Finite dimensional integrable structure in systems with infinite degree of freedom

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This research project aimed to search for various finite dimensional integrable systems in systems with infinite degrees of freedom, and to elucidate their mathematical structures. Related issues on geometry and non-integrable systems were also investigated.The head investigator obtained interesting results on Whitham deformation equations, isomonodromic deformations, systems arising in supersymmetric/topological gauge theories and Calogero-Moser systems, all of which are mutually related. Firsly, he could find an explicit form of the Whitham deformation equations for asymptoci description of isomonodromic systems on the Rie-mann sphere. Secondly, an extension, to a torus, of such isomonodromic systems was achieved on the basis of methods in solvable spin sysmtes and conformal field theories. Thirdly, he clarified the roles that the tau functions and the Whitham deformations play in four dimensional supersymmetric and topolotical gauge theories. Finally, he considered the elliptic Calogero-Moser systems, which are also closely related to four dimensional supersymmetric gauge theories, and discoveref that a non-autonomous analogues of those systems give an example of isomonodromic dermations on a torus.The achievement due to other members of the project group respectively ranges over low-dimensional topology, stochastic analysis of hypoelliptic operators, complex dynamical systems, transcendental number theory related to nonlinear dynamical systems, celular automata, intelligent agents, hypoelliptic and hyperbolic equations with infnite degeneracy,