In the past three decades, studies on the Painleve equations have rapidly developed through the efforts of many researchers in various fields. The Painleve equations have numerous applications to differential geometry, probability theory, soliton theory, topological field theory, and others. In the past ten years, the Painleve equations have been studied not only by analytic methods, but also by algebraic methods, such as rational algebraic surfaces, differential Galois theory, the affine Weyl groups, and representation theory. This book serves as a guide to algebraic studies on the Painleve equations and as an introduction to non-specialists. The book aims to be self-contained by presenting complete proofs of the theorems.
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