Computational Uniformization

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

The investigator applies emerging computational methods in circle packing to study conformal structures on surfaces, with particular emphasis on complex, nonplanar surfaces. The global realization of conformal structures which are local in nature is known classically as uniformization. Circle packing methods have led to notions of discrete uniformization which both mimic and approximate the classical notion. However, the geometric nature of the discretization raises computational issues outside traditional numerical mathematics. In particular, uniformization is a self-assembly process which is extremely challenging in practice --- for instance, with circle packings containing millions of circles. In the central computational work, the investigator implements a recursive framework for managing this self-assembly which avoids global distortions while accommodating efficient parallel implementation. The computational issues are not considered in isolation, but rather in relation to ongoing applications by the investigator and his collaborators to topics including brain imaging, conformal tiling, dessins d'enfants, and conformal welding. Of the many theoretical issues raised in applications and experiments, the investigator pays special attention to the notion of ``flow uniformization'' and to its potential use in discrete conformal welding and shape analysis. Surfaces --- a smooth soap film, the convoluted gray matter of the brain, a faceted crystal lattice --- are ubiquitous in the natural and physical sciences, engineering, computer visualization, and scores of other areas. The tools for studying and describing surfaces mathematically came out of work in the nineteenth century, with a particularly rich geometric vein associated with angular measure going by the name "conformal structure". Among the most celebrated results in mathematics is the Riemann mapping theorem of 1851 which proved that every surface with a conformal structure, no matter how complicated, can be identified with one of three very simple familiar surfaces --- a ball, a plane, or a disc --- in a way that preserves conformal structure, that is, that preserves angles. Wonderful as this theory is, and despite the availability of huge computational resources, it has been only in the last decade that new mathematics in the form of circle packing has provided a practical means for actually realizing Riemann's theorem. The investigator develops the mathematical and computational aspects of circle packing in the context of several applications. Among these are the flattening of human brain cortical surfaces to aid analysis by neuroscientists, the study of plane shapes through a process known as conformal welding for use in computer vision, and the construction of mathematical Riemann surfaces in various topics which are now amenable to experimentation for the first time.