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

1.全面体调和：发展了关于多面体调和函数空间的有限维多面体调和的一般理论，以及表征多面体调和函数的偏微分方程组的完整系统。写了一篇关于这一主题的综述文章，并将这一主题的最新进展作为日本数学学会2002年秋季会议的特别讲座。2.超几何方程：建立了与孤立奇点有关的扭曲de Rham上同调群的交理论。通过将Kumano-Go-Taniguchi-型伪微分演算推广到Witten‘s Laplace ian，证明了Hodge-Kodaira分解的一种形式和Poincare-Serre型对偶定理。作为应用，用斜舒尔多项式显式确定了广义艾里函数的交矩阵。Inhom-…系统的一个上同调理论构造了更均匀的有限差分方程组。将这一理论应用于合流超几何系统的邻接关系，计算了它们的Gevrey上同调群。3.Painleve方程：Painleve II方程有理数解的母函数已被显式地用Ary函数确定。Painleve VI方程的非线性单向性被实现为四参数仿射三次曲面族上模群的作用。Painleve VI方程和Gamier系统的相空间被代数几何地构造为稳定抛物联络的模空间。从Riemann-Hilbert对应的角度构造了Backlund变换的仿射-Weyl群对称性。4.逆分歧问题：研究了奇异Wiener-Hopf方程的可解性，并将其应用于分支现象的反问题以及数学生物学中的一些反应扩散模型。较少

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