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UID:20260601T033337EDT-7621Ub9fsK@132.216.98.100
DTSTAMP:20260601T073337Z
DESCRIPTION:\n	\n		\n			\n				\n					\n						\n							\n								\n									\n										\n											\n												 \n												TITLE / TITRE\n\n												The local-global co
 njecture for Apollonian circle packings is false\n													\n													ABSTRACT / RÉSUMÉ \n\n
 												Primitive integral Apollonian circle packings are fractal arrangements of 
 tangent circles with integer curvatures. The curvatures form an orbit of a
  'thin group\,' a subgroup of an algebraic group having infinite index in 
 its Zariski closure. The curvatures that appear must fall into one of six 
 or eight residue classes modulo 24. The twenty-year old local-global conje
 cture states that every sufficiently large integer in one of these residue
  classes will appear as a curvature in the packing. We prove that this con
 jecture is false for many packings\, by proving that certain quadratic and
  quartic families are missed. The new obstructions are a property of the t
 hin Apollonian group (and not its Zariski closure)\, and are a result of q
 uadratic and quartic reciprocity\, reminiscent of a Brauer-Manin obstructi
 on. Based on computational evidence\, we formulate a new conjecture. This 
 is joint work with Summer Haag\, Clyde Kertzer\, and James Rickards. Time 
 permitting\, I will discuss some new results\, joint with Rickards\, that 
 extend these phenomena to certain settings in the study of continued fract
 ions.\n\n												 \n\n												PLACE / LIEU \n													Hybride - CRM\, Salle / Room 5340\, Pavillon 
 André Aisenstadt\n													\n													ZOOM\n													https://us06web.zoom.us/j/84226701306?pwd=UEZ5NV
 pZaUlldW5qNU8vZzIvbEJXQT09\n											\n										\n									\n								\n							\n						\n					\n				\n			\n		\n	\n\n
DTSTART:20240202T203000Z
DTEND:20240202T213000Z
SUMMARY:Katherine Stange (University of Colorado\, Boulder)
URL:/mathstat/channels/event/katherine-stange-universi
 ty-colorado-boulder-355143
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