Conformal Mapping

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The main theme of Marshall's research program is to study conformal mappings generated by the Loewner differential equation, and related topics. The Loewner equation has as input an arbitrary continuous function and produces a continuous family of conformal mappings. Marshall plans to investigate properties of the solutions of Loewner's equation under various assumptions on the driving function, and conversely to investigate how properties of the boundaries of the associated regions are reflected in the driving function. This is a classical problem where progress has been made only recently. The Loewner equation is also related to an algorithm for numerical conformal mapping discovered by Marshall and K\"uhnau. Marshall will analyze convergence and error-estimates for the "zipper"' algorithm and improve the speed of convergence using "generational"' techniques. Conformal mappings have been used as a tool in science and engineering for many years. They are often used to change coordinates from a complicated region to a simpler region like a disc. A partial differential equation on the complicated region is then changed to a similar equation on the disc, a setting where it is easier to solve. Classically, this method was used for problems related to Laplace's equation, such as electrostatics and two dimensional fluid flow. Numerous non-classical applications have been developed in the last three decades such as electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, and heat transfer. The Loewner differential equation was introduced in 1923 to study extremal problems for conformal maps in the unit disc. Schramm's recently invention of stochastic Loewner evolution SLE, the fusion of Loewner's differential equation and probability, has formed a bridge between the important areas of conformal mapping in mathematics and conformal field theory in physics. It has led to the discovery of new results in percolation and random walks, for example, as well as the discovery mathematical proofs of results known to the theoretical physics community. This project is likely to increase the understanding of solutions to Loewner's equation, as foundational work, which should increase its usefulness in understanding stochastic processes. Broader impacts include the continued improvement and dissemination of the conformal mapping computer codes, which have been used by a number of investigators not in mathematics, as well as by mathematicians. Greater speed and new knowledge of convergence should lead to wider applicability and use of this algorithm. The mentoring of postdoctoral scholars and graduate students through our complex analysis "working seminar", has supported the work of several women. Support of our research increases the number of students interested in pursuing a career in this direction.