Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin.  

This talk is about the paper that came out of my first foray into "applied mathematics". As you'll see in the more formal abstract below, it turned out to be a lot less "applied algebra" than we expected, we had to follow where the project took us! The use of mathematical models in the sciences often require the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. We develop a precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold' in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric on parameter space. This opens up the possibility of alternative quantification of sloppiness beyond the traditional use of the Fisher Information Matrix, which implicitly assumes infinitesimal measurement error and an Euclidean parameter space. We illustrate the various concepts involved in the proper definition of sloppiness with examples of ordinary differential equation models with time series data arising in mathematical biology. (joint with Heather Harrington and Dhruva Raman)

We will give a brief introduction and overview of some recent results on differentially large fields (in characteristic zero). In particular, we note that a field admits a differentially large structure iff it is of infinite transcendence degree (over the rationals). We also give conditions on when a differential field can be extended to a differentially large one without changing the constants. We then turn our attention to the theory CODF (closed ordered differential fields). We observe that a real closed differential field has a prime model extension (in CODF) iff it is already a CODF. This extends a result of Singer showing that CODF has no prime model. Time permitting, we will discuss the question of when a real closed differential field has a CODF extension inside a differential closure. This is joint work with Marcus Tressl.

A singularity of the coefficients of a linear homogenous differential equation with rational function coefficients is called regular singular if the solutions are polynomially bounded in sectors centered at the singularity. There is a beautiful analogy between the classical Galois theory of the rational function field and the differential Galois theory of regular singular differential equations, i.e., linear differential equations who's singularities are all regular singular. The goal of this talk is to explore this analogy and to push it one step further.