The source of this mathematical research is a conjecture made by W. Thurston five years ago to the effect that if one packed circles inside planar simply connected domains and then associated the circles by a discrete map, then as the circles became smaller, the maps would converge to the analytic Riemann map between the two domains. Several proofs of the conjecture followed shortly thereafter. This project continues the study of connections between circle packings and analytic functions, with the aim of developing a geometrically faithful discrete analogue of parts of classical function theory. Initial developments in the application of circle packing relied on hexagonal packings. For reasons both aesthetic and practical, this restriction is being removed. Deeper insights into the fundamental behavior of circle packings is expected to result from this work. A departure from studies of circle packing in the plane to the setting of hyperbolic geometry is also leading to new connections between combinatorics and geometry. What has already emerged, for example, are hyperbolic analogues for the Schwarz lemma and Pick lemma in the context of circle packing maps. A new element recently introduced into these studies is that of using a random walk as a perspective from which to view the transition from one packing to the next. This model shows, for instance, how the effects of differential changes in boundary radii distribute themselves about the packing. The precision of the model is exact, reflecting the profound rigidity in circle packings. The first and most important goal of this work will be to determine the corresponding result of the Mostow rigidity theorem: Are infinite circle packings which fill the hyperbolic plane unique up to automorphisms? Work will also be done to answer the question of which compact surfaces support circle packings.
