Introduction, par les organisateurs Nous avons retracé la carrière et les travaux d'Emmanuel Guitter dans un exposé introductif à plusieurs voix. - Bertrand Duplantier : Emmanuel, Tintin et Descartes, from hexatic to ecstatic membranes - Philippe Di Francesco : Excursions with Emmanuel into foldings, colorings, loops and paths - Bertrand Eynard : Meandering through random cycles and random colorings with Emmanuel

Introduction, par les organisateurs (suite et surprise) - Jérémie Bouttier : L’aventure des cartes, an ongoing quest with Emmanuel and friends - Raphaël Guitter (surprise talk) : Certifying high-dimensional quantum entanglement using a time-stamping camera

Mark Bowick : Facets of Order We are trained to exploit symmetry but I will discuss examples in which the solution to an analysis problem violates the obvious symmetries of the problem. Particular features are the emergence of non-trivial topology and non-trivial geometry, most notably where sharp structures arise from a smooth-looking starting point.

Mireille Bousquet-Mélou : The 3-state Potts model on planar maps We consider the 3-state Potts generating function $T(\nu,w)$ of planar triangulations; that is, the series in $\nu$ and $w$ counting planar triangulations with vertices coloured in 3 colours, weighted by their size and by the number of monochromatic edges (variable $\nu$). This series was proved to be algebraic 15 years ago: this follows from its link with the solution of a discrete differential equation (DDE), and from general algebraicity results on such equations. However, despite recent progresses on the effective solution of DDEs, the exact value of $T(\nu,w)$ had remained unknown so far. We have determined at last this exact value, proving that $T(\nu,w)$ satisfies a polynomial equation of degree $11$ in $T$. From this we determine the critical value of $\nu$ and the associated exponent. Another approach, applied to the heavier case of general planar maps (still 3-coloured) yields an equation of degree 22. Joint work with Hadrien Notarantonio (IRIF, Paris)

Gilles Schaeffer : On the combinatorics of one variable catalytic equations Functional equations with divided differences, aka catalytic equations, or discrete differential equations, play a central role in map enumeration, but many such equations also appear in other enumeration problems. I will discuss how technics of bijective combinatorics that have been developed for maps extend to deal with some generic equations. Joint work with Enrica Duchi.

Charlotte Kristjansen : Quantum Quenches from Quantum Fields An integrable spin chain underlies the celebrated Maldacena duality. Overlaps between matrix product states and Bethe eigenstates of the chain contain information both about correlation functions of the quantum field theory entering the duality and about the behavior of the spin chain after a quantum quench. I will explain how such overlaps can be calculated exactly.

Grégory Miermont : 25 ans après, la bijection continue (The continuum Schaeffer - BDG bijection) Il ne sera sans doute pas besoin de convaincre le public de cette conférence, et en particulier son invité d'honneur, du fait que les bijections sont des outils d'une grande puissance pour étudier les propriétés fines des cartes. Par passage à la limite, il est maintenant bien connu que des espaces métriques aléatoires canoniques tels que la sphère brownienne peuvent être construits à l'aide de versions continues des bijections de Schaeffer et Bouttier-Di Francesco-Guitter. Plus précisément, c'est en général l'application qui à un arbre étiqueté associe une carte dont on définit une version continue, ce qui permet de voir la sphère brownienne comme fonction déterministe du serpent brownien. Ceci pose la question de savoir s'il existe une application mesurable qui à un espace métrique mesuré associe un arbre continu étiqueté, qui est un inverse de cette « bijection continue ». Dans un travail en collaboration avec Omer Angel, Emmanuel Jacob et Brett Kolesnik, nous montrons que c'est effectivement le cas, et que la réponse précise à cette question fait intervenir de façon cruciale l'orientation de la sphère brownienne. There's probably no need to convince the public of this conference, and in particular the guest of honour, that bijections are powerful tools for studying the fine properties of maps. By passing to the limit, it is now well known that canonical random metric spaces like the Brownian sphere can be constructed by using continuous versions of the Schaeffer and Bouttier-Di Francesco-Guitter bijections. More precisely, it is usually the function that associates a map to a labeled tree of which one defines a continuous version, which allows to see the Brownian sphere as a deterministic function of the Brownian snake. This raises the question of whether there exists a measurable function which, to a measured metric space, associates a continuous labeled tree, and which is an inverse of this “continuous bijection”. In this work with Omer Angel, Emmanuel Jacob and Brett Kolesnik, we show that this is indeed the case, and that the precise answer to this question crucially involves the orientation of the Brownian sphere.

