Computational Discrete Conformal Geometry and Applications

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9972769The investigator further develops the software tool CirclePack and uses it to explore mathematical questions that arise in several areas of application. Circle packing was introduced by Thurston, first in the context of orbifold constructions and then in the approximation of conformal mappings. The topic has been developed to the point that it provides a surprisingly comprehensive discrete theory that both parallels and approximates the classical theory of analytic functions, i.e., conformal mappings in two dimensions. Developments have been driven largely by classical connections --- discrete analyticity, extremal length, harmonic measure, Riemann mappings, and so forth --- and in turn those same connections account for circle packing's utility in a variety of other topics. In function theory, the research focuses on circle packing techniques in studies of a conjecture of Smale, conformal tilings, and construction of Grothendieck's dessins d'enfants. In applications, circle packing provides a unique new tool for graph embedding, and particular emphasis is placed on collaborations in brain-flattening and 2-dim quenching. Computability of circle packings is a central strength, but growing demands have placed considerable stress on the software package "CirclePack." The project addresses porting and modularization issues, and investigates strategies for implementing parallel packing algorithms.Many areas of science, technology, and mathematics rely in fundamental ways on combinatorial information, that is, patterns, both concrete and abstract --- crossing patterns in a twisted strand of DNA, atomic locations in a crystal, activation sites in the visual cortex. In 2-dimensional settings such patterns are often represented as planar graphs, which then need to be presented visually. A pleasing new approach has emerged from abstract topology. A "circle packing" is a configuration of circles having a specified pattern of tangencies. It has dual natures: "combinatoric" in the prescribed pattern, but "geometric" in the radii of the actual circles. One should think of the circles as bringing a "spontaneous" geometry to what began as a purely abstract combinatorial situation. Fortuitously, this geometry inherits key features of an important classical mathematical topic, conformal geometry, so it brings to applications over 150 years of theoretical development. However, unlike the classical theory, circle packings are both concrete and computable. This project involves the development and application of circle packing algorithms and display software, with particular emphasis on collaboration with an NIH project on "brain mapping," on the physics of 2-dimensional quenching, and on discrete analytic function theory in mathematical analysis.