Some years ago, one of my students, having enjoyed Galois theory, asked me if I could lead her in an independent study of the analogous theory for differential equations. She had become interested in that because of Michio Kuga’s Galois’ Dream (Birkhäuser, 1994). I told her I knew nothing about the subject, but since she was an excellent student, I decided I would try it. We would learn together.
We quickly found that Kuga’s book, while fascinating, was also very sketchy. Details were missing — a lot of details were missing, What’s worse, it wasn’t ever quite clear what the overall picture was, what the theorems were. We found ourselves casting around for other books on the subject, and soon found out that there weren’t very many. Since then, one more has been published: Galois Theory of Linear Differential Equations, by Marius van der Put and Michael F. Singer, (Springer Grundelhren, 2003). Nevertheless, this LMS Lecture Notes volume is very welcome.
MacCallum and Mikhailov have put together a set of expository accounts of various algebraic approaches to the theory of differential equations. The theory my student and I tried to learn that semester is covered in the first chapter, by Michael F. Singer. This is based on a series of lectures he gave at the International Center for Mathematical Sciences in Edinburgh. It is written at a fairly high level, but offers a good way to get started in a subject that is covered in more detail in such high-level monographs as the van der Put–Singer Grundlehren volume.
My one complaint is that the list of references (in this chapter, at least) is given in order of citation in the text, rather than in some sort of alphabetical order. It is eight pages long. So suppose someone wants to know whether Singer discusses a particular paper; the only way to find out is to either read the whole chapter or read the eight pages of bibliography!
The remaining chapters are based on shorter talks given in a workshop held in parallel to the main lectures. They deal with other algebraic approaches to differential equations, such as the theory of D-modules, nicely rounding out the book.
The result is a useful book that serves as an introduction to both the Galois theory of (linear) differential equations and several other algebraic approaches to such equations. Libraries will definitely want to have a copy.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.