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SUMMARY:Bicycle tracks with hyperbolic monodromy – results and conjectur
 es
DTSTART:20260513T120000Z
DTEND:20260513T130000Z
DTSTAMP:20260613T125700Z
UID:indico-event-16408@indico.math.cnrs.fr
DESCRIPTION:Speakers: Gil Bor (CIMAT)\n\nOne associates with every closed 
 plane curve its "bicycle monodromy" -- an element of the Mobius group $PSL
 (2\,R)$\, defined via a dynamical system associated with the curve\, that 
 keeps appearing in different guises for over 100 years. The Menzin Conject
 ure (1906) states that this monodromy is hyperbolic if the curve is simple
  and encloses an area more than π. It has been proven in 2006 under the a
 ssumption that the curve is convex. The general case is still open. I will
  present two recent related results: (1) another sufficient condition for 
 hyperbolicity of the monodromy is that the curvature of the curve is bound
 ed above pointwise by 1 (but is not identically 1). (2) A necessary condit
 ion for hyperbolicity is that the length of the curve is greater than 2π.
  The main tool for both results is a “hyperbolic development” interpre
 tation of the bicycling monodromy of plane curves (rolling without slippin
 g and twisting of the hyperbolic plane along the given curve). This is joi
 nt work with Luis Hernandez (CIMAT) and Sergei Tabachnikov (Penn State). h
 ttps://arxiv.org/abs/2412.18676\n \n\nhttps://indico.math.cnrs.fr/event/1
 6408/
LOCATION:Salle Pellos\, bât. 1R2 (IMT)
URL:https://indico.math.cnrs.fr/event/16408/
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