Journées du GDR EFI 2022

12 sept. 2022, 08:55 → 14 sept. 2022, 17:00 Europe/Paris

Toulouse

Journées du GDR EFI 2022

Description de la manifestation:

Le GDR EFI a pour but de fédérer les nombreux chercheurs qui travaillent en France dans des domaines dont les équations fonctionnelles sont soit l'objet d'étude soit des outils importants pour les applications. Ces domaines concernent les mathématiques « pures » et « appliquées », l'informatique et la physique théorique. On observe actuellement des interactions de plus en plus fortes entre la théorie et les applications, de nombreuses actions communes ayant eu lieu depuis une dizaine d'années. Par équations fonctionnelles, on entend principalement les équations différentielles ordinaires, aux différences, aux q-différences, mahlériennes, linéaires ou algébriques, éventuellement multivariées. Le cas différentiel algébrique non linéaire concerne par exemple les équations de Painlevé. Tous ces types d'équations fonctionnelles ont été et sont toujours très activement étudiés de nombreux points de vue : algébrique, algorithmique, arithmétique, combinatoire, logique, géométrique, physique, etc.

Les journées du GDR EFI 2022 s'articuleront autour de 2 mini-cours  donnés par Marie Albenque (CNRS et Ecole Polytechnique) et Jason P. Bell (University of Waterloo), ainsi que de 8 exposés courts par Mercedes Haiech, Rémi Jaoui, Mickael Matusinsky, Veronika Pillwein, Jacques Sauloy, Michael Wibmer, Sergey Yurkevich.

Ces journées seront suivies d'une journée d'exposés de recherche de l'ANR De Rerum Natura dont le programme est disponible ici

lundi 12 septembre

1 Ising model on random planar maps via Tutte’s invariants.

I will present a survey of recent and not so recent results about combinatorial random planar maps decorated (or not !) with a statistical physics model. I will put a special emphasis on the combinatorial aspects of this story. In particular, I will introduce and explain the method of Tutte’s invariants to solve some functional equations.

Orateur: Marie Albenque

2 Arithmetic dynamics

We give an introduction to arithmetic dynamics and the questions in the area intended for a broad audience with an emphasis on connections to other branches of mathematics.

Orateur: Jason P. Bell (University of Waterloo)

3 Towards a description of the algebraic closure of multivariate power series

We consider the algebraic closure of K((x)), x = (x1, ..., xr), char(K) = 0, namely what we call the field of algebroid Puiseux series, viewed as a subfield of the so-called field of rational polyhedral Puiseux series. Our target is to solve the following problems:
- given a polynomial equation P(x, y) = 0 for P ∈ K[[x]][y], provide a closed form formula for the coefficients of an algebroid Puiseux series solution y(x) in terms of the coefficients of P;
- given an algebroid Puiseux series y(x), reconstruct algorithmically the coefficients of a vanishing polynomial P ∈ K[[x]][y] using the coefficients of the series.
Our strategy involves the answers that we recently obtained to the same type of questions about algebraic Puiseux series, i.e. for the algebraic closure of K(x).
Joint work in progress with M. Hickel (U. Bordeaux)

Orateur: Dr Mickaël Matusinsky (Institut de Mathématiques de Bordeaux)

4 The Galois group of irregular $q$-difference equations

The Galois group of irregular $q$-difference equations with integral slopes was described by Ramis and Sauloy, along with a Riemann-Hilbert correspondence, based on classification results by Ramis-Sauloy-Zhang. The complete determination of a discrete Zariski-dense subgroup, the "wild fundamental group" allowed to solve the inverse problem in that case. For arbitrary slopes, the wild fundamental group has also been determined, but no corresponding progress has been made for the inverse problem. The talk will be purely descriptive.

