When solving a differential equation, one sometimes finds that solutions can be expressed using a finite number of fixed, particular solutions. As an example, the set of solutions of a linear differential equation is a finite-dimensional complex vector space. This is an incarnation of the model-theoretic phenomenon of internality to the constants in a differentially closed field of characteristic zero. In this talk, I will explain what this means, and discuss some recent progress, joint with Christine Eagles, on finding methods to determine whether or not the solution set of a differential equation is internal. A corollary of our method also gives a criteria for solutions to be orthogonal to the constants, and in particular not Liouvillian. I will show a concrete application to Lotka-Volterra systems.

The Picard-Vessiot differential Galois theory of ordinary homogeneous linear differential equations was codified in the work of Ellis Kolchin in the late 1940s. I will describe some of the results of my upcoming thesis on the model theoretic approach to differential Galois theory and the interactions with Pillay's definable Galois cohomology. The emphasis is on existence theorems for various flavors of differential Galois extension and the relationship to field arithmetic properties of the field of constants. I will also talk about an abstract Galois correspondence in the setting of a prime model of a totally transcendental theory. The aim is to give an accessible overview of the area and my results.

Abstract TBA