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UID:8570@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20250506T110000
DTEND;TZID=Europe/Paris:20250506T120000
DTSTAMP:20250430T084618Z
URL:https://www.i2m.univ-amu.fr/evenements/litman/
SUMMARY:Matthew Litman (University College\, Dublin): Markoff mod-p Graphs:
  Connectivity\, Maximal Divisors\, and Generalizations
DESCRIPTION:Matthew Litman: Markoff triples were introduced in 1879 and hav
 e a rich history spanning many branches of mathematics. In 2016\, Bourgain
 \, Gamburd\, and Sarnak answered a long standing question by showing there
  exist infinitely many composite Markoff numbers. Their proof relied on sh
 owing the connectivity for an infinite family of graphs associated to Mark
 off triples modulo p for infinitely many primes p\, whereas it was conject
 ured in 1991 by Baragar that these graphs are connected for all p.\n\nIn t
 his talk\, we discuss our recent work confirming this conjecture for all p
 rimes p&gt\;3448×10³⁹² by employing results of Chen and Bourgain\, Ga
 mburd\, and Sarnak. We introduce the notion of maximal divisors as a key t
 ool in our proof and prove sharp asymptotic and explicit upper bounds on t
 he number of them. After showing connectivity in the standard Markoff sett
 ing\, we discuss what happens for the projective analogue of Markoff tripl
 es\, that is surfaces W in P¹xP¹xP¹ cut out by the vanishing of a (2\,2
 \,2)-form that admit three non-commuting involutions and are fixed under c
 oordinate permutations and double sign changes. Inspired by the work of BG
 S\, we investigate such surfaces over finite fields\, specifically their o
 rbit structure under their automorphism group. For a specific one-paramete
 r subfamily Wₖ of such surfaces\, we construct finite orbits in Wₖ(C) 
 by studying small orbits that appear in Wₖ(Fₚ) for many values of p an
 d k. This shows that the connectivity property for the standard Markoff ca
 se fails for its projective counterpart.\n\nThis talk is based on separate
  joint works with J. Eddy\, E. Fuchs\, D. Martin\, and N. Tripeny\, as wel
 l as with E. Fuchs\, J. Silverman\, and A. Tran respectively.
CATEGORIES:Séminaire,Ernest
LOCATION:I2M Luminy - TPR2\, Salle 210-212 (2e étage)\, 163\, avenue de Lu
 miny\, Marseille\, 13009\, France
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=163\, avenue de Luminy\, Ma
 rseille\, 13009\, France;X-APPLE-RADIUS=100;X-TITLE=I2M Luminy - TPR2\, Sa
 lle 210-212 (2e étage):geo:0,0
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