Lectures on differential

Differential Galois theory has known an outburst of activity in the last decade. To pinpoint what triggered this renewal is probably a matter of personal taste; all the same, let me start the present review by a tentative list, restricted on purpose to “non-obviously differential” occurrences of the theory (and also, as in the book under review, to the Galois theory of linear differential equations in characteristic 0): —In 1984, J.-P. Ramis [16] discovered that the classical Stokes phenomenon one encounters in the resummation of divergent series has a Galoisian nature and can therefore be viewed as the effect of a generalized monodromy operator. —In 1986, F. Beukers, D. Brownawell, and G. Hekman [3] realized that the technical hypothesis upon which Siegel had based his classical generalization of the Lindemann-Weierstass theorem amounts to a simple condition on a differential Galois group. This rejuvenated his approach to the theory of Eand G-functions. —In 1988, N. Katz (cf. [10]) started a new way of investigating Sato-Tate conjectures on exponential sums by relating the measure involved in the associated law to a differential Galois group. —In 1990, P. Deligne [5] rewrote the fundamentals of tannakian categories. In this theory, Galois groups preexist Galois extensions. This enabled him to give a new construction of Picard-Vessiot extensions. —Algebraic extensions of function fields are particular cases of differential extensions. In this way, differential Galois theory can contribute to our knowledge of finite subgroups of classical groups (see, e.g., M. Singer and F. Ulmer [18]). —Difference equations too have their own Galois theory. A number of authors have recently applied it with success to the study of recurrence relations and of their q-analogues. Thus, interest in differential Galois theory is no longer restricted to specialists. But (strangely enough for a theory with so much historical appeal), textbook introductions are rare. Kolchin’s exhaustive book [12] covers a much broader area (including, for instance, the Galois theory of nonlinear differential equations), at the cost of a heavy machinery to set in place. Kaplansky’s excellent and brisk introduction to the linear theory [8] has one drawback (more on this in §4). As for Kuga’s delightful Galois’ dream [14], it is in fact concerned with Fuchsian equations and their monodromy groups (a finer object for such equations than the differential Galois group). So this new textbook is welcome. The core of the book is expressed in a remarkably concise way on the first page of its preface: “The structure of the differential Galois extension is a twisted form of the function field of the differential Galois group, with scalars the base differential field,” and it is on this fact that the whole theory will be based. We now explain these terms and describe at the same time what differential Galois theory is about, under the light of the tannakian approach (cf. [5, 10, 1]).