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SUMMARY:Arithmetic and Dynamics on Markoff-Hurwitz Varieties
DTSTART:20180710T134500
DTEND:20180710T144500
DTSTAMP:20260408T035011Z
UID:17ad1c507e46b3641dfe3eb2f91f475410e8e68ec7b9ed650a83d5a3
CATEGORIES:Conferences - Seminars
DESCRIPTION:Alexander Gamburd (The Graduate Center\, CUNY)\nMarkoff triple
 s are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in 
 Markoff's spectacular and fundamental work (1879) on diophantine approxima
 tion and has been henceforth ubiquitous in a tremendous variety of differe
 nt fields in mathematics and beyond. After reviewing some of these\, we wi
 ll discuss joint work with Bourgain and Sarnak on the connectedness of the
  set of solutions of the Markoff equation modulo primes under the action o
 f the group generated by Vieta involutions\, showing\, in particular\, tha
 t for almost all primes the induced graph is connected. Similar results fo
 r composite moduli enable us to establish certain new arithmetical propert
 ies of Markoff numbers\, for instance the fact that almost all of them are
  composite. We will also discuss recent joint work with Magee and Ronan on
  the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_
 1^2+x_2^2 + \\dots + x_n^2 = x_1 x_2 \\dots x_n$\, giving an interpretatio
 n for the exponent of growth in terms of certain conformal measure on the 
 projective space.  
LOCATION:MA A3 30 https://plan.epfl.ch/?room=MAA330
STATUS:CONFIRMED
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