This book consists of seven chapters, each containing a written version of one lecture series of the School held in Edinburgh in 2006. The first part (82 pages) presents lectures given by Michael F. Singer, containing a description of Galois theory for linear differential equations. It starts with an introduction to differential modules and their connections to linear differential equations. After a short review of the classical Galois theory for polynomials, the Galois group of a differential module is defined to be the group of differential isomorphisms. An analogue of Galois correspondence for differential equations is stated and proved and several examples and applications are discussed, including the notion of the monodromy group of a differential system, factorization of differential equations and connections to transcendental numbers. Finally, an algorithm for computing the differential Galois group of an equation is outlined and it is shown, with examples, how it depends on a parameter in the equation. The techniques introduced use Picard-Vessiot theory and its generalizations.
Further chapters are shorter and are mostly based on Galois differential theory. The second part (written by F. Ulmer) is about methods of how to obtain solutions of particular ordinary differential equations in a closed form. Most of the article consists of examples, computations and MATLAB algorithms. The third part (by S. Tsarev) is an exposition of the theory of factorization of ordinary differential equations and partial differential equations. Techniques used here include Gröbner bases and generalized Laplace and Dini transformations. The next part (by A. Leykin) is an introduction to D-modules. It describes D-modules over Weyl algebra and holonomic D-modules, the Bernstein-Sato polynomial, hypergeometric differential equations and local cohomology. It contains many examples computed using software for research in algebraic geometry.
The fifth part (written by Mikhailov, Novikov and Wang) contains a definition of symbolic representations of the ring of differential polynomials and classification of integrable homogenous evolutionary equations (which are symmetries of nonlinear partial differential equations of orders 2, 3 and 5). The sixth part (by J. Hietarinta) searches for integrable systems of (partial) differential equations by algebraic methods. There are several definitions of integrable systems (Liouville, super- and quantum integrability) and a discussion of two examples (connected with two dimensional point-particle dynamics and the soliton equation), both leading to an overdetermined set of equations. The last part (written by A. Pillay) deals with model theory and its connection with Galois theory, and with a nonlinear generalization of the Grothendieck conjecture. The book may be useful for graduate mathematicians working in differential systems and their invariants. The text covers a large area of research on relatively few pages and contains many examples. The reader is assumed to have a basic knowledge of classical theories (e.g. partial differential equations, ordinary differential equations, integrable systems and classical Galois theory).