Journées du GDR EFI 2022

Liste des Contributions

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

Marie Albenque

12/09/2022 09:00

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.

1. Arithmetic dynamics

Jason P. Bell (University of Waterloo)

12/09/2022 10:30

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.

5. Towards a description of the algebraic closure of multivariate power series

Dr Mickaël Matusinsky (Institut de Mathématiques de Bordeaux)

12/09/2022 13:30

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...

8. The Galois group of irregular $q$-difference equations

Jacques Sauloy

12/09/2022 14:30

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...

6. Algorithms for the holonomic and non-holonomic universe

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

12/09/2022 15:30

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...

12. Arithmetic dynamics

Prof. Jason P. Bell (University of Waterloo)

13/09/2022 09:00

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.

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

Marie Albenque

13/09/2022 10:30

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.

9. Matzat's conjecture in differential Galois theory

M. Michael Wibmer (Graz University of Technology)

13/09/2022 13:30

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...

3. The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry

Mercedes Haiech (Université de Limoges)

13/09/2022 14:30

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...

4. Abelian reduction of differential equations

Dr Rémi Jaoui (CNRS)

13/09/2022 15:30

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...

14. Arithmetic dynamics

M. Jason P. Bell (University of Waterloo)

14/09/2022 09:00

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.

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

Marie Albenque

14/09/2022 10:30

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.

10. Hypergeometric diagonals and a step towards Christol's conjecture

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

14/09/2022 11:40

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...

11. Arithmetic dynamics

Prof. Jason P. Bell (University of Waterloo)

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.