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.
Abstract:
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.