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Circle packings, thin orbits and the arithmetic of imaginary quadratic fields

Banff International Research Station for Mathematical Innovation and Discovery

Integral Apollonian circle packings are certain fractal packings of the plane with circles of disjoint interior, and integer curvatures. The set of curvatures which appears has been of recent interest as a challenging problem in the study of orbits of thin groups. Work of Bourgain, Fuchs and Kontorovich culminated in the demonstration that density one of the integers appear as curvatures, up to a congruence restriction. In this talk, we'll rediscover Apollonian circle packings as part of the essential nature of the Gaussian integers and their Diophantine approximation, generalize to other quadratic fields to discover new circle packings, and discuss the extension of results on curvatures to these and other Kleinian packings.

Mathematics; Number theory; Algebraic geometry

video/mp4

Author affiliation: University of Colorado Boulder

2017-09-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63040

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/