Local-global principle in circle packings

Lundi 28 janvier 10:15-11:45 - Xin Zhang - IHES

Résumé : Starting with four mutually tangent circles, one containing the other three, an Apollonian circle packing (ACP) is formed by recursively inscribing one circle into three neighbouring circles. A most spectacular result on the arithmetic aspect of ACP due to Bourgain and Kontorovich is an « almost » local-global principle, which gives precise information on integers appearing as curvatures of circles from a fixed integral ACP. In recent years, integral circle packings of different conformal types have been constructed as limit sets of geometrically finite Kleinian groups. We identify the keys of Bourgain and Kontorovich’s work, and obtain an almost local-global principle for a broad class of integral circle packings. We explain how tools from analytic number theory, dynamics on hyperbolic 3-spaces and spectral graph theory come into the proof. This is joint work with Fuchs and Stange.

Lieu : salle 3L8

Local-global principle in circle packings  Version PDF