Holonomic deformation and nonlinear integrable systems

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The aim of present project is to study the theory of holonomic deformation and that of nonlinear integrable systems associated with holonomic deformation. The main subjects of investigation are the Painleve equations and the Garnier systems, which are a generalization in the case of several variables of the Painleve equations. At the beginning of researches of the present project, we have completed studies on Painleve equations by considering the Hamiltonian structure of the two types of the degenerated Painleve equations. The head investigator has published a book, entitled as "Painleve equations", written in Japanese, concerning to the theory of the Painleve equations and the Garnier systems from a viewpoint of holonomic deformation. This contains a historical and mathematical meaning of these integrable systems, and recent development of studies on them. The second aim of the project is to support younger researchers, not only Japanese ones but also foreigners, on theory of integrable systems ; many of them have attended the three international conferences, mainly supported by the project, where they have read their own results.