Descartes’ circle theorem relates the curvatures of four mutually externally tangent circles, three “petal” circles around the exterior of a central circle, forming a “3-flower” configuration. We generalise this theorem to the case of an “n-flower”, consisting of n tangent circles around the exterior of a central circle, and give an explicit equation satisfied by their curvatures. The proof uses a spinorial description of horospheres in hyperbolic geometry.

## 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”.

## Oklahoma State topology seminar, November 2022

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”.

## Talk at Australian Conference on Science and Mathematics Education, September 2022

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

## Tsinghua topology seminar: A symplectic approach to 3-manifold triangulations and hyperbolic structures

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”.

## Monash topology talk on Symplectic approach to 3-manifold Triangulations, September 2022

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”.

## “There is always room for a new theory in mathematics”: Maryna Viazovska and her mathematics

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.

## A symplectic basis for 3-manifold triangulations

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.