Starting from a presentation of the many recent applications of Galois theory of functional equations to enumerative combinatorics, we will introduce the Galois theory of (different kinds) of difference equations. We will focus on the point of view of the applications, hence with little emphasis on the technicalities of the domain, but I'm willing to do an hour of « exercises » (i.e. to go a little deeper into the proofs), if a part of the audience is interested.[-]

In the mid-90's, generalising a theorem of Jouanolou, Hrushovski proved that if a D-variety over the constant field C has no non-constant D-rational functions to C, then it has only finitely many D-subvarieties of codimension one. This theorem has analogues in other geometric contexts where model theory plays a role: in complex analytic geometry where it is an old theorem of Krasnov, in algebraic dynamics where it is a theorem of Bell-Rogalski-Sierra, and in meromorphic dynamics where it is a theorem of Cantat. I will report on work-in-progress with Jason Bell and Adam Topaz toward generalising and unifying these statements.[-]

The field of Laurent series (with real coefficients, say) has a natural derivation but is too small to be closed under integration and other natural operations such as taking logarithms of positive elements. The field has a natural extension to a field of generalized series, the ordered differential field of transseries, where these defects are remedied in a radical way. I will sketch this field of transseries. Recently it was established (Aschenbrenner, Van der Hoeven, vdD) that the differential field of transseries also has very good model theoretic properties. I hope to discuss this in the later part of my talk.[-]

In the recent years, the nature of the generating series of walks in the quarter plane has attracted the attention of many authors in combinatorics and probability. The main questions are: are they algebraic, holonomic (solutions of linear differential equations) or at least hyperalgebraic (solutions of algebraic differential equations)? In this talk, we will show how the nature of the generating function can be approached via the study of a discrete functional equation over a curve E, of genus zero or one. In the first case, the functional equation corresponds to a so called q-difference equation and all the related generating series are differentially transcendental. For the genus one case, the dynamic of the functional equation corresponds to the addition by a given point P of the elliptic curve E. In that situation, one can relate the nature of the generating series to the fact that the point P is of torsion or not.[-]

To any algebraic differential equation, one can associate a first-order structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions.To describe the structure associated to a given algebraic (nonlinear) differential equation (E), typical questions are:- Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations?- Is it possible to express the general solutions of (E) from successive resolutions of algebraic differential equations of lower order than (E)?- Given distinct initial conditions for (E), under which conditions are the solutions associated to these initial conditions algebraically independent?In my talk, I will discuss in this setting one of the first examples of non-completely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative model-theoretic description of the associated structure based on the global hyperbolic dynamical properties identified by Anosov in the 70's for the geodesic motion in negative curvature.[-]