Bijections between planar maps and tree-like structures have been proven to play a crucial role in understanding the geometry of large random planar maps. Perhaps the most popular (and useful) bijections fit into two categories: bijections between maps and labeled trees and bijections between maps and blossoming trees. They were popularized in the late nineties in the important contribution of Schaeffer and they have been widely developed since then. It is natural to ask whether these bijections still hold when the underlying surface is no longer the sphere but any two-dimensional compact manifold? In this case trees are replaced by maps on a given surface with only one face and while the construction of Schaefer of the labeled-type bijection works independently on genus (but crucially depending on the assumption of orientability) his construction of the blossoming-type bijection was known only in the planar case. We will discuss a (recent?) development of these bijections that extends them to all compact two-dimensional manifolds. I will quickly review my previous joint work with Chapuy and its extension due to Bettinelli which treats the labeled-type bijection and will focus on a more recent work joint with Lepoutre which extends the blossoming-type bijection to non-oriented surfaces.[-]

In this talk, we consider two models of random geometry of surfaces in high genus: combinatorial maps and hyperbolic surfaces. The geometric properties of random hyperbolic surfaces under the Weil–Petersson measure as the genus tends to infinity have been studied for around 15 years now, building notably on enumerative results of Mirzakhani. On the other hand, random combinatorial maps were first studied in the planar/finite genus case, and then in the high genus regime starting 10 years ago with unicellular (i.e. one-faced) maps. In a joint work with Svante Janson, we noticed some numerical coincidence regarding the counting of short closed curves in unicellular maps/hyperbolic surfaces in high genus (comparing it to results of Mirzakhani and Petri). This leads us to conjecture some similarities between the two models in the limit, and raises several other open questions.[-]

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.
Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length $O(g^{3/2}n^{1/2})$ for any triangulated combinatorial surface of genus g with n triangles, and describe an $O(gn)$-time algorithm to compute such a decomposition.
Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
systolic geometry - computational topology - topological graph theory - graphs on surfaces - triangulations - random graphs[-]

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner[-]

We analyze random labelled cubic planar graphs according to the uniform distribution. This model was analyzed first by Bodirsky et al. in a paper from 2007. Here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs where incomplete with respect to the singularity analysis and we provide full proofs. Secondly, we obtain new results that considerably strengthen those known before. For instance, we show that the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance, while formerly it was only known that it is linear with high probability.
This is based on a joint work with Marc Noy (UPC) and Clément Requilé (FU Berlin - BMS).[-]

Angel and Schramm ont étudié en 2003 la limite locale des triangulations uniformes. La loi limite, appelée UIPT (pour Uniform Infinite planar Triangulation) a depuis été pas mal étudiée et est plutôt bien comprise. Dans cet exposé, je vais expliquer comment on peut obtenir un résultat analogue à celui d'Angel et Schramm mais lorsque les triangulations ne sont plus uniformes mais distribuées selon un modèle d'Ising. Une partie importante de la preuve consiste à étudier une équation sur des séries génératrices à deux variables catalytiques et repose sur la méthode des invariants de Tutte (introduite par Tutte et popularisée par Bernardi et Bousquet-Mélou). L'objet limite est pour le moment très mal compris et soulève un grand nombre de questions ouvertes ![-]