# Mathematics Colloquia and Seminars

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### Rubber Bands and Rational Maps

**Distinguished Lecture Series**

Speaker: | Dylan Thurston, Indiana University |

Related Webpage: | https://www.math.ucdavis.edu/research/seminars/thurston/ |

Location: | 1147 MSB |

Start time: | Wed, May 3 2017, 4:10PM |

Given a topological branched covering

fof the sphereS^2over itself, with branch values contained in a finite subsetPinS^2, canfbe realized as a rational map on the Riemann sphere? William Thurston gave a criterion in 1982: If the orbifold type offis hyperbolic, then it can be realized as a rational map if and only if there is no invariant multi-curve satisfying certain conditions. This condition is hard to apply in practice, since it involves checking infinitely many multi-curves.We give a complete positive combinatorial condition, using the notion of domination of graphs, a stricter condition than being 1-Lipschitz. We also give a physical interpretation of this condition in terms of elasticity of rubber-band spines .