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

微分方程和反射群（多面调和、超几何方程和 Painleve 方程）

• 批准号: 12440043
• 负责人: IWASAKI Katsunori
• 金额: $ 5.76万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440043/
• 关键词: Painleve equations; hypergeometric equations; polyhedral harmonics; reflection groups; moduli spaces; twisted cohomology; modular groups; stable parabolic connections; 安定放物接続; エアリー関数; 超幾何関数; モノドロミー群; ウィナー・ホップ方程式; ウイナー・ホップ方程式; 逆分岐問題

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

1.全面体有害物质：本文对多面体调和函数空间的有限维性和刻画多面体调和函数的完整偏微分方程组，建立了多面体调和函数的一般理论。2002年日本数学学会秋季会议的特别演讲中，发表了关于这一问题的综述文章，并介绍了这一问题的最新进展。2.超几何方程：“与孤立奇点相关的扭曲de Rham上同调群的相交理论已经建立。通过对维滕拉普拉斯算子发展Kumano-go-Taniguchi型伪微分演算，证明了Hodge-Kodaira分解的一种形式和Poincare-Serre型对偶定理.作为应用，本文用斜Schur多项式明确地确定了广义Airy函数的交矩阵。一类非齐次系统的上同调理论 关于我们 构造了齐次有限差分方程。将该理论应用到合流超几何系统的邻接关系中，计算了合流超几何系统的Gevrey上同调群。3. Painleve方程：Painleve II方程的有理解的生成函数已明确地用Airy函数确定。Painleve VI方程的非线性单值性被实现为模群在四参数仿射三次曲面族上的作用。Painleve VI方程和Gamier系统的相空间已被代数几何地构造为稳定抛物联络的模空间。从Riemann-Hilbert对应的观点出发，构造了Backlund变换的仿射-Weyl群对称性。4.逆分支问题：研究了奇异Wiener-Hopf方程的可解性，并将其应用于分支现象的逆问题以及数学生物学中的一些反应扩散模型。少

项目成果

Kazuo Okamoto, Kyoichi Takano: "The proof of the Painleve property by Masuo Hukuhara"Funkcialaj Ekvacioj. 44(2). 201-217 (2001)

Kazuo Okamoto、Kyoichi Takano：“Masuo Hukuhara 的 Painleve 性质的证明”Funkcialaj Ekvacioj。

T.Sasaki, M.Yoshida: "Invariant theory for linear differential systems modeled after the Grassmannian Gr(n,2n)"Nagoya Journal of Mathematics. 171. 163-186 (2003)

T.Sasaki，M.Yoshida：“根据格拉斯曼 Gr(n,2n) 建模的线性微分系统的不变理论”名古屋数学杂志。

K.Iwasaki, Y.Kamimura: "Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology"Journal of Mathematical Biology. 43. 101-143 (2001)

K.Iwasaki、Y.Kamimura：“逆分岔问题、奇异维纳-霍普夫方程和生态学数学模型”数学生物学杂志。

J.Kamimoto: "Non-analytic Bergman and Szego Kernels for weakly pseudoconvex tube domains in C^2"Mathematische Zeitschrift. 236. 585-603 (2001)

J.Kamimoto：“C^2 中弱赝凸管域的非解析 Bergman 和 Szego 核”Mathematische Zeitschrift。

M.Yoshida et al.: "Recent progress of intersection theory for twisted (co) homology groups"Advanced Studies in Pure Mathematics. 27. 217-237 (2000)

M.Yoshida 等人：“扭曲（同）同调群的交集理论的最新进展”纯数学高级研究。