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Title:
Combinatorics and Geometry to Arithmetic of Circle Packings
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Abstract:
Combinatorics and Geometry to Arithmetic of Circle Packings
Abstract: The Koebe-Andreev-Thurston/Schramm theorem assigns a conformally rigid fi-
nite circle packing to a convex polyhedron, and then successive inversions yield a conformally
rigid infinite circle packing. For example, starting with the tetrahedron, we take a configuration
of four pairwise tangent circles and invert successively to obtain the classical Apollonian
Circle Packing. The latter, an object of much recent study, is ”arithmetic”, in that there
are realizations for which all circles have curvatures in the rational integers. Our aim, in
joint work with Alex Kontorovich, is to classify polyhedra with this property and study the
integral curvatures of the resulting circle packings. We will start from scratch and report on
work toward this goal, with some emphasis on Archimedean and Catalan polyhedra.
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