Differential Galois Theory through Riemann-Hilbert Correspondence: An Elementary Introduction

We all know what Galois theory is, and we know the meaning of “differential,” as in “differential equations,” but what does “differential Galois theory” mean? Isn’t Galois theory first and foremost concerned with roots and factoring of polynomials, with special attention paid to the overfields of the smallest field in which these polynomials’ coefficients live? What’s differential about that? What could this mean?

Well, the truth of the matter is that differential Galois theory really talks about groups attached to linear differential equations, somewhat analogously to how our familiar Galois groups are attached to polynomials. Here the paradigm is Picard-Vessiot theory, which Wikipedia tells us “was initiated by Emile Picard and Ernest Vessiot from about 1883 to 1904.” However, the book under review is emphatically not about Picard-Vessiot theory; in fact these names don’t even appear in the index. But two other names do — indeed, they already appear on the title page: we’re dealing with the Riemann-Hilbert Correspondence. So this raises the question: what is the connection here?

Well, Part 3 of the book is devoted to this topic, after Parts 1 and 2 deal with, respectively, a pretty thorough, if dense, review of complex function theory, complex linear differential equations (CLDE’s for now), and monodromy (about which more presently). It transpires, then, that Riemann-Hilbert comes in both a local and a global form. Roughly, we have the following characterizations. Locally, the idea is to attach to a CLDE, together with one of its monodromy representations, a bijection between “meromorphic equivalence classes of regular singular systems on the one hand and isomorphism classes of linear representations of the fundamental group on the other hand” (p. 127). All right, what’s monodromy, then, and what fundamental group are we talking about? Well, (p. 46 ff.) monodromy has to do with uniqueness of analytic continuations of holomorphic functions, working with respect to homotopic paths inside the domain where this is all taking place. And it is for such paths, starting at, say, a point a, that we get the aforementioned fundamental group based at a. So much for the local situation.

Going over to the global case, to which Sauloy devotes his entire twelfth chapter, we in fact run up against nothing less that Hilbert’s 21st Paris Problem, to wit (again from Wikipedia, I’m afraid): “[Give a] proof of the existence of linear differential equations having a prescribed monodromic [sic, I think: it’s “monodromy”] group.” So one attaches monodromy to DEs in a natural way, and of course this has to do with the behavior of solutions in neighborhoods of singular points for the DE. The fact that monodromy has a group structure testifies to the fundamentally geometric nature of such singular behavior: see p. 92 ff. for more details, but roughly speaking what’s going on is that the earlier fundamental group attached to analytic continuations from the earlier given point a is represented in GL(germs at a), and the image of this representation is, by definition, our monodromy group. The clincher in Chapter 12 (p. 163) is that all finite dimensional representations of the fundamental group (still based at a) are conjugate to some monodromy representation attached to a well-chosen CLDE. Big stuff.

The book’s last part, Part 4, is titled “Differential Galois Theory,” so it’s the culmination of what went before it, i.e. the first ca. 170 pages. Here are some snapshots of what Sauloy presents here: differential Galois groups as linear algebraic groups, actual calculations of these groups using Schlesinger density theorems, the universal Galois group (a fuchsian local group), and even a tantalizing section titled, “the inverse problem in differential Galois theory.” This last theme is particularly evocative given the parallel with the inverse problem in regular Galois theory.

All right, then, the foregoing gives the flavor of what this book is about. It’s an excellent book about a beautiful and deep subject, and the point of view adopted by Sauloy is very apposite: monodromy occupies central stage, and rightly so. But it’s developed very thoroughly, using pretty modern methods, with sheaves (e.g. solution sheaves to analytic LCDE’s) heavily featured. This makes for maximum bang for your buck: the results are very general, as are the methods. And, consistent with this, Saulay goes categorical on us, too, but not too much so. There are loads of exercises, and I think the book is very well-paced, as well as very clearly written. It’s a fabulous entry in the AMS GSM series.

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.