Le Séminaire de Combinatoire Enumérative et Analytique, rebaptisé Séminaire Philippe Flajolet le 7 avril 2011, a pour objectif de couvrir un large spectre de recherche en combinatoire, et est ouvert à tous les chercheurs et étudiants intéressés.

Il se tient un jeudi tous les deux mois à l'IHP, plus de détails ici.

Traditionnellement, après chaque exposé, un volontaire se charge de rédiger
une petite synthèse des résultats présentés. Voir la page des archives.

Les séances de l'année 2016-2017 sont fixées au: 29 septembre 2016,
1er decembre 2016, 2 février 2017, 30 mars 2017, 1er juin 2017.

###### Prochaine séance : 30 mars 2017, Amphi Darboux, IHP

- 10h30 - 11h30:
**Gaetan Borot (Max Planck Institute for Mathematics, Bonn)**,

*The ABCD of topological recursion*,

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I will give a short introduction to the formalism of topological recursion, which has been simplified and generalised by a recent proposal by Kontsevich and Soibelman: no geometry is necessary to present it, only algebra and combinatorics. The initial data is now a collection (L_i)_i of differential operators in many variables, which are atmost quadratic in derivatives and satisfy Lie algebra commutation relations. The topological recursion then produces the Taylor coefficients of the common solution to the equations {L_i Z = 0}_{i}. For the simplest initial data, Z just counts trivalent graphs with certain conditions, and can be exactly computed in terms of Whittaker functions. In general, the topological recursion appears as a sum over these graphs.

In fact, finding initial data is not a priori an easy task ! I will present 3 classes of initial data, respectively obtained from the data of a Frobenius algebra, of a non-commutative Frobenius algebra, and of the vector space of formal series with coefficients in a Frobenius algebra. This third class of examples contains the topological recursion from the former Eynard and Orantin perspective -- so, I will review some already known applications of topological recursion to various problems of enumerative geometry, in light of Kontsevich-Soibelman approach, which is hopefully combinatoralist-friendly.

If time permits I will pose and motivate open questions about large genus asymptotics.

*Based on joint work with J.E. Andersen, L. Chekhov and N. Orantin.*

- 13h45 - 14h45:
**Rinat Kedem (University of Illinois at Urbana-Champaign)**,

*T-systems and discrete integrable systems*,

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The T-system for type A is a restricted octahedron relation, which first appeared in the context of quantum integrable spin chains with quantum affine algebra symmetries in the 80's. It has many beautiful combinatorial properties and applications, such as Frieze patterns, pentagram maps and Zamolodchikov periodicity. As a cluster algebra, it also admits a quantization. In this talk, I'll demonstrate some of the manifestations of the integrability of this system and explain the combinatorial solutions in the classical and quantum case. This talk is based on joint work with Philippe Di Francesco.

- 14h45 - 15h45:
**Cédric Boutillier (LPMA, Paris 6)**,

*The Z-invariant Ising model on isoradial graphs*,

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The Ising model is a mathematical model for ferromagnetism in which spins are located at vertices of a graph. For a large class of embedded planar graphs, called isoradial graphs, local weights for this model can be found so that an integrability condition (the Yang-Baxter relation) is satisfied. The model is then said to be Z-invariant.

In a joint work with Béatrice de Tilière (Créteil) and Kilian Raschel (Tours), we study the Z-invariant Ising model on infinite isoradial graphs. We show that certain probabilistic quantities have a local
expression in terms of the geometry of the embedding of the graph. Our main tool is Fisher's bijection between the Ising model and a dimer model on a decorated graph. We will discuss the properties of this model in connection with the geometry of an algebraic curbe naturally
associated to the dimer model.

- 16h00: Pause café