Starting from a presentation of the many recent applications of Galois theory of functional equations to enumerative combinatorics, we will introduce the Galois theory of (different kinds) of difference equations. We will focus on the point of view of the applications, hence with little emphasis on the technicalities of the domain, but I'm willing to do an hour of « exercises » (i.e. to go a little deeper into the proofs), if a part of the audience is interested.[-]

Let $X_{n}$ be an ensemble of combinatorial structures of size $N$, equipped with a measure. Consider the algorithmic problem of exactly sampling from this measure. When this ensemble has a ‘combinatorial specification, the celebrated Boltzmann sampling algorithm allows to solve this problem with a complexity which is, typically, of order $N(3/2)$. Here, a factor $N$ is inherent to the problem, and implied by the Shannon bound on the average number of required random bits, while the extra factor $N$.[-]

Recently several papers appears on ArXiv, on various topics apparently unrelated such as: spin system observable (T. Helmuth, A. Shapira), Fibonacci polynomials (A. Garsia, G. Ganzberger), fully commutative elements in Coxeter groups (E. Bagno, R. Biagioli, F. Jouhet, Y. Roichman), reciprocity theorem for bounded Dyck paths (J. Cigler, C. Krattenthaler), uniform random spanning tree in graphs (L. Fredes, J.-F. Marckert). In each of these papers the theory of heaps of pieces plays a central role. We propose a walk relating these topics, starting from the well-known loop erased random walk model (LERW), going around the classical bijection between lattice paths and heaps of cycles, and a second less known bijection due to T. Helmuth between lattice paths and heaps of oriented loops, in relation with the Ising model in physics, totally non-backtracking paths and zeta function in graphs. Dyck paths, these two bijections involve heaps of dimers and heaps of segments. A duality between these two kinds of heaps appears in some of the above papers, in relation with orthogonal polynomials and fully commutative elements. If time allows we will finish this excursion with the correspondence between heaps of segments, staircase polygons and q-Bessel functions.[-]

There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is, despite many efforts, an open problem for about $40$ years now. So far, four pairs of statistics that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with $n+3$ pairs of statistics that have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the $(-1)$enumerations of these extended objects with respect to one pair of the $n+3$ pairs of statistics. One important tool of our proof is a multivariate generalization of the operator formula for the number of monotone triangles with prescribed bottom row that generalizes Schur functions. Joint work with Florian Aigner.[-]

In this talk, I will present recent results, obtained in collaboration with Laurent Ménard, about the geometry of spin clusters in Ising-decorated triangulations, and build on previously work obtained in collaboration with Laurent Ménard and Gilles Schaeffer.
In this model, triangulations are sampled together with a spin configuration on their vertices, with a probability biased by their number of monochromatic edges, via a parameter nu. The fact that there exists a combinatorial critical value for this model has been initially established in the physics literature by Kazakov and was rederived by combinatorial methods by Bousquet-Mélou and Schaeffer, and Bouttier, Di Francesco and Guitter.
Here, we give geometric evidence of that this model undergoes a phase transition by studying the volume and perimeter of its monochromatic clusters. In particular, we establish that, when nu is critical or subcritical, the cluster of the root is finite almost surely, and is infinite with positive probability for nu supercritical.[-]

In this talk I will provide a brief and gentle introduction to Witten's conjecture, which predicts that the generating series of certain intersection numbers on the moduli space of curves is a tau function of the KdV integrable hierarchy, as a motivation for r-spin Witten's conjecture that concerns much more complicated geometric objects and specialises to the original conjecture for r=2. The r=2 conjecture was proved for the first time by Kontsevich making use of maps arising from a cubic hermitian matrix model with an external field. Together with R. Belliard, S. Charbonnier and B. Eynard, we studied the combinatorial model that generalises Kontsevich maps to higher r. Making use of some auxiliary models we manage to find a Tutte-like recursion for these maps and to massage it into a topological recursion. We also show a relation between a particular case of our maps and the r-spin intersection numbers, which allows us to prove that these satisfy topological recursion. Finally, I will explain how, in joint work with G. Borot and S. Charbonnier, we relate another specialisation of our models to fully simple maps, and how this identification helps us prove that fully simple maps satisfy topological recursion for the spectral curve in which one exchanges x and y from the spectral curve for ordinary maps. This solved a conjecture from G. Borot and myself from '17.[-]

Several operations of combinatorial surgery can be used to relate maps of a given genus g to maps of genus g' is smaller than g. One of them is the Tutte/Lehman-Walsh decomposition, but more advanced constructions exist in the combinatorial toolbox, based on the Marcus-Schaeffer/ Miermont or the trisection bijections.
At the asymptotic level, these constructions lead to surprising relations between the enumeration of maps of genus g, and the genus 0 Brownian map. I will talk about some fascinating identities and (open) problems resulting from these connections, related to Voronoi diagrams, 'W-functionals', and properties of the ISE measure. Although the motivation comes from 'higher genus', these questions deal with the usual Brownian map as everyone likes it.
This is not very new material, and a (mostly French) part of the audience may have heard these stories one million times. But still I hope it will be interesting to advertise them here. In particular, I do not know if recent 'Liouville-based' approaches have anything to say about all this.[-]

Les chemins du plan confinés dans un quadrant, ou plus généralement dans un cône convexe, ont été beaucoup étudiés ces dernières années, et ont donné lieu à de jolis résultats. Le plus remarquable dit que, pour les chemins à petits pas, la série génératrice est différentiellement finie si et seulement si un certain groupe de transformations rationnelles, construit à partir des pas autorisés, est fini. Les méthodes employées, allant de l'algèbre élémentaire sur les séries formelles à l'analyse complexe, en passant, entre autres, par le calcul formel, sont variées, ce qui participe au charme du sujet. Mais quid des chemins dans un cône non convexe, et, typiquement, des chemins évitant un quadrant ? On étudiera les deux cas les plus naturels (pas NSEO, quadrant négatif ou quadrant Ouest interdit), en esquissant avec optimisme ce que pourrait être une classification pour ce problème.[-]