2023 – 2024

28 septembre 2023 

  • Valentin Bonzom (LIGM, Université Gustave Eiffel),
    Propriétés d’invariance des b-poids pour les cartes déformées de Chapuy et Dołęga.
    Transparents.
Les cartes combinatoires sont des collages de polygones formant des surfaces de genre arbitraire et orientables ou non. On s'intéressera ici à des modèles de cartes, orientables ou non, introduits par Chapuy and Dołęga. Les cartes y sont pondérées par des b-poids, qui sont des polynômes en une variable b, déterminés récursivement par élimination des arêtes à partir de la racine. Ces modèles sont motivés par le lien avec la combinatoire algébrique, au sens où leurs séries génératrices se développent naturellement sur les polynômes de Jack pour la variable alpha=1+b. Néanmoins, la définition des b-poids les rend délicats à manipuler. Nous prouverons des propriétés d'invariance de la moyenne des b-poids qui sont bien connues pour les cartes orientables mais non-triviales ici. Ce sont des résultats à paraître avec Victor Nador (LaBRI - LIPN).
  • Cédric Boutillier (LPSM, Sorbonne Université)
    Limites de tableaux de Young aléatoires : une approche déterminantale.
    Transparents.
Nous présentons des résultats pour la limite d'échelle et la limite locale de grands tableaux de Young standards uniformes dont la forme (renormalisée) λ0 est fixée. Le point de départ est l'analyse asymptotique d'un processus déterminantal étudié par Gorin et Rahman, qui permet en particulier :
- de calculer la surface limite pour la limite d'échelle à partir d'une équation polynomiale dépendant de λ0;  
- de décrire la limite locale de ces grands tableaux de Young par une famille de tableaux de Young infinis, construits à partir du processus du collier de perles.
Collaboration avec Jacopo Borga, Valentin Féray et Pierre-Loïc Méliot.

7 décembre 2023 

  • Mireille Bousquet-Mélou (LaBRI, Université de Bordeaux)
    Computer algebra in my combinatorics life.
  • Transparents.

Many of my papers would just not exist without computer algebra. I will describe how CA has become an essential tool in my research in enumerative combinatorics. The point of view will be that of a (sometimes naive) user, not of an expert. Many examples and questions will be taken from a joint paper with Michael Wallner dealing with the enumeration of king walks avoiding a quadrant (arXiv 2021, to appear). My hope is that some of the questions that I will raise will have an immediate answer ("yes, this is done") and/or that some people in the audience will find a question interesting enough to take it back home.
  • Tony Guttmann (University of Melbourne)
    Self-avoiding walks in a square and the gerrymander sequence.
  • Transparents.

We give an improved algorithm for the enumeration of self-avoiding walks and polygons within an N×N square as well as SAWs crossing a square. We present some proofs of the expected asymptotic behaviour as the size N  of the square grows, and then show how one can numerically estimate the parameters in the asymptotic expression. We then show how the improved algorithm can be adapted to count gerrymander sequences (OEIS A348456), and prove that the asymptotics of the gerrymander sequence is similar to that of SAWs crossing a square. This work has been done in collaboration with Iwan Jensen, and in part with Aleks Owczarek.
  • Arvind Ayyer (Indian Institute of Science, Bangalore)
    Computer algebra for the study of two-dimensional exclusion processes.
  • Transparents.

We define a new disordered asymmetric simple exclusion process (ASEP) with two species of particles, first-class and second-class, on a two-dimensional toroidal lattice. The dynamics is controlled by first-class particles, which only move horizontally, with forward and backward hopping rates $p_i$ and $q_i$ respectively if the particle is on row $i$. The motion of second-class particles depends on the relative position of these with respect to the first-class ones, and can be both horizontal and vertical.

In the first part of my talk, I will illustrate how we used computer algebra software, specifically Mathematica and SageMath, to understand the stationary distribution of this process. We computed the partition function, as well as densities and currents of all particles in the steady state. We observed a novel mechanism we call the Scott Russell phenomenon: the current of second class particles in the vertical direction is the same as that of first-class particles in the horizontal direction. In the second part of my talk, I will show how we simulated the process and realized that the Scott Russell phenomenon also holds out of equilibrium.

This is partly joint work with P. Nadeau (European Journal of Combinatorics, 2022).
  • Dan Romik (University of California, Davis)
    A new proof of Viazovska’s modular form inequalities for sphere packing in dimension 8.
  • Transparents.

Maryna Viazovska in 2016 found a remarkable application of the theory of modular forms to a fundamental problem in geometry, obtaining a solution to the sphere packing problem in dimension 8 through an explicit construction of a so-called "magic function" that she defined in terms of classical functions, the Eisenstein series and Jacobi thetanull functions. The same method also led shortly afterwards to the solution of the sphere packing in dimension 24 by her and several collaborators. One component of Viazovska's proof consisted of proving a pair of inequalities satisfied by the modular forms she constructed. Viazovska gave a proof of these inequalities that relied in an essential way on computer calculations. In this talk I will present a new proof of Viazovska's inequalities that uses only elementary arguments that can be easily checked by a human.