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CATEGORIES:Geometric Group Theory (GGT) Seminar
SUMMARY:Arithmetic and Dynamics on Markoff-Hurwitz Varieti
es - Alexander Gamburd (CUNY)
DTSTART;TZID=Europe/London:20180615T134500
DTEND;TZID=Europe/London:20180615T144500
UID:TALK98320AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/98320
DESCRIPTION:Markoff triples are integer solutions of the equat
ion x^{2}&plus\;y^{2}&plus\;z^{
2}=3xyz which arose in Markoff's spectacular
and fundamental work (1879) on diophantine approxi
mation and has been henceforth ubiquitous in a tre
mendous variety of different fields in mathematics
and beyond. After reviewing some of these\, we w
ill discuss joint work with Bourgain and Sarnak on
the connectedness of the set of solutions of the
Markoff equation modulo primes under the action of
the group generated by Vieta involutions\, showin
g\, in particular\, that for almost all primes th
e induced graph is connected. Similar results for
composite moduli enable us to establish certain n
ew arithmetical properties of Markoff numbers\, fo
r instance the fact that almost all of them are co
mposite.\nWe will also discuss recent joint work w
ith Magee and Ronan on the asymptotic formula for
integer points on Markoff-Hurwitz surfaces x_{1
}^{2}&plus\;x_{2}^{2}
&plus\;...&plus\;x_{n}^{2} = x_{1} x_{2} ... x_{n}\, giving
an interpretation for the exponent of growth in te
rms of certain conformal measure on the projective
space.
LOCATION:CMS\, MR13
CONTACT:Richard Webb
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