This award supports mathematical research focusing on analytic and geometric problems associated with functions and domains defined in the complex plane. The work combines methods of harmonic analysis, quasiconformal mapping and geometric function theory. Certain domains which have been identified as being particularly suited to questions of extensions of analytic maps to quasiconformal maps, chord-arc domains, form a class whose generic properties will be analyzed. This will include the question of whether the class is connected. In addition, work will be done on the problem of factorizing bi-Lipschitz planar maps, using the boundedness of a the Cauchy integral to obtain geometric information about domains and obtaining distortion estimates for quasiconformal maps in higher dimensions. Included in the research will be work to extend the recent results of Bishop-Jones on quasicircles to the problem of determining which quasiconformal maps send the line onto a chord-arc curve. The theories of geometric function theory and quasiconformal mapping have contributed significantly to the understanding of geometric transformations of plane domains and domains in higher dimensions where the transformations are constrained by the amount of twisting or shrinking that is permitted (bounded dilatation). This project continues this tradition by taking up the study of several new concepts which have arisen in the past few years.
