Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Nearly 20 years ago, a paper of Graham-Lagarias-Mallows-Wilks-Yan on the number theory of Apollonian circle packings sparked an interest in the number theory community, which was just developing tools to handle arithmetic problems involving so-called thin groups. At the time, these packings were the only naturally occurring example of such an arithmetic problem, and naturally number theorists sprang upon the opportunity to discover all their rich properties, thinness notwithstanding. In 2010, building upon conjectures of Graham-Lagarias-Mallows-Wilks-Yan, the speaker together with her co-author Katherine Sanden gave evidence to what they called the Local to Global Conjecture for Apollonian circle packings, stating that in any integral packing, any large enough integer that satisfied certain congruence conditions modulo 24 must appear as a curvature in the packing. For 13 years, most everyone believed this conjecture to be true. In this talk, we will explore the history of this conjecture, and its fascinating downfall after Haag-Kertzer-Rickards-Stange proved that, in fact, infinitely many integral Apollonian packings fail to abide by the local to global principle, and come with extra obstructions from quadratic and quartic reciprocity.