Abstract: 

The absolute Galois group of a (perfect) field is the automorphism group of the algebraic closure, it is naturally a profinite group and is a central object of study in number theory. In analogy, one can study the automorphism group of the differential closure of a differential field. In contrast with usual Galois theory, where the Galois group of a polynomial equation is a finite discrete group, the Galois group of a linear differential equation is an algebraic group. The automorphism group of the differential closure then carries the structure of an iterated extension of proalgebraic groups. In recent work with Pillay, extending work of Magid, we give an infinite Galois correspondence for the differential closure which generalizes the usual model-theoretic infinite Galois correspondence for acl^eq(emptyset) introduced by Poizat to the setting of a prime model of a totally transcendental theory. We describe both this correspondence and related recent basic results on differential Galois cohomology. Time permitting we will discuss the existence of measures on automorphism groups and future directions. The hope is that the talk will be an approachable introduction to model theoretic Galois theory and to field arithmetic issues in the context of differential Galois theory.

Zoom: 717 463 6082