Studies on Differential and Difference Equations by Means of Geometric and Algebraic Methods

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1. Polyhedral harmonics and invariant theory : K.Iwasaki settled a longstanding conjecture concerning the finite dimen- sionality of the space of polyhedral harmonic functions. He discovered new basic invariants of finite reflection groups in order to explicitly determine the polyhedral harmonic functions for polytopes with ample symmetry. In collaboration with A.Kenma and K.Matsumoto, he also made an explicit determination of polyhedral harmonic functions for the exceptional regular polytopes. He is planning to write a book on polyhedral harmonics and invariant theory.2. Cohomology for reccurence relations and hypergeometric functions : K.Iwasaki obtained a sharp asymptotic formula for solutions to certain difference equations. Based on this formula, he is planning to develop a cohomology theory for recurrence relations. K.Iwasaki and M.Kita discovered an exterior power structure on the twisted de Rham cohomology groups associated to hypergeometric functions. K.Iwasaki and K.Matsumoto … More obtained a conjecture that the intersection matrix of the twisted cohomology groups associated to generalized Airy functions can be expressed in terms of skew-Schur polynomials. Attempts to prove this are now in progress.3. Invese bifurcation problem and singular integral equations : K.Iwasaki and Y.Kamimura established the solvability of a class of singular integral equations. They applied this result to prove the existence of solutions to the inverse bifurcation problem for nonlinear Sturm-Liouville equations. Y.Kamimura is writing a book on integral equations which gives a detailed account of their results.4. Painlev_ equations and combinatorics : K.Iwasaki and H.Kawamuko discovered a combinatorial formula of Leibniz type associated to the Hamiltonian structure of the fourth Painlev_ equation in several variables. As an application, they obtained a new quadratic relation among Gegenbauer's orthogonal polynomials. H.Watanabe found birational transformations of solutions to the sixth Painlev_ equation. He also determined classical solutions to that equation. Less