorich
Articles de conférences
Présentations
- lundi 21 août 2017 (IMA, Minneapolis, USA):
- Conférence: Sage Days: Opening Workshop for a Year of Coding Sprints
- (Page de la présentation avec une vidéo)
- pdf présentation
- feuille de travail sur l'optimisation linéaire ipynb pdf
- feuille de travail sur les fractions continues ipynb pdf
- feuille de travail sur les problèmes avec les expressions symboliques ipynb pdf
Thèse
J'ai soutenu ma thèse sous la direction d'Arnaldo Nogueira le 16 Novembre 2011 présentation au format pdf. Mon mémoire est disponible en version longue (173 pages) et en version courte (57 pages).
Conférences, groupes de travail, etc
- Février 2026: Journées de combinatoire de Bordeaux 2026
- Janvier 2026: Sage days 130 (le Teich)
- Février 2025: Sage days 128 (le Teich)
- Février 2025: Journées de combinatoire de Bordeaux 2025
- Janvier 2024: Sage days 125 (le Teich)
- Février 2023: Sage days 117 (le Teich)
- Janvier 2023: Journées de combinatoire de Bordeaux 2024
- Juin 2022: Journées de combinatoire de Bordeaux 2022
- En 2022-2023, j'ai animé un groupe de travail sur les groupes modulaires de surface
- Février 2021: Journées de combinatoire de Bordeaux 2021
- Organisation des 17e journées montoises d'informatique théorique à Bordeaux en septembre 2018.
- Organisation de l'école MathExp 2018 à Saint-Flour fin mai 2018.
- En 2017-2018, j'ai animé un groupe de travail sur les cartes combinatoires et les géométries des surfaces.
- Au printemps 2014, j'ai animé des séances d'introduction au logiciel Sage à l'université de Bordeaux. Pour plus d'informations voir la page wiki.
- J'ai coorganisé avec Pascal Hubert et Erwan Lanneau un groupe de travail sur les travaux de Benoist-Quint (la page contient quelques notes de cours et des références).