The theory of transformation groups and its application

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The present project aimed studies on transformation groups, appearing in the various aspects of analysis. Investigating transformation groups of completely integrable systems defined in a complex domain, we obtaind a new point of view and various resultes particularly on this subject. The group of this project in the University of Tokyo concerned researchs on nonlinear integrable systems, in particular from the point of view of the algebraic theory. The main objects of our investigations is losted as follows :(1) Only special cases of partial differential equations admit fruitfull algebraic structure of their transformation groups. Since few of such equations, we attempt to construct the examples systematically.(2) We determine the transformation groups and their realization of nonlinear integrable system, in partcular these of the Garnier systems.(3) One of the most important points of studies on completely integrable systems is an application to theoretical physics. We try to establish the general method of determination of transformation groups of integrable systems.The investgators of this research project have continued their studies on transformation groups of integrable systems and announced their own results obtained during two years, 1994-95, in various occasins They have published some of results in journals.