Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue des États-Unis 78035 Versailles cedex, France
e-mail: lucia.di.vizio[at]math.cnrs.fr          
Office: bâtiment Fermat, office 3305

GROUPE DE TRAVAIL «Transcendance et combinatoire» - Programme 2020-2021

Le groupe de travail «Transcendance et combinatoire» a débuté en janvier 2018. Il bénéficie du soutien de la bourse ERC COMBINEPIC.

Organisateurs : Alin Bostan, Lucia Di Vizio et Kilian Raschel 

Lieu et horaires : Le groupe de travail se déroulera en ligne jusqu'à nouvel ordre. Il aura lieu 1 ou 2 vendredis par mois, entre 15h et 17h sauf mention contraire. 

Pour revoir quelques exposés : c'est par ici... ou sur cette page 

  1. Vendredi 18 juin 2021, 15h-17h.
    • Orateur : Thomas Dreyfus (CNRS et Université de Strasbourg)
    • Titre : Differential algebraic series of walks in the quarter plane
    • Résumé : Given a model of walk, the study of the algebraic and differential relations satisfied by the trivariate generating series has attracted the attentions of many authors, coming from various fields of mathematics. In this talk, we focus on the differential algebraic cases (the generating series satisfies an algebraic differential equation). In the unweighted case, it was proved that the generating series satisfies an algebraic differential equation in one of its variable if and only if it satisfies an algebraic differential equation in every of its variables. This result is a mix of case by case argument and general reasoning. In the present talk, we explain how we may extend this result for weighted models of walks.
  2. Vendredi 21 mai 2021, 15h-17h.
    • Orateur : Andrew Elvey Price (CNRS et Université de Tours)
    • Titre : The six vertex model on random lattices using Jacobi theta functions [SLIDES]
    • Résumé : I will describe our solution to the six vertex model on random lattices. Our solution to this problem is a purely combinatorial reformulation of the non-rigorous solution of Kostov in mathematical physics literature, along with a proof that it is the solution. The method involves exactly solving a system of functional equations for the associated generating function in terms of Jacobi theta functions. We observe that the solution satisfies certain modular properties implying that it simplifies when the ratio of the two weights is 2\cos(\alpha) with \alpha/\pi rational, thereby explaining the simple expressions found by E.P. and Bousquet-Mélou in two such cases. In all cases the generating function is D-algebraic, however we do not know if the multivariate generating functions involved in the solution are D-algebraic. This is joint work with Paul Zinn-Justin.
  3. Vendredi 7 mai 2021, 15h-17h.
    • Orateur : Bruno Salvy (Inria et LIP, Lyon)
    • Titre : Computation of tight enclosures for Laplacian eigenvalues [SLIDES]
    • Résumé : Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the 3D Kreweras model that has been the object of previous efforts.
  4. Vendredi 9 avril 2021, 14h-16h.
  5. Vendredi 2 avril 2021, 15h-17h.
    • Orateur : Philippe Biane (CNRS et Université Paris-Est)
    • Titre : Mating of discrete trees and walks in the quarter-plane  [SLIDES]
    • Résumé : We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.
  6. Vendredi 5 mars 2021, 15h-17h.
    • Orateur : Manfred Buchacher (Johannes Kepler Universität, Linz)
    • Titre : On Restricted Lattice Walks [SLIDES]
    • Résumé : In this talk, we will present two recent contributions to the topic of restricted lattice walks. In the first part, titled 'Quadrant Walks Starting Outside the Quadrant’, we investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its solution is not algebraic. The solution can be interpreted as the generating function of lattice walks in $\mathbb{Z}^2$ starting at $(-1,-1)$ and subject to the restriction that the coordinate axes can be crossed only in one direction. We also consider certain variants of the equation, all of which seem to have transcendental solutions. In one case, the solution is perhaps not even D-finite. This is joint work with Manuel Kauers and Amélie Trotignon. In the second part, titled 'Inhomogeneous Restricted Lattice Walks’, we consider inhomogeneous lattice walk models in a half-space and show by a generalization of the kernel method to linear systems of functional equations that their generating functions are always algebraic. The nature of generating functions of inhomogeneous lattice walks restricted to the non-negative orthant is more diverse. We show how some of the methods employed in the homogeneous setting generalize to the inhomogeneous one at the example of time-inhomogeneous lattice walks restricted to $\mathbb{Z}_{\geq 0}^2$. This is joint work with Manuel Kauers.
  7. Vendredi 26 février 2021, 15h-17h.
    • Orateur : Michael Singer (North Carolina State University) 
    • Titre : Differentially Algebraic Generating Series for Walks in the Quarter Plane  [SLIDES] 
    • Résumé : I will present a refinement of necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, I will discuss algorithms based on the theory of Mordell-Weil lattices that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series. I will include an overview of work concerning algebraic and differential properties of these generating series and give an exposition of the necessary facts concerning Mordell-Weil lattices. No previous knowledge of these latter subjects will be assumed. This is joint work with C. Hardouin and appears in arXiv:2010.00963.
  8. Vendredi 12 février 2021, 15h-17h.
    • Orateur : Camilo Sanabria Malagon (Universidad de los Andes, Bogota)
    • Titre : Linear differential equations with finite differential Galois group [SLIDES]
    • Résumé : Let $G\subseteq SL_2(\mathbb{C})$ be a finite primitive group. A classical result of Klein states that there exists a hypergeometric equation such that any second order linear ordinary differential equation whose differential Galois group is $G$ is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this talk I present a generalization of this result. Let $G\subseteq SL_n(\mathbb{C})$ be a finite primitive group. I will show that there is a positive integer $d=d(G)$ and a standard equation such that any linear ordinary differential equation whose differential Galois group is $G$ is gauge equivalent over a field extension $F$ of degree $d$ to an equation projectively equivalent to the pullback by a map in $F$ of this standard equations. For $n=3$, the standard equations can be chosen so that they are hypergeometric.
  9. Vendredi 8 janvier 2021, 10h-12h.
  10. Vendredi 4 décembre 2020, 11h-12h (!). Séance en commun avec le séminaire de l'équipe PolSys.
    • Orateur : Gleb Pogudin
    • Titre : Elimination problem for ODE systems and parameter identifiability
    • Résumé : Elimination of unknowns in systems of polynomial equations is a standard tool of computational algebra and algebraic geometry going back to the pioneers of the subject. An analogous question of elimination of unknowns in differential-algebraic equations can be stated as follows: given a system of nonlinear differential equations in two groups of variables, $x$'s and $y$'s, describe all the consequences of the system depending solely on $x$'s
      Such a differential elimination problem has been studied already by Joseph Ritt, the founder of differential algebra, who proposed the first algorithm for solving it in 1930's. Collective effort of many researchers in the field aimed at understanding and improving this algorithm culminated in modern differential elimination algorithms such as the Rosenfeld-Groebner algorithm which is now a part of the Maple standard library.
      I will talk about differential elimination from the point of view of applications to modeling. Differential elimination is the key step in one of the approaches to the parameter identifiability problem (to decide whether or not parameters can in principle be recovered from the observations). A big portion of dynamical models used in sciences are ODE system, that is, are of the form $x' = f(x)$. General algorithms like the Rosenfeld-Groebner algorithm are applicable in this situation but there is a price to pay for their generality and versatility: many interesting instances cannot be tackled by them in reasonable time. I will describe a new projection-based elimination algorithm tailored to ODE systems which allows to perform elimination (and then assess parameter identifiability) for systems that could not be tackled before. I will conclude by speculating about potential applications of this algorithm in other domains where ODE systems often appear, for example, for computations with differential-algebraic functions.
      This is based on joint work in progress with Ruiwen Dong, Christian Goodbrake, and Heather Harrington.
  11. Vendredi 27 novembre 2020, 15h-17h. 
    • Orateur : Colin Faverjon
    • Titre : La méthode de Mahler et ses conséquences sur les nombres automatiques (II) [SLIDES]
    • Résumé : Dans ce second exposé nous verrons comment la méthode de Mahler se généralise pour étudier simultanément des fonctions de plusieurs variables associées à plusieurs opérateurs différents. Cette généralisation permet d’aborder la question suivante : existe-t-il un nombre réel irrationnel dont les développements dans deux bases multiplicativement indépendantes sont produits par des automates finis ?
  12. Vendredi 20 novembre 2020, 15h-17h. 
    • Orateur : Colin Faverjon 
    • Titre :  La méthode de Mahler et ses conséquences sur les nombres automatiques (I) [SLIDES]
    • Résumé : En 1929, à l’époque où Siegel écrit son article fondateur sur la théorie des $E$-fonctions, Mahler développe une méthode permettant de démontrer la transcendance et l’indépendance algébrique de valeurs de fonctions vérifiant certaines équations aux différences, pour l’opérateur $z\mapsto z^q$, $q\geq 2$ un entier. On dit ainsi qu’une fonction $f(z) \in \mathbb Q\{z\}$ est $q$-mahlérienne s’il existe un $\mathbb Q(z)$-espace vectoriel de dimension finie contenant $f(z)$ et stable par l’opérateur $z\mapsto z^q$.

