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SUMMARY:Arithmetic and Dynamics on Markoff-Hurwitz Varieties - Alexander G
 amburd (CUNY)
DTSTART:20180615T124500Z
DTEND:20180615T134500Z
UID:TALK98320@talks.cam.ac.uk
CONTACT:Richard Webb
DESCRIPTION:Markoff triples are integer solutions of the equation x<sup>2<
 /sup>&plus\;y<sup>2</sup>&plus\;z<sup>2</sup>=3xyz which arose in Markoff'
 s spectacular and fundamental work (1879) on diophantine approximation and
  has been henceforth ubiquitous in a tremendous variety of different field
 s in mathematics and beyond.  After reviewing some of these\, we will disc
 uss 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 g
 roup generated by Vieta involutions\, showing\, in particular\,  that for 
 almost all primes the induced graph is connected.  Similar results for com
 posite moduli enable us to establish certain new arithmetical properties o
 f Markoff numbers\, for instance the fact that almost all of them are comp
 osite.\nWe will also discuss recent joint work with Magee and Ronan on the
  asymptotic formula for integer points on Markoff-Hurwitz surfaces x<sub>1
 </sub><sup>2</sup>&plus\;x<sub>2</sub><sup>2</sup>&plus\;...&plus\;x<sub>n
 </sub><sup>2</sup> = x<sub>1</sub> x<sub>2</sub> ... x<sub>n</sub>\, givin
 g an interpretation for the exponent of growth in terms of certain conform
 al measure on the projective space.
LOCATION:CMS\, MR13
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