Computational Uniformization

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The investigator applies emerging computational methods in circlepacking to study conformal structures on surfaces, with particularemphasis on complex, nonplanar surfaces. The global realization ofconformal structures which are local in nature is known classically asuniformization. Circle packing methods have led to notions of discreteuniformization which both mimic and approximate the classical notion.However, the geometric nature of the discretization raisescomputational issues outside traditional numerical mathematics. Inparticular, uniformization is a self-assembly process which isextremely challenging in practice --- for instance, with circlepackings containing millions of circles. In the central computationalwork, the investigator implements a recursive framework for managingthis self-assembly which avoids global distortions while accommodatingefficient parallel implementation. The computational issues are notconsidered in isolation, but rather in relation to ongoingapplications by the investigator and his collaborators to topicsincluding brain imaging, conformal tiling, dessins d'enfants, andconformal welding. Of the many theoretical issues raised inapplications and experiments, the investigator pays special attentionto the notion of ``flow uniformization'' and to its potential use indiscrete conformal welding and shape analysis.Surfaces --- a smooth soap film, the convoluted gray matter of thebrain, a faceted crystal lattice --- are ubiquitous in the natural andphysical sciences, engineering, computer visualization, and scores ofother areas. The tools for studying and describing surfacesmathematically came out of work in the nineteenth century, with aparticularly rich geometric vein associated with angular measure goingby the name "conformal structure". Among the most celebrated results inmathematics is the Riemann mapping theorem of 1851 which proved thatevery 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 conformalstructure, that is, that preserves angles. Wonderful as this theoryis, and despite the availability of huge computational resources, ithas been only in the last decade that new mathematics in the form ofcircle packing has provided a practical means for actually realizingRiemann's theorem. The investigator develops the mathematical andcomputational aspects of circle packing in the context of severalapplications. Among these are the flattening of human brain corticalsurfaces to aid analysis by neuroscientists, the study of planeshapes through a process known as conformal welding for use incomputer vision, and the construction of mathematical Riemann surfacesin various topics which are now amenable to experimentationfor the first time.