Mathematical Sciences: Topics in Geometric Function Theory

9401504 Minda该奖项支持将微分几何方法应用于复函数理论中的各种问题的持续数学研究计划。共形几何，特别是双曲、欧几里得和球面三种经典几何的使用，是连接研究各部分的共同主线。将寻求比较结果，即将一个保形几何中的量与另一个保形几何中相同的量联系起来。我们还将分析全纯函数，将其视为从一个几何到另一个几何的映射。例如，Bloch函数正是那些从双曲几何到欧几里得几何的具有有界失真的全纯函数。正在进行的研究也集中在保形几何的两点比较上。这将直接应用于理解单叶函数的两点扭曲。最后，将使用从属原则作为系统研究Bloch函数的基础。这一方法尚未得到认真应用。这个项目代表了微分几何和经典复变函数理论的边界研究。掌握了这两个领域的技术的调查人员手中有相当多的可用工具。***

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. ***