We give an explicit bijective correspondence between between nonzero pairs of complex numbers, which we regard as spinors or spin vectors, and horospheres in 3-dimensional hyperbolic space decorated with certain spinorial directions. This correspondence builds upon work of Penrose–Rindler and Penner. We show that the natural bilinear form on spin vectors describes a certain complex-valued distance between spin-decorated horospheres, generalising Penner’s lambda lengths to 3 dimensions.
From this, we derive several applications. We show that the complex lambda lengths in a hyperbolic ideal tetrahedron satisfy a Ptolemy equation.
We also obtain correspondences between certain spaces of hyperbolic ideal polygons and certain Grassmannian spaces, under which lambda lengths correspond to Plücker coordinates, illuminating the connection between Grassmannians, hyperbolic polygons, and type A cluster algebras.
On 8 November 2022 I gave a zoom talk in the Oklahoma State topology seminar (although it was the 9th in Oklahoma). It was entitled “Symplectic structures in hyperbolic 3-manifold triangulations”.
On 29 September 2022, my Monash colleague Karen Hogeboom and I presented a poster and gave a talk at the 28th Australian Conference on Science and Mathematics Education (ACSME). The ACSME conference is an annual conference for tertiary science educators
On 20 September 2022 I gave a zoom talk in the Topology seminar at Tsinghua University, Beijing, entitled “A symplectic approach to 3-manifold triangulations and hyperbolic structures”.
On 14 September 2022 I gave a talk (in person!) in the Monash Topology seminar, entitled “A symplectic approach to 3-manifold triangulations and hyperbolic structures”.
In August 2022 I gave a talk about recent Fields medallist Prof Maryna Viazovska and some of her mathematical work. This was a Monash LunchMaths seminar.
In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give a geometric interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis.
On May 12, 2022, I gave a talk at Monash for undergraduate students on some of the mathematics of Maryam Mirzakhani.
On 22 April 2022 I gave a (virtual) talk at the 49.75th (!) Knots in Washington conference.
On 8 December 2021 I gave a (virtual) talk in the Topology session of the 2021 meeting of the Australian Mathematical Society.