BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Josiah Sugarman (Hebrew University of Jerusalem) DTSTART:20250520T180000Z DTEND:20250520T182500Z DTSTAMP:20260423T041655Z UID:CANT2025/20 DESCRIPTION:Title: Explicit spectral gap for the quaquaversal operator\nby Josiah Sugar man (Hebrew University of Jerusalem) as part of Combinatorial and additive number theory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Scien ce Center (4th floor).\n\nAbstract\nThe spectral gap of an operator is the gap between the largest eigenvalue and the rest of the spectrum. In the m id 90s\, John Conway and Charles Radin introduced a three dimensional subs titution tiling\, the Quaquaversal Tiling\, with the property that the ori entations of its tiles equidistribute faster than what is possible for two dimensional substitution tilings. Conway and Radin showed that the orient ations of the tiles were dense in $SO(3)$ and implicity introduced an oper ator (later explicitly studied by Draco\, Sadun\, and Van Wieren) whose sp ectral gap controls the equidistribution rate.\nDraco\, Sadun\, and Van Wi eren studied the eigenvalues of this operator numerically and conjectured that it has a spectral gap bounded below by approximately $0.0061697$. \nW e exploit a fact\, due to Serre\, that the group of orientations for this tiling is $2$-arithmetic and follow a strategy similar to Lubotzky\, Phill ips\, and Sarnak's in order to obtain a lower bound of about $0.0061711$\, resolving the conjecture.\n LOCATION:https://researchseminars.org/talk/CANT2025/20/ END:VEVENT END:VCALENDAR