UIC MATH COLLOQUIUM
November 13, 1995
Speaker: Anand Pillay (University of Notre Dame)
Title: Differential Galois Theory
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.