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.[-]

In this talk I will discuss a bijection between the moduli space of genus-0 hyperbolic surfaces with a distinguished cusp and certain labeled trees, analogous to known tree bijections in the combinatorics of planar maps. The Weil-Petersson measure on the moduli space takes a simple form at the level of the trees, and gives a bijective interpretation to the coefficients in the Weil-Petersson volume polynomials. The labels on the trees give precise information about geodesic distances in the surface, which can be used to study the geometry of random hyperbolic surfaces sampled from the Weil-Petersson measure. In particular, the random genus-0 hyperbolic surface with $n$ cusps is shown to converge as a metric space, after rescaling by $n^{-1/4}$, to the Brownian sphere.This talk is based on work with Nicolas Curien and with Thomas Meeusen and Bart Zonneveld.[-]

Let $G$ be an infinite locally finite and transitive graph. We investigate the relation between supercritical transient branching random walk (BRW) and the Martin boundary of its underlying random walk. We show results regarding the typical (and some atypical) asymptotic directions taken by the particles. We focus on the behavior of BRW inside given subgraphs by putting into relation geometrical properties of the subgraph itself and the behavior of BRW on it. We will also present some examples and counter examples. (Based on joint works with T. Hutchcroft,D. Bertacchi and F. Zucca.)[-]

I will announce the proof, with Thomas Budzinski and Baptiste Louf, of the following fact: a uniformly random triangulation of size n whose genus grows linearly with $n$, has diameter $O(log(n))$ with high probability. The proof is based on isoperimetric inequalities built from enumerative estimates strongly built on the (celebrated) previous work of my two coauthors.
But before this, I will try to review a little bit the questions surrounding random maps on surfaces, in either fixed genus or high genus.[-]

Mirzakhani's recursion for Weil-Petersson volumes was shown by Eynard and Orantin to be equivalent to Topological Recursion with a specific choice of spectral curve. However, such a recursion is known to produce formal power series with factorially growing coefficient which, according to the theory of Resurgence, should be upgraded to “transseries” via the computation of non-perturbative contributions (i.e. instantons). In this talk I will show how a non-perturbative formulation of Topological Recursion allows for the computation of such contributions which, through simple resurgent relations, allow to obtain large genus asymptotics of Weil-Petersson volumes.[-]

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.[-]

The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any planar drawing of a graph in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to other questions about arcs and curves on surfaces. Here is one: how many crossings do a collection of m homotopically distinct curves on a surface of genus $g$ induce? The talk will be about joint work with Alfredo Hubard where we explore some of these, using tools from the hyperbolic geometry of surfaces in the process.[-]

We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum - the set of lengths of all closed geodesics - of a 3-manifold constructed under this model.[-]

Consider a rooted tree whose vertices will be interpreted as free parking spots, each spot accommodating at most one car. On top of that tree, we consider a non-negative integer labeling representing the number of cars arriving on each vertex. Each car tries to park on its arrival vertex, and if the spot is occupied, it travels downwards in direction of the root of the tree until it finds an empty vertex to park. If there is no such vertex on the path towards the root, the car exits the tree, contributing to the flux of cars at the root. This models undergoes an interesting phase transition which we will analyze in detail. After an overview of the case where the underlying tree is a critical Bienayme-Galton-Watson tree, we will concentrate on the case where the underlying tree is the in finite binary tree, where the phase transition turns out to be "discontinuous".
The talk is based on a joint work with David Aldous, Nicolas Curien and Olivier Hénard.[-]

Integrals on the space U(N) of unitary matrices have a large N expansion whose coefficients count factorisations of permutations into "monotone" sequences of transpositions. We will show how this classical story can be adapted to integrals on the complex Grassmannian Gr(M,N), which leads to a 1-parameter deformation of the aforementioned enumeration. The resulting polynomials obey remarkable properties, some known and some conjectural. The notion of topological recursion inspired this work and we will briefly attempt to explain how and why. (This is joint work with Xavier Coulter and Ellena Moskovsky.)[-]