9401504 Minda This award supports a continuing program of mathematical research applying differential geometric methods to a variety of problems in complex function theory. The common thread joining the parts of the investigation is the use of conformal geometries, especially the three classical geometries: hyperbolic, euclidean and spherical. Comparison results will be sought, that is, relating quantities in one conformal geometry to like quantities in another. Work will also be done analyzing holomorphic functions viewed as mappings from on geometry to another. For example, Bloch functions are precisely those holomorphic functions from hyperbolic to euclidean geometry which have bounded distortion. Ongoing research is also focusing on two-point comparisons for conformal geometries. This will directly apply to the understanding of two-point distortions for univalent functions. Finally, work will be done using subordination principles as the basis for a systematic study of Bloch functions. This approach has yet to be employed seriously. This project represents research at the boundary of differential geometry and classical complex function theory. The tools available are considerable, in the hands of investigators who have mastered the techniques of both areas. ***
