Differential Galois theory in Strasbourg
from
Wednesday, 4 September 2019 (14:00)
to
Friday, 6 September 2019 (12:00)
Monday, 2 September 2019
Tuesday, 3 September 2019
Wednesday, 4 September 2019
14:00
Notions of difference closures of difference fields.

Zoé Chatzidakis
Notions of difference closures of difference fields.
Zoé Chatzidakis
14:00  15:00
Room: Salle séminaire
It is well known that a differential field K of characteristic 0 is contained in a differential field which is differentially closed and has the property that it Kembeds in every differentially closed field containing K. Such a field is called a differential closure of K, and it is unique up to Kisomorphism. The difference closure is what modeltheorists call a "prime model". One can ask the same question about difference fields: do they have a difference closure, and is it unique? The immediate answer to both these questions is no, for trivial reasons: in most cases, there are continuum many ways of extending an automorphism of a field to its algebraic closure. Therefore a natural requirement is to impose that the field K be algebraically closed. Similarly, if the subfield of K fixed by the automorphism is not pseudofinite, then there are continuum many ways of extending it to a pseudofinite field, so one needs to add the hypothesis that the fixed subfield of K is pseudofinite. In this talk I will show by an example that even these two conditions do not suffice. There are two (and more) natural strengthenings of the notion of difference closure, and we show that in characteristic 0, these notions do admit unique "closures" over any algebraically closed difference field K, provided the subfield of K fixed by the automorphism is large enough. In characteristic p, no such result can hold. All definitions will be introduced.
15:00
Break
Break
15:00  15:30
Room: Salle séminaire
15:30
Integration on Darbouxian foliations

Thierry Combot
Integration on Darbouxian foliations
Thierry Combot
15:30  16:30
Room: Salle séminaire
Consider a Darbouxian function $f=F_0+\sum \lambda_i \ln F_i$ with $F_i$ rational functions in two variables, and the foliation of curves $\mathcal{C}_h=\{f(x,y)=h\}$. We consider the problem of symbolic integration of a rational function $G$ along $\mathcal{C}_h$. If the monodromy of the integral satisfies a differential equation in $h$, then it is linear with constant coefficients, and the integral can be expressed in terms of Liouvillian functions restricted to $\mathcal{C}_h$. Such situation is exceptional, but is however more general than elementary integration. We present an algorithm to test the existence of such differential equation and return the Liouvillian expression of the integral if it exists.
Thursday, 5 September 2019
09:30
Quotients and equations

Martin MartinPizarro
Quotients and equations
Martin MartinPizarro
09:30  10:30
Room: Salle séminaire
Quotients are ubiquitous in Mathematics, and a general question is whether a certain category of sets allows quotients. For the category of definable sets in a given structure, the model theoretic approach is called elimination of imaginaries. For algebraically closed fields, Chevalley’s theorem and the existence of a field of definition of a variety imply that a quotient of a Zariski constructible set by a Zariski constructible equivalence relation is again constructible. Similar results hold for other classes of fields, such as differentially closed fields. In this talk, we will focus on separably closed fields and differentially closed fields of positive characteristic. In joint work with Martin Ziegler, we will provide a natural expansion of the language to achieve elimination of imaginaries, by showing that these theories are equational. Equationality, introduced by Srour, and later considered by Srour and Pillay, is a generalisation of local noetherianity. We will present the main ideas of the proof, without assuming a deep knowledge of model theory.
10:30
Break
Break
10:30  11:00
Room: Salle séminaire
11:00
On Miyake's algorithm for formal reduction of linear differential systems

Joelle Saade
On Miyake's algorithm for formal reduction of linear differential systems
Joelle Saade
11:00  12:00
Room: Salle séminaire
14:00
The RiemannHilbert mapping in genus two

Viktoria Heu
The RiemannHilbert mapping in genus two
Viktoria Heu
14:00  15:00
Room: Salle séminaire
15:00
Becker’s conjecture on Mahler functions

Frédéric Chyzak
Becker’s conjecture on Mahler functions
Frédéric Chyzak
15:00  16:00
Room: Salle séminaire
In 1994, Becker conjectured that if $F(z)$ is a $k$regular power series, then there exists a $k$regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahlertype functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z) = 1$. In this work, we prove Becker’s conjecture in the best possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^γ Q(z)$ for some explicit nonnegative integer $γ$ and such that $1/Q(z)$ is $k$regular. (This is joint work with Jason P. Bell, Michael Coons, and Philippe Dumas.)
16:00
Break
Break
16:00  16:30
Room: Salle séminaire
16:30
Inverse problem for germs of parabolic diffeomorphisms of the complex line

Loïc Teyssier
Inverse problem for germs of parabolic diffeomorphisms of the complex line
Loïc Teyssier
16:30  17:30
Room: Salle séminaire
Friday, 6 September 2019
09:30
AxLindemannWeierstrass theorem for the genus 0 Fuchsian groups

Guy Casale
AxLindemannWeierstrass theorem for the genus 0 Fuchsian groups
Guy Casale
09:30  10:30
Room: Salle séminaire
10:30
Break
Break
10:30  11:00
Room: Salle séminaire
11:00
The asymptotic proprieties of solutions of a qdifference equation as q tends to a root of unity

Changgui Zhang
The asymptotic proprieties of solutions of a qdifference equation as q tends to a root of unity
Changgui Zhang
11:00  12:00
Room: Salle séminaire