Introduction

I will give a short account of some of my talks at the very interesting mini-programme on the algebraic theory of differential equations. I have recently written several expository and research papers on the relevant topics, so I refer the reader to these papers for more details and restrict myself here to just conveying a few key points. In the first week I gave three talks, originally intended to consist of (I) model theory and differential algebra, (II) nonlinear differential Galois theory, and (III) differential algebra and diophantine geometry. As it turned out I only really covered (I) and (II) and I will discuss these below. References are [9] which gives an introduction to model theory and applications to differential algebra, and [8] which gives among other things an accessible account of a differential Galois theory going beyond both the Picard-Vessiot (linear) theory and Kolchin's strongly normal theory. Both these papers are written with an eye to the non logician. I also recommend the appendix to Hrushovski's paper [4] for another discussion of model theory and Galois theory.

In the second week of the programme I gave a talk on nonlinear generalizations of Grothendieck's conjecture on the arithmetic of differential equations.