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Integration of differential equations is a central problem in mathematics and several approaches have been developed by studying analytic, algebraic, and algorithmic aspects of the subject. One of these is Differential Galois Theory, developed by Kolchin and his school, and another originates from the Soliton Theory and Inverse Spectral Transform method, which was born in the works of Kruskal, Zabusky, Gardner, Green and Miura. Many other approaches have also been developed, but there has so far been no intersection between them. This unique introduction to the subject finally brings them together, with the aim of initiating interaction and collaboration between these various mathematical communities. The collection includes a LMS Invited Lecture Course by Michael F. Singer, together with some shorter lecture courses and review articles, all based upon a mini-programme held at the International Centre for Mathematical Sciences (ICMS) in Edinburgh.

Differential equations --- Algebraic number theory --- Differential calculus --- Differential algebra --- Equations différentielles --- Théorie des nombres algébriques --- Calcul différentiel --- Algèbre différentielle --- Congresses --- Congrès --- Algebraic theory --- Calculus, Differential --- Calculus --- 517.91 Differential equations --- Algebra, Differential --- Differential fields --- Algebraic fields