SUMS: Tang -- Twisting the torus

This talk is about the interplay between closed loops on a torus,
rational numbers and continued fractions. I will prove a simple
formula for the number of intersections between two loops using
properties of certain elementary "cut, twist and glue" moves on the
torus. I will then introduce the Farey graph which encodes basic
intersection information on the set of loops and show how its geometry
relates to the Euclidean algorithm and lengths of continued fractions.

This talk will lead naturally to my postgraduate seminar talk later on
in the afternoon, so I encourage everyone to attend both.

Prerequisite knowledge: fractions, knowledge of 2x2 matrices, vectors,
and the ability to visualise donuts.