Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g-k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.Comment: 15 pages, 7 figure

Systoles and diameters of hyperbolic surfaces

In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent) upper bound.Comment: 13 pages, 9 figure

Flipping Geometric Triangulations on Hyperbolic Surfaces

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation

Experimental analysis of Delaunay flip algorithms on genus two hyperbolic surfaces

International audienceWe give experimental evidence that the only known upper bound on the diameter of the flip graph of a hyperbolic surface recently proven by Despré, Schlenker, and Teillaud (SoCG'20), is largely overestimated. To this aim, we develop an experimental framework for the storage of triangulations of hyperbolic surfaces and modifications through twists. We show that the computations with algebraic numbers can be overcome, and we propose ways to generate surfaces that are meaningful for the experiments

Experimental analysis of Delaunay flip algorithms on genus two hyperbolic surfaces

Guided by insights on the mapping class group of a surface, we give experimental evidence that the upper bound recently proven on the diameter of the flip graph of a surface by Despré, Schlenker, and Teillaud (SoCG'20) is largely overestimated. To obtain this result, we propose a set of techniques allowing us to actually perform experiments. We solve arithmetic issues by proving a density result on rationally described genus two hyperbolic surfaces, and we rely on a description of surfaces allowing us to propose a data structure on which flips can be efficiently implemented

Sovereign Credit Risk Analysis for Selected Asian and European Countries

We analyze the nature of sovereign credit risk for selected Asian and European countries through a set of sovereign CDS data for an eighty-year period that includes the episode of the 2008-2009 financial crisis. Our principal component analysis results suggest that there is strong commonality in sovereign credit risk across countries after the crisis. The regression tests show that the commonality is linked to both local and global financial and economic variables. Besides, we also notice intriguing differences in the sovereign credit risk behavior of Asian and European countries. Specifically, we find that some variables, including foreign reserve, global stock market, and volatility risk premium, affect the of Asian and European sovereign credit risks in the opposite direction. Further, we assume that the arrival rates of credit events follow a square-root diffusion from which we build our pricing model. The resulting model is used to decompose credit spreads into risk premium and credit-event components

A structural and algorithmic study of combinatorial maps and their curves.

Dans cette thèse, nous nous intéressons aux propriétés topologiques des surfaces, i.e. celles qui sont préservées par des déformations continues. Intuitivement, ces propriétés peuvent être imaginées comme étant celles qui décrivent le forme générale des surfaces. Nous utilisons des cartes combinatoires pour décrire les surfaces. Elles ont le double avantage d'être de naturels objets mathématiques et de pouvoir être transformées naturellement en structure de données.Nous étudions trois problèmes différents. Premièrement, nous donnons des algorithmes pour calculer le nombre géométrique d'intersection de courbes dessinées sur des surfaces. Nous avons obtenu un algorithm quadratique pour calculer le nombre minimal d'auto-intersections dans une classe d'homotopie, un algorithme quartique pour construire un représentant minimal et un algorithme quasi-linéaire pour décider si une classe d'homotopie contient une courbe simple. Ensuite, nous donnons des contre-exemples à une conjecture de Mohar et Thomassen au sujet de l'existence de cycles de partage dans les triangulations. Finalement, nous utilisons les travaux récents de Lévèque et Gonçalves à propos des bois de Schnyder toriques pour construire une bijection entre les triangulations du tore et certaines cartes unicellulaires analogue à le célèbre bijection de Poulalhon et Schaeffer pour les triangulations planaires.Plusieurs points de vue sont utilisés au cours de cette thèse. Nous proposons donc un important chapitre préliminaire où nous insistons sur les connections entre ces différents points de vue.In this thesis, we focus on the topological properties of surfaces, i.e. those that are preserved by continuous deformations. Intuitively, it can be understood as the properties that describe the general shape of surfaces. We describe surfaces as combinatorial maps. They have the double advantage of being well defined mathematical objects and of being straightforwardly transformed into data-structures.We study three distinct problems. Firstly, we give algorihtms to compute geometric intersection numbers of curves on surfaces. We obtain a quadratic algorithm to compute the minimal number of self-intersections in a homotopy class, a quartic one to construct a minimal representative and a quasi-linear one to decide if a homotopy class contains a simple curve. Secondly, we give counter-examples to a conjecture of Mohar and Thomassen about the existence of splitting cycles in triangulations. Finally, we use the recent work of Gonçalves and Lévèque about toiroidal Schnyder woods to describe a bijection between toroidal triangulations and toroidal unicellular maps analogous to the well known bijection of Poulalhon and Schaeffer for planar triangulations.Many different points of view are involved in this thesis. We thus propose a large preliminary chapter where we provide connections between the different viewpoints