Mathematical Sciences: Nielsen Extensions and Kernels of Riemann Surfaces

这个项目的目的是改进几个关于有限黎曼曲面的Nielsen扩张的不等式，推广一些关于Nielsen扩张的定理，并得到在Nielsen扩张过程中保持不变的曲面的模。这些结果将有助于更好地理解曲面与其Nielsen扩张之间的关系，以及黎曼曲面上的共形结构的分类。使用数学软件如数学和Geomview的数值方法将通过获得将尼尔森凸区映射到上半平面的保形函数的数值和图形近似来研究问题。

The purpose of the project is to refine several inequalities and to extend some theorems pertaining to Nielsen extensions of finite Riemann surfaces and to obtain moduli for surfaces which remain constant in the Nielsen extension process. These results would lead to a better understanding of the relationship between a surface and its Nielsen extension and the classification of conformal structures on Riemann surfaces. Numerical methods using mathematical software such as Mathematica and Geomview will be utilized in investigating the problems by obtaining numerical and graphical approximations of a conformal function mapping a Nielsen convex region onto the upper half-plane.