• affine group schemes
• differential Galois theory
• embedding problems
• regular singular differential equations
• tannakian formalism

We describe the differential Galois group of a particular family of differential equations over the differential field (C(z),d/dz), namely, regular singular differential equations. The main result is the following theorem, presented by Michael Wibmer a few years ago:

The differential Galois group of the family of all regular singular differential equations over C(z) is the free proalgebraic group on a set of cardinality |C|.

This can be seen as an analogue of the following Douady's theorem in the differential setting, where polynomials with coefficients in C(z) are replaced by regular singular differential equations over the same field:

The absolute Galois group of the rational function field C(z) is the free profinite group on a set of cardinality |C|.

This work is developed into four chapters.

In the first part of Chapter 1, we review the theory of Hopf algebras and affine group schemes over a field k. The latter will be the objects of an appropriate category, which is shown to be anti-equivalent to the category of Hopf k-algebras.
Using this anti-equivalence, among other tools, we present a list of important objects and prove several results about them. A special role is played by quotient maps: morphisms G->H of affine group schemes over k such that, denoting by k[G] and k[H] the Hopf algebras corresponding to G and H, the corresponding morphism k[H]->k[G] of Hopf algebras is injective.
We also study rank and dimension of affine group schemes, quantities encoding algebraic and geometric information.

In the second part of this chapter, we begin the study of embedding problems for a given affine group scheme. An embedding problem for an affine group scheme \Gamma is the data of two affine group schemes G,H and two quotient maps G->H, \Gamma-> H.
A solution for such an embedding problem is a quotient map \Gamma->G that makes the diagram formed using the given two morphisms commutative.

We last introduce saturated groups, providing different characterisations for them. They will be affine group schemes for which every embedding problem respecting certain conditions has a solution. We also prove that they are uniquely determined up to isomorphism by the rank.

In Chapter 2, we study the categories of representations of affine group schemes and introduce the Tannakian formalism. After defining neutral Tannakian categories over a field and providing a detailed explanation of the various structures involved, we show that we can associate to them a group functor, which turns out to be an affine group scheme. This is related to the proof of the following theorem:

Let G be an affine group scheme, and let \omega: k-Rep(G) -> k-VecF be the forgetful functor from the category of its representations to the category of finite-dimensional k-vector spaces. There exists a natural isomorphism between G and the affine group scheme of tensor automorphisms of \omega.

We conclude with a list of examples, among which the proalgebraic completion of an abstract group.

In Chapter 3, we introduce the special family of free proalgebraic groups, whose definition encompasses a universal property. We show that, under certain simple conditions, free proalgebraic groups coincide with the saturated groups introduced in Chapter 1. To establish this, we study the rank of free proalgebraic groups and we consider their embedding problems, specialising the work done in Chapter 1. This will play a crucial role in the proof of Wibmer's theorem.

In Chapter 4, we turn to differential Galois theory. In the first part of the chapter, we outline the basics of differential algebra: we introduce derivations and differential fields, and we provide several approaches to describing differential equations. One of these, matrix differential equations, is particularly well-suited to introduce Picard-Vessiot extensions.

Another approach, based on differential modules, allows for a natural categorical formulation. We consider categories of differential modules, which will be endowed with a neutral Tannakian structure.

The differential Galois group of a family of differential equations is given by the Tannakian fundamental group of the corresponding category of differential modules. Here, we state the differential Galois correspondence, which generalises the classical one by relating closed subgroups of the differential Galois group to subextension of the Picard-Vessiot extension corresponding to the differential equations under consideration.

In the final part of the chapter, we delve into the context of Wibmer's theorem, using all the tools presented so far to provide a proof.

We introduce regular singular differential equations and study the category of the corresponding differential modules. At this point, we recall the Riemann-Hilbert correspondence, stated in the Tannakian setting, which provides a bridge with transcendental tools such as monodromy and topological fundamental groups. This approach will be enough when dealing with regular singular differential equations whose singularities lie in a proper subset of the Riemann sphere. A crucial aspect of the reasoning is that we can choose a basepoint for the fundamental groups that is not a singularity of any equation.

This is no longer possible when considering the family of all regular singular differential equations. To address this issue, embedding problems become relevant. We overcome this difficulty by using one of the characterisations of saturated groups proved in Chapter 1, and the relation with free proalgebraic groups established in Chapter 3.