Local to global principles in integral circle packings
Recent developments in Analytic Number Theory May 01, 2017 - May 05, 2017
Location: MSRI: Simons Auditorium
Apollonian circle packing
11J70 - Continued fractions and generalizations [See also 11A55, 11K50]
37A45 - Relations with number theory and harmonic analysis [See also 11Kxx]
52C15 - Packing and covering in $2$ dimensions [See also 05B40, 11H31]
One of the most spectacular results on arithmetic of Apollonian circle packings is the "almost" local to global principle for curvatures in any given integral Apollonian packing as described by Bourgain-Kontorovich in 2014. The methods in their work, inspired originally by an observation of Sarnak’s in his letter to Lagarias on Apollonian circle packings, apply to a much larger class of circle packings. In this talk, we clarify what "almost" local to global means, and describe what the larger class is, as well as what aspects of the packings in this class seem necessary in order to conclude an "almost" local to global result and how they enter the proof. This is joint work with Stange and Zhang.
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