Mathematical Sciences: Geometric Properties of Domains, Extremal Quasiconformal Mappings and Integrability of Conformal Mappings

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This award provides support for postdoctoral research on problems arising in the general area of geometric function theory. The work focuses on the geometric properties of quasiextremal distance domains. These are domains characterized by the property that points and closed sets can be joined by curves within the domain whose lengths do not be arbitrarily long. They are singled out because they have the characteristics believed to be fundamental for the study of quasiconformal mappings. Work on this project will concentrate on showing how the modulus of quasiextremal distance domains carries over to the dilatation constant for quasiconformal mapping defined on them. A second line of research will consider the length of level sets of univalent mappings and the question of the largest power to which such a mapping's derivative has a finite integral. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications of the theory to potential theory and fluid dynamics is now standard in engineering circles