Differential Equations and Reflection Groups(Polyhedral Harmonics, Hypergeometric Equations, and Painleve Equations)

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1.Holyhedral harminics : A general theory for polyhedral harmonics has been developed concerning the finite dimensionality of the space of polyhedral harmonic functions and those holonomic systems of partial differential equations which characterize the polyhedral harmonic functions. A survey article on the subject was written and the state of the art of the subject was addressed as a special lecture of the 2002 autumn meeting of the Mathematical Society of Japan.2.Hypergeometric equation :' An intersection theory for twisted de Rham cohomology groups associated with isolated singularities has been established. By developing Kumano-go-Taniguchi-type pseudodifirential calculus for Witten's Laplacian, a version of Hodge-Kodaira decomposition and Poincare-Serre-type duality theorems have been proved. As an application, the intersection matrices of generalized Airy functions have been determined explicitly in terms of skew-Schur polynomials. A certain cohomology theory for systems of inhom … More ogeneous finite difference equations has been constructed. The theory was applied to contiguity relations of confluent hypergeometric systems to compute their Gevrey cohomology groups.3.Painleve equations : The generating function for the rational solutions to the Painleve II equation has been determined explicitly in terms of the Airy function. The nonlinear monodromy of the Painleve VI equation has been realized as an action of the modular group on the four-parameter family of affine cubic surfaces. The phase spaces of the Painleve VI equation and the Gamier systems have been constructed algebro-geometrically as moduli spaces of stable parabolic connections. The affine-Weyl group symmetry of Backlund transformations has been constructed from the viewpoint of Riemann-Hilbert correspondence.4.Inverse bifurcation problems : The solvability of singular Wiener-Hopf equations has been investigated and applied to the inverse problem for bifurcation phenomena as well as to some reaction-diffsion models in mathematical biology. Less