The research proposal concerns nonlinear Galois theory and applications to study of continuous and discrete dynamical systems. Recently, two theories were proposed by H. Umemura (with an algebraic and arithmetical flavour) and B.Malgrange (with a geometrical and dynamical flavour).

The purpose of this proposal is to:
- develop the theory of Lie groupoid using fruitful interrelations between the formal theory of partial differential equations and differential algebra,
- apply Galois theory to irreducibility problems, especially for Painleve equations and Garnier systems,
- unify integrability notions of both continuous and discrete dynamical systems using differential Galois theory and to give integrability and non integrability criteria for hamiltonian systems,
- study analytic and algebraic geometry of holomorphic singular foliations with a view toward deformations of geometric structures,
-give a Tannakian unification of these two theory.
This will provide numbers of new relations between arithmetic and dynamic.

This research is important to set up a complete differential Galois theory as important as the classical one. The numbers of applications and used tools give a concrete view of the transversal and interdisciplinary nature of the proposal. Research methodology contains regular discussion sessions with the scientific in charge and other local mathematicians and intermediate goals leading to the objectives. Scientific credibility and feasibility of the project is increase by many examples already computed and numerous discussions with international level mathematicians.

Practical arrangements for the management of the project include organisation of a work shop on differential Galois theory at the end of the first year, invitation of international level mathematicians. Algebraic and geometric studies of differential equations are fields in which Europe and Japan excel. This project contributes to the enhancement of European scientific excellence.