Geometric study of the hypergeometric function

我得到了以下关于超几何函数的结果。1)研究了Selberg型积分上的（co）同调群，求出了交数，发现了Selberg函数的组合性质。2)提出协变函数理论。找到了kappa函数，以及与经典多项式有很大不同的三参数超几何多项式族。3)建立了一个新的椭圆模函数lambda的无穷积公式。4)从组合拓扑上研究了内角为一般情况下Schwarz三角形的形状。5)研究Whitehead-link-complement群，构造该群的自同构函数，并将商空间嵌入到欧几里得空间。6)研究了指数差为纯虚数时超几何方程解的性质，研究了参数空间与2属曲线的Teichmuler空间的关系。7)发明了双曲施瓦茨图理论。施瓦茨地图的目标一直是球体。我们的超几何模型以三维双曲空间为目标。研究了三维李群作用的曲面，特别是SL（2，R）作用的曲面。

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.