Abstract: Connections on algebraic schemes generalize linear differential equations. Their study is motivated by Hilbert's 21^st problem and Galois theory of solving polynomial equations. Hilbert's 21^st problem asks about a correspondence between linear differential equations on a punctured complex plane and representations of its fundamental group. This is nowaday known as the Riemann-Hilbert correspondence. Some geometric features of a connection are captured in its monodromy group, which relates to the fundamental group mentioned above and Galois' theory. This group is henceforth called the differential Galois group of the connection.
In my talk I will report about our recent works on connections over a family of smooth schemes parameterized by a curve and their differential Galois group.