This project will be concerned with problems related to analytic functions of a complex variable and associated geometric questions. The principal investigator is particularly concerned with univalent functions defined in a disc or in the exterior of a disc. By virtue of their analyticity, these functions have power series expansions in the variable or its reciprocal. Objectives of this work include the analysis of extremal problems in terms of the series coefficients and a description of the image boundary for extremal maps. Techniques to be used involve methods of extremal metrics and quadratic differentials (techniques which are actually closely related) The principal investigator has displayed considerable expertise in using these methods in his past work. The coefficient function of the quadratic differential often gives precise information concerning the boundary image of an extremal map. Work will be done investigating this function further. In particular, the principal investigator will seek to establish whether or not this function must have simple roots. A new direction of research undertaken by the principal investigator focuses on iteration of analytic maps and Julia sets. These are sets representing the infinite past history of points carried by mapping iterates. Evidence suggests an interesting connection between the nature of Julia sets (simple or extremely irregular) and the decay of coefficients of univalent mappings. Work on such questions is still at the computer-experimental stage but appears to hold promise for interesting and unexpected relationships with dynamical systems.
