Mathematics

In this talk I will present some applications of the theory of foliations to the theory of differential equations in the complex domain. I will start by recalling the definitions and basic properties of foliations. I will then consider the case in which the total space of a fibre bundle is foliated in a way which is compatible, in a suitable sense, with the bundle projection, and discuss the notions of Painleve' foliations of the first and second kind, which were introduced by Gerard and Sec in order to give a geometric interpretation and to generalise some of the theorems of Painleve' about complex differential equations. One of the authors' main results concerns the case of a holomorphic foliation compatible with a holomorphic locally trivial bundle with compact fibers, and I will show how, after performing suitable compactifications, one can apply this theorem to obtain some results about complex ODE, usually proved by analytic methods.