Colloquium, Ian Short, "Classifying SL2-tilings with the Farey graph"
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Classifying SL2-tilings with the Farey graph
In the 1970s, Coxeter studied certain arrays of integers that form
friezes in the plane. He and Conway discovered an elegant way of classifying
these friezes using triangulated polygons. Recently, there has been a good
deal of interest in expanding Conway and Coxeter’s ideas and relating them to
other mathematical structures, such as SL2-tilings. In this talk we
demonstrate how several significant theorems in this field can be explained
using the geometry and arithmetic of an infinite graph embedded in the
hyperbolic plane called the Farey graph. We also discuss how certain quotients
of the Farey graph can be used to obtain results on integer tilings modulo n.
It will be an accessible talk – for master's students and above.