We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations. ∗This paper is an expanded version of a talk presented at the conference Singularités des équations

In this thesis we deal with the model theory of differentially closed fields of characteristic zero with several commuting derivations. The questions we consider belong to the area of geometric stability theory. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types. Then we show that the generic type of the heat variety, which is one of these new types, is locally modular. So, unlike the case of ordinary differential fields, the additive group of a partial differential field has locally modular subgroups. We also classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are nonorthogonal to fields. iii Karma Police, arrest this man, he talks in maths... iv

Abstract. This is a survey of a recent work done by the three authors, in which an analysis of geometric properties of a structure relative to a reduct is initiated. In particular, definable groups and fields in this context are considered. In a relatively 1-based theory every group is definably isogenous to a subgroup of a group definable in the reduct. For relatively CM-trivial theories (which encompass certain Hrushovski’s amalgams, such as the fusion of two strongly minimal theories or coloured fields), we prove that every group can be mapped by a homomorphism with central kernel to a group definable in the reduct. 1.

When studying type-definable sets in closed Hasse fields, we observe a certain analogy with the case of algebraically closed fields. We show here in particular that connected type-definable groups can be described as “D-algebraic groups”, which is the natural equivalent for algebraic groups. However, many problems arise when we try to give a description of these geometric objects in the language of schemes. We will mainly focus on the relationship between a D-ring A and the D-ring  of global sections for the D-scheme defined by A. We will give some minimal conditions for these two D-rings to define the same D-scheme, and exhibit an example where really bad things occur. 1