MATH 87800: Differential Galois Theory:

In ordinary Galois Theory one studies the roots of a polynomial equation by means of the Galois group, i.e., the automorphism group of the extension of the coefficient field generated by those roots. In Differential Galois Theory one studies the solutions of linear differential equations in an analogous way, i.e., by means of an automorphism group of the extension of the coefficient field generated by the solutions. The flavor of the subject is algebraic, and students who register should have been completed a first-year graduate algebra course which included a treatment of the usual Galois theory. A knowledge of basic point-set topology and elementary complex variables will also be assumed, but familiarity with differential equations is not required. Everything else needed will be developed in the course.

To get an idea of the flavor of the subject see my lecture notes Introduction to the Galois Theory of Ordinary Differential Equations in the posted papers section of the Kolchin Seminar website: http://www.sci.ccny.cuny.edu/~ksda/