Type: Seminar

Distribution: World

Expiry: 22 Sep 2011

Auth: duncans@p80714.pc.maths.usyd.edu.au

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.