Orateur: Jacques Sauloy

5 Algorithms for the holonomic and non-holonomic universe

A univariate sequence is called holonomic, if it satisfies a linear difference equation with polyonomial coefficients. Likewise, a univariate holonomic function satisfies a linear differential equation with polynomial coefficients. In the multivariate (mixed) case, holonomic objects are also characterized through systems of linear difference-differential equations. These equations give a way to finitely represent holonomic objects on the computer. It is well known that based on this representation identities on holonomic expressions can be discovered and proven automatically. Recently with Antonio Jimenez Pastor and Philipp Nuspl, we have studied certain extensions of, e.g., the class of holonomic functions to objects that satisfy linear differential equations with holonomic function coeffiicients and of computational properties that carry over. In this talk, I want to give an overview on the use of the classical algorithms as well as these recent extensions.

Orateur: Dr Veronika Pillwein (Research Institute for Symbolic Computation Johannes Kepler University)

mardi 13 septembre

6 Arithmetic dynamics

We give an introduction to arithmetic dynamics and the questions in the area intended for a broad audience with an emphasis on connections to other branches of mathematics.

Orateur: Prof. Jason P. Bell (University of Waterloo)

7 Ising model on random planar maps via Tutte’s invariants

I will present a survey of recent and not so recent results about combinatorial random planar maps decorated (or not !) with a statistical physics model. I will put a special emphasis on the combinatorial aspects of this story. In particular, I will introduce and explain the method of Tutte’s invariants to solve some functional equations.

Orateur: Marie Albenque

8 Matzat's conjecture in differential Galois theory

Determining the absolute differential Galois group of interesting differential fields is a central problem in differential Galois theory. For the fields of formal and convergent Laurent series the solution is well-known, but the classical case of rational functions has long resisted a solution. Matzat's conjecture predicts the structure of the absolute differential Galois group of the rational function field, and more generally, of one-variable function fields. In this talk, I will review recent progress towards Matzat's conjecture.

Orateur: M. Michael Wibmer (Graz University of Technology)

9 The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry

Given a partial differential equation (PDE), its solutions can be difficult, if not impossible, to describe.
The purpose of the Fundamental theorem of tropical (partial) differential algebraic geometry is to extract from the equations certain properties of the solutions.
More precisely, this theorem proves that the support of the solutions in $k[[t_1, \cdots, t_m]]$ (with $k$ a field of characteristic zero) of differential equations can be obtained by solving a so-called tropicalized differential system.

Orateur: Mercedes Haiech (Université de Limoges)

10 Abelian reduction of differential equations

Kolchin’s differential Galois theory is a generalization of Picard-Vessiot theory for which the Galois groups are algebraic groups but not necessarily linear. In the one dimensional case, Kolchin’s theory can be applied to the study of elliptic differential equations and Riccati equations.

I will describe some structural results concerning the higher-order differential equations (and in particular a full classification of the second-order autonomous equations) to which Kolchin’s theory can be applied. This is joint work with Rahim Moosa.

Orateur: Dr Rémi Jaoui (CNRS)

mercredi 14 septembre

11 Arithmetic dynamics

We give an introduction to arithmetic dynamics and the questions in the area intended for a broad audience with an emphasis on connections to other branches of mathematics.

Orateur: M. Jason P. Bell (University of Waterloo)

12 Ising model on random planar maps via Tutte’s invariants

I will present a survey of recent and not so recent results about combinatorial random planar maps decorated (or not !) with a statistical physics model. I will put a special emphasis on the combinatorial aspects of this story. In particular, I will introduce and explain the method of Tutte’s invariants to solve some functional equations.

Orateur: Marie Albenque

13 Hypergeometric diagonals and a step towards Christol's conjecture

Even though diagonals of multivariate rational functions have been studied from various viewpoints, they still remain quite mysterious objects. An example for this is the widely open conjecture by Christol which characterizes diagonals inside the class of all D-finite functions. In 2012 Bostan, Boukraa, Christol, Hassani, and Maillard created a list with 116 potential counter examples for this conjecture. As of today, using new kinds of identities involving diagonals and hypergeometric functions, 40 of these examples were resolved by the starting work of Abdelaziz, Koutschan and Maillard and the generalization by Bostan and the speaker.
In the talk I will explain how the key identities were found and proven, indicate their various implications, and finally mention limitations and possible extensions. The talk is based on joint work with A.~Bostan.

Orateur: Sergey Yurkevich (University Paris-Saclay and University of Vienna)