Bernard Malgrange
On differential Galois theory for non-linear equations.
Let X be a smooth complex analytic manifold,and let Aut(X)be the set of
germs of automorphism of X (the source and the target are point of X,may be
differents).
Roughly speaking,a Lie groupoid on X is a subgroupoid of Aut(X) defined by
a system of partial differential equations ;typical examples are the
symplectic germs ,the germs preserving a volume etc.
Given a foliation with singularities on X,I consider the smallest groupoid
whose Lie algebra contains the vectors fields tangent to this foliation.In
the case of a foliation defined by a linear differential equation, one
proves that this groupoid is essentially the same object as the
differential Galois group of the equation.In more general examples,its
determination seems to be an interesting problem,but very difficult.