On 9 December 2024 I gave a talk in the NUS Topology, Geometry and Dynamics seminar. It was entitled “Spinors and lambda lengths”.
Spinors and horospheres
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.
Spinors and Horospheres, Monash topology seminar, April 2023
On 26 April 2023 I gave a talk in the Monash topology seminar. It was entitled “Spinors and horospheres”.
The geometry of spinors in Minkowski space, ANZAMP February 2023
On 7 February I gave a talk at the 2023 ANZAMP meeting, entitled “The geometry of spinors in Minkowski space”.