On 9 December 2020 I gave a (virtual) talk in the Topology session of the 2020 meeting of the Australian Mathematical Society.

Title: Ptolemy vs Thurston in Hyperbolic Geometry and Topology.


Bill Thurston (1946-2012 CE) developed a system of great simplicity and power for understanding hyperbolic 3-manifolds. In particular, he introduced equations whose variables encode the shapes of ideal hyperbolic tetrahedra and whose solutions describe hyperbolic structures on 3-manifolds.

Claudius Ptolemy (c.100-170 CE), better known for developing a rather different system, proved in his Almagest an equation about the lengths of sides and diagonals in a cyclic quadrilateral. Such Ptolemy equations arise in numerous places across mathematics, including in 3-dimensional hyperbolic geometry and representation theory.

In this talk I’ll discuss how Ptolemy and Thurston equations provide complementary perspectives on hyperbolic geometry and topology.


Ptolemy vs Thurston in Hyperbolic Geometry and Topology, AustMS 2020

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