2005 Fiscal Year Final Research Report Summary
Geometric study of the hypergeometric function
| Project/Area Number |
14340049
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
YOSHIDA Masaaki KYUSHU UNIVERSITY, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30030787)
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| Co-Investigator(Kenkyū-buntansha) |
SASAKI Takeshi Kobe University, Faculty of Science, Professor, 理学部, 教授 (00022682)
IWASAKI Katsunori KYUSHU UNIVERSITY, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00176538)
MIMACHI Katsuhisa Tokyo Inst of Tech., Faculty of Science and engineering, Professor, 大学院・理工学研究科, 教授 (40211594)
MATSUMO Keiji Hokkaido University, Faculty of Science, Associate Professor, 大学院・理学研究科, 助教授 (30229546)
CHO Koji KYUSHU UNIVERSITY, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (10197634)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Selberg integral / twisted (co)homology / intersection number / hypergeometric / Whitehead-link / Lambda function / Schwarz triangle / hyperbolic Schwarz map |
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| Research Abstract |
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 natural
8)Studied the surfaces on which 3-dimensional Lie group acts, especially ones on which SL(2,R) acts.
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