Every proper (Gromov) negatively curved metric space whose boundary
contains a nontrivial continuum admits a $(2,C)$-quasi-isometric
embedding of a uniform binary tree. We apply this result to
various ``type'' problems, including questions of recurrence or
transience
of random walks on graphs and questions of parabolicity or hyperbolicity
of circle packings. Though our graphs and circle packings are locally
finite, there is no assumption of bounded degree, nor of any
isoperimetric condition. (20 pages)
Michigan Math. J. 45(1998), 31-53.
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A conformally regular pentagonal tiling of the plane with mesmerizing
combinatorics is constructed and dissected. Cool pictures!
Conformal Geom. Dynam.
1(1997),
58-86
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We study the structures on compact surfaces determined combinatorially in
Grothendieck's theory of dessins d'enfants; namely, affine,
reflective and conformal structures. A parallel discrete theory is
introduced based on circle packings and is shown to be geometrically
faithful, even at its coarsest stages, to the classical theory. The
resulting discrete structures converge to their classical counterparts
under a hexagonal refinement scheme. In particular, circle packing offers
a general approach for uniformizing dessin surfaces and approximating
their associated Belyi meromorphic functions. (64 pages with 3 tables
and 28 figures)
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with M. Hurdal, K. Stephenson, D. Sumners, K. Rehm, K. Schaper, D.
Rottenberg
We present a novel approach to creating flat maps of the brain. It is
impossible to flatten a curved surface in 3D space without metric and
areal distortion; however, the Riemann Mapping Theorem implies that it is
theoretically possible to preserve conformal (angular) information under
flattening. Our approach attempts to preserve the conformal structure
between the original cortical surface in 3-space and the flattened
surface. We demonstrate this with data from the human cerebellum and we
produce maps in the conventional Euclidean plane, as well as in the
hyperbolic plane and on a sphere. Since our flat maps exhibit
quasi-conformal behavior, they offer several advantages over existing
approaches. In particular, conformal mappings are determined canonically,
meaning that they are uniquely determined once certain normalizations have
been chosen, and this allows one to impose a coordinate system on the
surface when flattening in the hyperbolic or spherical setting. Unlike
existing methods, our approach does not require that cuts be
introduced into the original surface. In addition, hyperbolic and
spherical maps allow the map focus to be transformed interactively to
correspond to any anatomical landmark and adjust the locations of map
distortion. (8 pages with 3 color figures)
MICCAI'99, Lecture Notes in Computer Science 1679(1999),
279-286.
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Mathematical Reviews of Phil Bowers's papers
Home sweet home.