ABSTRACT Proposal: DMS-9622650 PI: Yang This proposal is concerned with two types of problems. One deals with the problems of characterizing domains (and/or their boundaries) in spaces on which certain quasiconformal maps exist. Examples include quasiconformal balls, quasiconformal reflection domains and fixed point sets of periodic quasiconformal maps. The study of these problems will lead to better understandings of the general problem regarding where exactly quasiconformal maps fit as an intermediate class of maps lying between homeomorphisms and diffeomorphisms, an interesting subject across several branches of mathematics including analysis, topology and differential geometry. The methods that will be employed include the usage of the newly introduced concept of strong uniformity in both homotopy and homology categories. This project will also involve research to explore more properties and applications of strongly uniform domains. Another type of problems deals with some conformal invariants of planar domains related to quasiconformal maps and the integrability of derivatives of conformal maps of the unit disk. Existing results indicate that there are close connections between these problems and the theory of quasiconformal maps in the plane. It is Yang's intention to explore more connections of this type. The study of these problems will shed new lights on how quasiconformal mapping theory can be applied to classical complex analysis. The major theme of this proposal is the investigation of the interplay between the geometry and topology of geometric figures and the analysis of various mappings living on those geometric figures. We study how the geometric shapes of domains, curves, or surfaces affect the analytic properties of certain functions defined on those objects and vise versa. This may lead to better understandings of the geometric and analytic structures of objects such as domains, curves, and surfaces that relate to the physical world.
