Geometric study of the hypergeometric function

1)研究了Selberg-TPE积分的(上)同调群，计算了交点数，发现了Selberg函数的一个组合性质。2)提出了协变函数理论。3)给出了椭圆模函数Lambda的一个新的无穷积公式。4)研究了内角一般时Schwarz三角形的形状。5)研究了Whitehead-link-补群，构造了该群的自同构函数，并将商空间嵌入到欧氏空间中。6)研究了指数差为纯虚数时超几何方程的解的行为，并研究了参数空间与亏格2曲线的Teichmuler空间之间的关系。7)提出了双曲Schwarz映射理论。施瓦茨地图的目标一直是球体。我们的双几何模型以三维双曲空间为目标。群理论上更自然的是研究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.