BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:William Chen (Columbia University) DTSTART:20210325T210000Z DTEND:20210325T220000Z DTSTAMP:20260423T005826Z UID:JNTS/18 DESCRIPTION:Title: MA RKOFF TRIPLES\, NIELSON EQUIVALENCE\, AND NONABELIAN LEVEL STRUCTURES\ nby William Chen (Columbia University) as part of Columbia CUNY NYU number theory seminar\n\n\nAbstract\nFollowing Bourgain\, Gamburd\, and Sarnak\, we say that the Markoff equation $x^2 + y^2 + z^2 − 3xyz = 0$ satisfies strong approximation at a prime p if its integral points surject onto its $F_p$ points. In 2016\, Bourgain\, Gamburd\, and Sarnak were able to esta blish strong approximation at all but a sparse (but infinite) set of prime s\, and conjectured that it holds at all primes. Building on their results \, in this talk I will explain how to establish strong approximation for a ll but a finite and effectively computable set of primes\, thus reducing t he conjecture to a finite computation. Using the connection between the Ma rkoff surface and the character variety of SL(2) representations of the fu ndamental group of a punctured torus\, this result becomes a corollary of a more general divisibility theorem on the cardinalities of Nielsen equiva lence classes of generating pairs of finite groups\, which in turn follows from a simple observation regarding the degree of a certain line bundle o n the moduli stack of elliptic curves with nonabelian level structures. As time allows we will also describe some applications.\n LOCATION:https://researchseminars.org/talk/JNTS/18/ END:VEVENT END:VCALENDAR