Course description: Differential Galois theory is the study of linear differential equations with coefficients in a field. After initial contributions by several mathematicians, including E. Picard and E. Vessiot, differential Galois theory was developped systematically by E. Kolchin in the middle of the 20th century. In more recent times, some of the key results of the theory have been reformulated in the framework of Tannakian categories.

The goal of this course is to give a first introduction to differential Galois theory using a rather elementary approach. One of the primary goals will be the "main theorem of differential Galois theory" which is analogous to the classical correspondence between intermediate fields of Galois extensions and subgroups of the corresponding Galois group. If time permits, we will reinterpret some of the results described in this course using Tannakian categories.

Prerequisites: An undergraduate course in (commutative) algebra should be enough to follow most of the course. In some places, familiarity with basic notions of algebraic geometry (roughly at the level of chapter one of Hartshorne) will be useful.

FAQ:

Lecture notes: (apologies for poor quality)
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5, extra notes
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16

Literature: The main reference for this course will be:

• A. Magid: Lectures on differential Galois theory. University Lecture Series, 7. American Mathematical Society, Providence, RI, 1994. xiv+105 pp.

• Y. André: Différentielles non commutatives et théorie de Galois différentielle ou aux différences. Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 685--739.
  
  
• F. Beukers: Differential Galois theory. From number theory to physics (Les Houches, 1989), 413--439, Springer, Berlin, 1992.
  
  
• P. Deligne: Catégories tannakiennes. The Grothendieck Festschrift, Vol. II, 111--195, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990.
  
  
• M. van der Put, M. Singer: Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328. Springer-Verlag, Berlin, 2003. xviii+438 pp.
  
  
• T. Szamuely: Galois groups and fundamental groups. Cambridge Studies in Advanced Mathematics, 117. Cambridge University Press, Cambridge, 2009. x+270 pp.