November 13, 1995

Speaker: Anand Pillay (University of Notre Dame)

Title: Differential Galois Theory

Abstract: We generalise the Picard-Vessiot-Kolchin Galois theory of differential fields. The context is a pair K/k of differential fields (fields equipped with a derivation) where K is finitely generated over (as a differential field) and of finite transcendence degree over k. We can view K as being generated over k by a solution of a system of differential equations over k . The problem is to find a class of such extensions supporting a Galois theory. For Picard-Vessiot extensions the automorphism group of K over k has naturally the structure of a linear algebraic group over the constants of k. For Kolchin's strongly normal extensions, the Galois group can have the structure of an arbitrary algebraic group over the constants of k . In the new theory a classof extensions is defined, where the Galois group can have the structure of an arbitrary "finite-dimensional differential algebraic group". Groups of this kind, which are not algebraic groups over the constants, arise from Manin's work on the Mordell conjecture for function fields. We also study some inverse problems for this new differential Galois theory.