Mathematical Sciences: Teichmuller Spaces & Analysis

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This project extends previous studies in complex function theory and its generalizations to quasiconformal mappings, quasiregular mappings and Teichmuller theory. All are tied by a geometric point of view regarding smooth transformations with restricted distortion. The main themes of this particular activity involve newly developed methods of conformal welding. One of the primary goals will be to study the uniqueness of conformally welded domains with complements of zero area. Preliminary results have also been obtained in the theory of iteration. The object of this investigation is to determine the types of Julia sets one can expect to get through welding by inner functions. Work will also be done on the question of whether one can gain insight into the Hilbert transform on finitely connected domains by maximizing the Fredholm determinant of the transform to obtain an equivalent domain all of whose boundary components are discs. 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.