Elective Subject: Algebraic theory of differential equations

Lecture: Tuesday 10:15-11:45 and Wednesday 10:15-11:00

Exercises: Wednesday 11:00-11:45

Location: Online via webex in my personal room

First meeting: Tuesday March 2 at 10:15

This course is an introduction to the algebraic theory of differential equations (also known as differential algebra). The main focus will be on the Galois theory of linear differential equations. While differential equations are commonly considered to belong to the world of analysis, algebra does have a lot of interesting things to say about differential equations and their solutions. For example, if you ask your favorite computer algebra system to solve a differential equation, it is symbolic-algebraic methods doing the work.

This class offers the opportunity to get to know many fundamental definitions and results from commutative algebra and algebraic geometry (e.g., Hilbert's Nullstellensatz, Krull dimension, algebraic groups, Yoneda lemma,...) and to see how these concepts are applied in the study of differential equations. To get a better idea of the content of this class, please take a look at the preliminary lecture notes. These notes will be completed and further expanded as the class progresses. The most current version of the lecture notes is available in the teach center. You might also find it helpful to take a look at this overview article by Andy Magid that appeared in the Notices of the AMS.

Literature:

• Galois theory of linear differential equations, M. van der Put and M.F. Singer, Springer, 2003, xviii+438 pp. ISBN: 3-540-44228-6

• Lectures on differential Galois theory, A. Magid, American Mathematical Society, 1994, xiv+105 pp. ISBN: 0-8218-7004-1

• Algebraic groups and differential Galois theory, T. Crespo and Z. Hajto, American Mathematical Society, 2011, xiv+225 pp. ISBN: 978-0-8218-5318-4

• Differential Galois theory through Riemann-Hilbert correspondence, J. Sauloy, American Mathematical Society, 2016, xxiii+275 pp. ISBN: 978-1-4704-3095-5