Geometric study of the hypergeometric function

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。

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.