Mathematical Sciences: Conformal Mapping, Riemann Surfaces, and Circle Packings

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This project combines the mathematics of classical geometric function theory with newly discovered applications of circle packing. The source of this connection lies in a conjecture of W. Thurston in 1985 that one should be able to obtain good approximations to the Riemann map of a simply connection planar domain through a circle packing of the domain following by a mapping of the circles by a prescribed algorithm. This turned out to be the case and is now known as the Finite Riemann Mapping Theorem. The proof spawned a new line of investigation within the field of complex analysis: what type of theorems can be proved by circle packing and, more importantly, what new discoveries are opened up by means of this new technique? Work now continues as the breadth of possibilities broadens. Topics to be investigated include hexagonal packing and the asymptotics and the ratios of maximum and minimum radii, the classical problem of canonical conformal maps of multiply connected regions - whether arbitrary domains are conformally equivalent to domains with boundary components of circles or line. Work will also be done in extending results on osculating Mobius transformations of a circle packing which are known to converge and give the first and second derivatives of the Riemann map. The question of whether higher order derivatives can also be approximated will be examined. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications to potential theory and fluid dynamics is now standard in engineering circles.