New approaches to potential theory and conformal mapping

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The Riemann mapping theorem is an old, beautiful, and tremendously useful device which can be used to pull back the explicit formulas for the classical objects of potential theory on the unit disc to arbitrary simply connected regions in the plane. The unit disc is the quintessential example of a double quadrature domain,i.e., a quadrature domain both with respect to area measure and boundary arc length. Professor Bell and his collaborators have been working on an improvement to the Riemann mapping theorem which replaces the unit disc by a double quadrature domain which, rather than being far away like the unit disc, is close to the original domain. Bell has proved that many of the classical objects of potential theory associated to a double quadrature domain are elementary functions which are simple combination of the coordinate variable and the algebraic Schwarz function, and consequently may be easily computed. Thus, it becomes possible to make minor, but subtle, alterations in a region to make it have many of the desirable properties of the unit disc. This approach has the virtue of being potentially applicable even in the multiply connected and Riemann surface setting, where the Riemann mapping theorem is unavailable.The objects of potential theory and conformal mapping are pervasive in Mathematics, Science, and Engineering. Although these objects are well studied, they continue to be a source of exciting and potentially applicable new mathematics. Professor Bell will express the classical objects of potential theory associated to a two dimensional surface with holes in terms of simple and computable analytic objects. These results may give rise to new and practical methods for expressing and zipping the solutions to classical problems in differential equations, conformal mapping, and potential theory. Because humans best perceive higher dimensional objects by taking a series of two dimensional slices, the tools developed could find important applications. Some of the key ideas behind this program grew out of a summer undergraduate research project that Bell directed in 2008, and a noteworthy component of the project is a significant commitment to mentoring and training undergraduates and graduate students involved for a career in mathematics.