Mathematical Sciences: The Geometry of Image Surfaces of Complex Functions

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This project will involve work on complex function theory using potential theory and Riemann surface methods. The two areas of investigation are distinct but related. First, work will be done on the geometry of image surfaces of analytic functions - particularly inner functions defined in a disc, and entire functions. The research derives from older work of Gross on the structure of singularities of functions in the little Bloch space. The methods involve the construction of Riemann surfaces using Brownian motion and potential theory. A second thrust will be made into problems of circle packings in the plane, motivated by a recent conjecture of Thurston. It is now established that one may use circle packings to approximate classical conformal mappings. An alternate approach to this theme will be pursued based on random walks which are expected to provide a link between the graph and approximations to Brownian motion. This work is expected to have application to numerical approximations to partial differential equations as well as to potential theory and probability.