Geometric study of the hypergeometric function

超几何函数的几何研究

• 批准号: 14340049
• 负责人: YOSHIDA Masaaki
• 金额: $ 4.42万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340049/
• 关键词: Selberg integral; twisted (co)homology; intersection number; hypergeometric; Whitehead-link; Lambda function; Schwarz triangle; hyperbolic Schwarz map; 捩表裏路地群; 一寸来群; 背負ってる回路; 交叉形式; 黒写像; 又曲幾何; ベタ関数; 三次曲面; 又曲構造

I got the following results concerning the hypergeometric functions.1)Studied the (co)homology groups attached to Selberg-tpe integrals, evaluated the intersection numbers, and discovered a combinatorial properties of the Selberg functions.2)Presented co-variant function theory. Found the kappa function, and a 3-parameter families of hypergeometric polynomials, which are very different from the classical ones.3)Found a new infinite-product formula for the elliptic modular function Lambda.4)studied combinatorial-topologically the shape of the Schwarz triangles when the inner angles are general.5)Studied the Whitehead-link-complement group, constructed automorphic functions for this group, and embedded the quotient space to a Euclidean space.6)Studied the behavior of the solutions of the hypergeometric equation when the exponent-diffences are pure-imaginary, and studied the relation between the space of parameters and the Teichmuler space of genus 2 curves.7)Invented the theory of hyperbolic Schwarz map. The target of the Schwarz map has been the sphere. Our hypergeometric one has the 3-dimensional hyperbolic space as its target. Group theoretically it is more natural8)Studied the surfaces on which 3-dimensional Lie group acts, especially ones on which SL(2,R) acts.

研究了Selberg型积分的（上）同调群，计算了Selberg型积分的交数，发现了Selberg型积分的一个组合性质;提出了协变函数理论.发现了与经典超几何多项式不同的kappa函数和一个三参数超几何多项式族。3）发现了椭圆模函数λ的一个新的无穷乘积公式。4）组合拓扑地研究了内角为一般内角时施瓦茨三角形的形状。5）研究了Whitehead环补群，构造了该群的自守函数，6）研究了指数差为纯虚时超几何方程解的性质，研究了亏格为2的曲线的参数空间与Teichmuler空间的关系; 7）提出了双曲施瓦茨映射理论.施瓦茨地图的目标是球体。我们的超几何算法以三维双曲空间为目标。8）研究了三维李群作用的曲面，特别是SL（2，R）作用的曲面。

项目成果

Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory

与塞尔伯格积分相关的扭曲循环的交点数及其在共形场论中的应用

• 发表时间: 2004
• 期刊: Commun.Math.Phys. 250
• 作者: Mimachi;Yoshida
• 通讯作者: Yoshida

Intersection numbers for loaded cycles associated with Selberg-type integrals

与 Selberg 型积分相关的负载循环的交点号

• 发表时间: 2004
• 期刊: Tohoku J Math 56
• 作者: K.Mimachi;K.Ohara;M.Yoshida
• 通讯作者: M.Yoshida

T.Sasaki, M.Yoshida: "Schwarzian derivatives and uniformization"CRM Proc. And Lect Notes. 32. 271-286 (2002)

T.Sasaki，M.Yoshida：“Schwarzian 导数和均匀化”CRM Proc。

K.Mimachi, H.Ochiai, M.Yoshida: "Intersection theory of loaded cycles IV--resonant cases"Math. Nach. (to appear).

K.Mimachi、H.Ochiai、M.Yoshida：“加载循环的交叉理论 IV - 共振案例”数学。

Intersection numbers of twisted cycles and the correlation functions of the conformal field theory

扭曲循环的交数与共形场论的相关函数

• 发表时间: 2003
• 期刊: Commun.Math.Phys. 234
• 作者: K.Mimachi;M.Yoshida
• 通讯作者: M.Yoshida