Circle Packing: Discrete Conformal Geometry

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ABSTRACT Proposal: DMS-962280 PI: Stephenson Stephenson proposes to study fundamental properties of circle packings and the geometric structures they induce. Circle packings are configurations of circles with prescribed patterns of tangency. Until recently Stephenson has concentrated on the development of connections with classical analytic functions, contributing to what is now a very comprehensive and geometrically faithful "discrete" analytic function theory. His attention has now turned in other directions --- partly in towards the geometric foundations and partly out towards applications. The current proposal concentrates on three main issues: (1) existence and uniqueness of branched packings on the sphere (equivalently, of branched hyperbolic polyhedra in the ball); (2) properties of and uses for tailored random walks on circle packings; and (3) circle induced geometries in combinatoric settings, such as combinatorial Riemann mapping, Grothendieck dessins, knot projections, hyperbolic groups, and graph embedding. Of particular note is the effort by the Stephenson and his collaborators to introduce "inversive distances" into circle packing. Also, he continues numerical and experimental approaches to investigating phenomena in circle packing through use of a sophisticated software package, "test bench". Circle packing is a relatively new topic in mathematics which involves the study of configurations of circles with specified patterns of tangency. It seems an odd and artificial topic at first, but turns out to be surprisingly rich. Without giving technical details, one can say that in a circle packing the local combinatoric information about contacts between circles turns into rigid global geometric information on the overall configuration. The unique value in this approach lies in the fact that these packings turn out to carry in "discreet" packets much of the key geometric information normally described via "continuous" equations and formulas. This is especially important as we move towards the more discrete model of the world required by computers. Among the important areas of study are analytic and harmonic function theory -- crucial, for instance, in modeling of fluid flow, electrical circuits, diffusion processes, and so forth. Results are now being proven using circle packing which could not be established or were even unanticipated in the classical theories. Moreover, circle packing gives opportunities for new results and insights because it permits computer experimentation. Applications have begun to emerge in diverse areas of mathematics and in computer science; for instance, circle packings are good for graph embedding and for various computer visualization tasks, such as 3-D modeling. And the classical and discrete mathematical theory which underlies these advances is of importance.