In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin’s theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay’s argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic D-groups.

In a series of papers from the 1990s and early 2000s, Pillay used the machinery of model-theoretic binding groups to give a slick geometric account and generalization of Kolchin’s theory of strongly normal extensions and constrained cohomology. This series of two talks is intended to be expository, with its main goal being to introduce and frame the relevant model-theoretic notions of internality and binding groups within the context of differential algebra, as well as to go through Pillay’s argument that his generalized strongly normal extensions arise from logarithmic differential equations defined over algebraic D-groups.

The material is taken from a joint paper with M. Kamensky, “Interpretations and differential Galois extensions.” Given a differential field K with field of constants k, and a logarithmic differential equation over K, the strongly normal extensions of K for the equation correspond (up to isomorphism over K) with the connected components of G(k) where G is the Galois groupoid of the equation. This generalizes to other contexts (parameterized theory,….), and is also the main tool in existence theorems for strongly normal extensions with prescribed properties.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.