This project seeks to develop further the methods and theory of numerical techniques applied to the study of problems arising in geometric function theory. While the basic numerical methods are generally available, their application to specific problems such as seeking global trajectory structures for quadratic differentials are far from perfected. Many fundamental questions can be reduced to those of solving for properties of a particular conformal map which is known to satisfy a particular differential equation. These equations, the result of applying variational techniques to single out extremal functions, are of apparent simple type. But the parameters they contain are defined by awkward side conditions. This makes the equations into functional differential equations - equations which first appeared in the theory of automorphic functions and now occur in the study of Riemann surface moduli and in various questions concerning ordinary differential equations of complex arguments. In addition to the development of general methods and, eventually, programs, work will be done on specific problem areas where new approaches should flow from the computational development. These areas include studies of the space of bounded nonvanishing univalent functions defined in a disc. This subject has been the focus of intense research in recent years but now is short of fresh new approaches to the resolution of many important questions. Work will also be done analyzing the class of univalent functions whose range covers a fixed disc. Finally efforts will be made to sharpen upper and lower estimates for the universal Bloch constants. There is known to be considerable room for progress here.
