Journées 2022 : du mercredi 29 juin 14h au vendredi 1 juillet 12h

Les journées se tiendront à l'IHP en salle 314 les mercredi et vendredi, et salle 201 le jeudi.

Programme :

  • mercredi 29 juin après-midi :
    • 14h-15h : Guillaume Chapuy : On the number of coloured triangulations of d-manifolds
    • 15h15-16h15 : Bram Petri : Random 3-manifolds with boundary

I will speak about joint work with Jean Raimbault on a model for random 3-manifolds with boundary. These manifolds are obtained as a random gluing of a finite number of truncated tetrahedra. I will discuss what we know about the geometry and topology of these manifolds, how this relates to the combinatorics of the underlying complex and what we don't yet know but would like to know.

  • 16h30- : discussions
  • jeudi 30 juin matin :
    • 10h-11h : Lionel Pournin : Distance, strong convexity, flagness, and associahedra

One can always transform a triangulation of a convex polygon into another by performing a sequence of edge flips, which amounts to follow a path in the graph G of the associahedron. The least number of flips required to do so is then a distance in that graph whose estimation is instrumental in a variety of contexts, as for instance in computational biology, in computer science, or in algebraic topology. On the other hand, it is known that paths in G correspond to a certain kind of 3-dimensional triangulation. This talk is about the recent proof that these 3-dimensional triangulations are flag when the corresponding path is a geodesic. This result, that provides a new powerful tool to study the geometry of G, can be thought of as a 3-dimensional analogue of a well-known strong convexity property of G. Several consequences on the computation of distances in G and on strong convexity in related graphs will be discussed. This talk is based on joint work with Zili Wang (Dartmouth College).

A permutation of size $n$ can be identified to its diagram in which there is exactly one point per row and column in the grid n^2$. In this paper we consider multidimensional permutations (or $d$-permutations), which are identified to their diagrams on the grid n^d$ in which there is exactly one point per hyperplane $x_i=j$ for $i\in[d]$ and $j\in[n]$. We first investigate exhaustively all small pattern avoiding classes. We provide some bijection to enumerate some of these classes and we propose some conjectures for others. We then give a generalization of well-studied Baxter permutations into this multidimensional setting. In addition, we provide a vincular pattern avoidance characterization of Baxter $d$-permutations.

This is a joint work with Pierre-Jean Morel.

  • 15h15-16h15 : Ariane Carrance : Bicolored maps with boundaries: a recursive structure on top of topological recursion
  • 16h30- : discussions

Tensor models are generalization to more than $2$ dimensions. As such, many techniques developed for matrix models can be generalized to tensor models. In particular, tensor models admit a large N expansion governed by a parameter called the degree. For all tensor models to this day, the leading order of this expansion is given by the same family of graphs known as the melonic graphs. I will then explain how the double-scaling limit mechanism, corresponding to a continuum limit for matrix models, can be implemented in tensor models using a scheme decomposition.

Plusieurs familles de cartes comme les triangulations et les cartes biparties comptées par taille et genre admettent des formules de récurrence remarquablement simples qui permettent de générer ces nombres efficacement. De manière intrigante, on ne sait prouver ces récurrences que par des équations issues des systèmes intégrables appelées hiérarchie KP. Dans cet exposé, j'expliquerai en termes de développement sur les fonctions de Schur ce que signifie pour une série génératrice de satisfaire la hiérarchie KP, et je donnerai plusieurs manières de la prouver dans le cas bien connu des factorisations de permutations (qui correspond également aux constellations et aux nombres de Hurwitz pondérés). Je présenterai le défi du passage aux modèles de cartes non-orientées par une tension entre développement sur les fonctions de Schur et sur les polynomes zonaux et j'évoquerai le problème connexe de la b-conjecture de Goulden et Jackson. Enfin je montrerai comment nous avons pu prouver une hiérarchie intégrable dans un cas assez résistant aux méthodes établies, celui de la série génératrice des nombres de Hurwitz monotones non-orientés. Ce sont des travaux en collaboration avec Guillaume Chapuy et Maciej Dolega.

Participants :

- Nicolas Bonichon (LaBRI, U-Bordeaux)

- Valentin Bonzom (LIPN, Univ. Sorbonne Paris Nord)

- Ariane Carrance (CMAP, Ecole Polytechnique)

- Luca Castelli-Aleardi (LIX, Ecole Polytechnique)

- Guillaume Chapuy (IRIF, Université de Paris)

- Eric Colin de Verdiere (LIGM, Univ. Gustave Eiffel)

- Wenjie Fang (LIGM, Univ. Gustave Eiffel)

- Eric Fusy (LIGM, Univ. Gustave Eiffel)

- Thomas Krajewski (CPT, Aix- Marseille Univ.)

- Thomas Muller (LaBRI, U-Bordeaux)

- Victor Nador (LaBRI, U-Bordeaux)

- Bram Petri (IMJ-PRG, Sorbonne Université)

- Lionel Pournin (LIPN, Univ. Sorbonne Paris Nord)

- Anna Roig-Sanchis (IMJ-PRG, Sorbonne Université)

- Zéphyr Salvy (LIGM, Marne-la-Valée)

- Adrian Tanasa (LaBRI, U-Bordeaux)

- Fabien Vignes-Tourneret (ICJ, Univ. Lyon)

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