      Bien qu’ayant rencontré un intérêt moindre que les travaux de Siegel, les résultats obtenus pour les $E$-fonctions et pour les fonctions mahlériennes convergent aujourd’hui en ce qui concerne la nature des relations algébriques entre les valeurs de ces fonctions aux points algébriques. Ainsi le théorème de Siegel-Shidlovskii (1969) a un analogue pour les fonctions mahlériennes, le théorème de Nishioka (1990). De même, on dispose dans les deux cadres d’algorithmes permettant, étant donnée une fonction mahlérienne ou une $E$-fonction et un point algébrique, de dire si la fonction prend une valeur algébrique ou transcendante en ce point.

      Une spécificité de la méthode de Mahler, c’est qu’elle permet de travailler avec des fonctions de plusieurs variables et avec plusieurs opérateurs différents, simultanément. Dans ce cadre là, les développements récents de la méthode permettent d’obtenir de puissants résultats d’indépendance algébrique.

      Outre son intérêt en théorie de la transcendance, la méthode de Mahler apporte un éclairage sur la complexité du calcul du développement des nombres réels. En effet, toute série génératrice d'une suite produite par un automate fini est une fonction mahlérienne.

      Dans un premier exposé, nous présenterons la méthode de Mahler en une variable. Nous expliquerons comment elle permet de démontrer que le développement dans une base entière d’un nombre algébrique ne peut pas être engendré par un automate fini.

  13. Vendredi 16 octobre 2020, 16h-18h (!).
  14. Vendredi 25 septembre 2020, 14h-16h (!). 

 

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