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.

我得到了有关超几何函数的以下结果。1）研究了附加到Selberg-TPE积分的（CO）同源组，评估了交叉数字，并发现了Selberg函数的组合特性。2）提出的共变量函数函数理论。找到了Kappa功能，以及一个与经典多项式的3个参数家族。该组，并将商嵌入到欧几里得空间中。6）研究了指数 - 散析是纯粹的象征性的，研究了超代方程的溶液的行为，并研究了参数空间与属2曲线的Teichmuler空间之间的关系。7）发明了超质量schwarz映射理论。 Schwarz地图的目标是球体。我们的高几何体具有3维双曲空间作为目标。从理论上讲，这是更自然的8）研究了三维谎言组的表面，尤其是SL（2，r）作用的表面。

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。

T.Ichikawa, M.Yoshida: "On Schottky groups arising from the hypergeometric equation with imaginary exponents"Proc. AMS. (to appear).

T.Ichikawa，M.Yoshida：“论由虚数指数超几何方程产生的肖特基群”Proc。