Mathematical Sciences: Geometric Analysis: Research and Conformal Mapping, Extremal Length, and Riemann Surfaces

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Burton Rodin will continue investigations into a newly developed computational approach to the Riemann mapping theorem. This follows from the discovery of connections between conformal mapping and circle packing. Such investigations should lead to new insights into both areas and to new techniques in numerical conformal mapping. In addition research will be carried out into other areas of geometric function theory. This research will involve investigations into the following areas: the relationship between the boundary behavior of the Riemann mapping function and the geometry of the region; the application of Brownian excursion laws in the study of conformal mapping; holomorphic and quasiconformal families of Riemann mapping functions; harmonic measure estimates; the conformal mapping of curvilinear polygons; extremal length and harmonic measure relationships.