xiv Introduction classical notions such as the monodromy group, Fuchsian equations and hy- pergeometric equations. The last section is devoted to Kovacic’s powerful algorithm to compute Liouvillian solutions to linear differential equations of order 2. Each chapter ends with a selection of exercise statements ranging in diﬃculty from the direct application of the theory to dealing with some topics that go beyond it. The reader will also find several illuminating ex- amples. We have included a chapter with a list of further reading outlining the different directions in which differential Galois theory and related topics are being developed. As guidance for teachers interested in using this book for a postgraduate course, we propose three possible courses, depending on the background and interests of their students. (1) For students with limited or no knowledge of algebraic geometry who wish to understand Galois theory of linear differential equa- tions in all its depth, a two–semester course can be given using the whole book. (2) For students with good knowledge of algebraic geometry and alge- braic groups, a one–semester course can be given based on Part 3 of the book using the first two parts as reference as needed. (3) For students without a good knowledge of algebraic geometry and eager to learn differential Galois theory more quickly, a one–semester course can be given by developing the topics included in the fol- lowing sections: 1.1, 3.1, 3.2, 3.3, 4.4 (skipping the references to Lie algebra), 4.6, and Part 3 (except the proof of Proposition 6.3.5, i.e. that the intermediate field of a Picard-Vessiot extension fixed by a normal closed subgroup of the differential Galois group is a Picard-Vessiot extension of the base field). This means introducing the concept of aﬃne variety, defining the algebraic group and its properties considering only aﬃne ones, determining the subgroups of SL(2, C) assuming as a fact that a connected linear group of dimension less than or equal to 2 is solvable, and developing differ- ential Galois theory (skipping the proof of Proposition 6.3.5).

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