Mathematical Sciences: Complex and Harmonic Analysis

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Work will focus on recent developments in the theory of quasiconformal groups, the universal Teichmuller space and domains arising in approximation theory termed John domains. Some classical questions regarding quasiconformal mappings will also be reexamined in light of new results obtained by several researchers. Among these are questions about the approximation of quasiconformal maps in higher dimensions by differentiable maps of the same dilatation. Efforts to characterize the quasiconformal groups (discrete ones, first) have energized several top investigators including the principal investigator. The fundamental question is to determine when a group is effectively a Mobius group. Since this is not always the case, the problem of filling in the middle should generate some exciting research. Other work oriented more toward functional-analytical structures in complex analysis will focus on the study of cyclic vectors in spaces of analytic functions. These are elements of the spaces whose powers span the space. General conditions for cyclicity are only known for special spaces. However, for Bloch spaces only partial results are available and work will continue to complete the picture here. In addition, investigations will be carried out on injectivity criteria for local homeomorphisms, quasisymmetric groups and Markov processes. Analysis in the field of several complex variables will focus on outer functions in the polydisk and ball in two complex variables, especially when the zero set is small. An old problem of B. Levin on integrable functions on the quarter plane will be pursued. This work relates to a number of important areas of mathematics including three-manifold theory, partial differential operators and Toeplitz transformation spaces of analytic functions. It is also used in engineering applications involving systems theory.