Timothy Budd : A bijection between rigid and integer-labeled quadrangulations In this talk I will introduce the combinatorial class of rigid quadrangulations, which form a subclass of flat quadrangulations of the disk, meaning that all non-boundary vertices are of degree 4. Rigid quadrangulations are shown to be in bijection with certain integer-labeled quadrangulations of the sphere, that were enumerated recently by Bousquet-Mélou and Elvey Price. The bijection relates several natural statistics on one side to equally natural, but rather different, statistics on the other. Finally, I will touch upon the question of scaling limits of large rigid quadrangulations and similar models of random flat metrics on the disk, and their physics motivation.

François David : Quantum Walks on Random Combs The continuous time quantum walk on an infinite comb with infinite teeth has been shown to differ from the classical random walk on the same object. I will show that introducing randomness in the geometry of the comb leads to interesting effects and new questions. Some can be related to the standard theory of localization, and some are specific to the infinite comb. Based on joint work (some in progress) with T. Jonsson (U. of Iceland)

Jason Miller : The scaling limit of the intrinsic metric and simple random walk on 2D critical percolation clusters We show that the CLE(κ) gasket for each κ in (4,8), the range of κ values where the loops can hit each other, themselves, and the domain boundary, can be equipped with: 1) A canonical intrinsic metric and 2) A canonical "Brownian motion" (a continuous Markov process living in the gasket). We also consider critical percolation on the triangular lattice T and show that: 1) The shortest path distance and 2) The simple random walk on large clusters jointly converge in the scaling limit to our continuum metric and Brownian motion on CLE(6). Based on joint works with Valeria Ambrosio, Irina Dankovic, Maarten Markering, and Yizheng Yuan.

Nicolas Curien : On the uniqueness of the infinite noodle When trying to understand the wonderful (and still unproved) conjectures of Emmanuel and co-authors on the enumeration of meanders, we design a much much (much) simpler question which is still open: Is there an infinite cluster in the gluing of two independent uniform noncrossing matchings? I will discuss the partial progresses we made and discuss another related problem on loop model on random quadrangulations. Based on a joint work with G. Kozma, L. Tournier, and V. Sidoravicius.

Enzo Orlandini : Combinatorics and topological weights of chromatin loop networks The 3D folding of mammalian DNA (chromatin) is tightly linked to its transcriptional activity, hence, understanding constitutes an important goal in biophysics. In this talk I will present a polymer model to study the 3D folding of a chromatin segment. By using combinatorial arguments, one can enumerate the emergent chromatin loop networks, both in the case where transcription factors are labeled and where they are unlabeled. These mathematical results, once combined with those of computer simulations, show that networks featuring local loops are statistically more likely with respect to networks including more nonlocal contacts. This surprising result, which can be rationalised by analytically computing the Boltzmann weight of different loop networks, lays down some basic rules to understand the topological alphabet of chromatin folding in mammalian genomes.

Éric Fusy : Schnyder orientations for d-irreducible maps I will review a bijective approach for planar maps developed in joint work with Olivier Bernardi. It relies on certain orientations and draws its inspiration from the Bouttier Di Francesco Guitter bijection between maps and mobiles, and from a bijection by Bernardi for tree-rooted maps. We have applied this method to simple triangulations, relying on orientations with vertex outdegrees 3, so-called Schnyder orientations, and more generally to d-angulations of girth d, and to planar maps of girth d with controlled face-degrees. I will also describe how the method can be extended to d-irreducible maps (recent joint work with Olivier Bernardi and Shizhe Liang), which have been previously counted bijectively by Bouttier and Guitter via slice decompositions.

Sofia Tarricone : Distance dans les cartes planaires et polynômes orthogonaux Dans cet exposé nous revisitons des propriétés de la fonction à deux points pour les cartes planaires avec des faces de degrés bornés. En particulier, nous verrons comment la relier à une famille de polynômes orthogonaux et comment cela permet de donner des preuves analytiques alternatives de ses propriétés intégrables, comme son écriture déterminantale et les équations discrètes qu'elle satisfait, qui avaient été prouvés originellement via des moyens combinatoires par Emmanuel Guitter et ses collaborateurs Jérémie Bouttier et Philippe Di Francesco. Basé sur un travail en cours avec Jérémie Bouttier.

Jean-François Le Gall : Local times of Brownian motion indexed by the Brownian tree, and volumes of spheres in the Brownian plane We discuss local times for the model called Brownian motion indexed by the Brownian tree, which is the building block of the scaling limits of random planar maps known as the Brownian sphere and the Brownian plane. As an application, we prove that the process of volumes of spheres in the Brownian plane has a continuous derivative, and moreover the pair consisting of this process and its derivative is Markovian and satisfies an explicit stochastic differential equation whose coefficients involve the classical Airy function. This is based in part on a joint work with Ed Perkins (UBC)