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.
