Internal symmetry of soliton equations and its applications

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An algebro-geometric interpretation and construction (a la Krichever theory) of a general form of string equation, and some "genus zero" property it possesses in the situation which is closer to the bispectral problem, are shown.Calogero-Moser type KP solutions, related to bispectral problem and Krichever theory, is studied. Results include a proof, which does not assume bispectrality etc., of the fact that the spectral curve of any rank one ordinary differential operator with rational coefficients is unicursal.Various topics related to matrix integrals and combinatorics has been studied jointly with Adler and van Moerbeke. A formula counting the number of some vicious walks is given.BKP solutions of hypergeometric type is observed to be related to the sum of products of Schur Q-functions over all the strict partitions λ = (λ_1 > λ_2 > ・・・) with λ_1 【less than or equal】 h, studied by Tracy and Widom in the context of combinatorics problems (result of Sasha Orlov) .Elementary approach to the Schottky problem - Based on the Krichever theory and yet without using complex analysis on principally polarized abelian varieties, we worked out details of characterization of Jacobian varieties in terms of finitely many differential equations in the KP hierarchy.