Analytic and algebraic aspects of q-difference equations – Q-DIFF

Whereas the theory of differential equations has known a considerable development during the past centuries, the theories of finite difference or $q$-difference equations have been relatively neglected. Recently, in particular, since $q$-difference equations appears, expected or not, in numerous mathematical and physics research domains these domains have enjoyed a new boom from an analytic point of view as well as from an algebraic point of view,.

The aim of this project is the study of the analytic, algebraic and arithmetic aspects of q-difference equations.

Concerning the algebraic aspects, we propose to develop the Galois theories of linear difference equations, possibly parametrized. On the one hand, we tackle the theoretic aspects : unification of the existing theories, iterative theory when q is a root of the unity, Galoisian confluence, differential Galois theory of difference equations, ...We are also concerned with the concrete applications : explicit computations of Galois groups, problems of transcendency and hypertranscendency of classical $q$-functions... In the non linear case, we plan to pursue the galoisian approach of a non linear $q$-difference equation through the Malgrange D-groupoid by studying for instance its connections with the linear theories as well as its numerous applications, specially the questions of irreducibility and integrability of q-Painlevé equations.

Concerning the analytic aspects, we plan to investigate deeply the analytic classification of $q$-difference equations for $|q| =1$, for instance its connections with the non-commutative elliptic curves. Intimately related with these problems, we wish to pursue the study of the structure of the singularities of mixed equations (q-difference/differential) and of the singular perturbation problems of these equations.

Finally, from an arithmetic point of view, our project aims, for the galoisian aspects, to give an arithmetic description of the Galois groups following a $q$-analogous of the Grothendieck conjecture, for the special values of classical functions, on the one hand to adapt to the complex case some recent works on the p-adic case and, on the other hand, to study the existence of a quantification of mirror maps (preserving the integrality, in a suitable sense, of the Taylor coefficients for instance).

With its multiples lightings and their interactions, this project intends to draw a rich and unified picture of the world of $q$-differences and through its numerous applications, connects this research domain to various mathematical areas such as the theory of discrete dynamical systems (non-integrability), knot theory (hyperbolic volume conjecture), quantum groups (associated to Kac-Moody algrebras), 12 th Hilbert problem, etc....