Joint with Nguyen Thanh Tung Le – (62 pages)
Abstract: Kauffman’s clock theorem provides a distributive lattice structure on the set of states of a four-valent graph in the plane. We prove two distinct generalisations of this theorem, for four-valent graphs embedded in more general compact oriented surfaces. The proofs use results of Propp providing distributive lattice structures on matchings on bipartite plane graphs, and orientations on graph with fixed circulation.
On Monday 11 August 2025, I gave a talk in a series of Lightning Talks in the Melbourne University Topology Seminar.
As a “Lightning Talk”, it was 5 minutes only. In which I attempted to explain the idea of our recent result with Orion Zymaris giving some equations for circle packings.
Short, but fun.
Joint with Dionne Ibarra and Jessica Purcell – (27 pages) – on the arxiv
Abstract: In this paper we construct two different explicit triangulations of the family of double twist knots K(p,q) using methods of triangulating Dehn fillings, with layered solid tori and their double covers. One construction yields the canonical triangulation, and one yields a triangulation that we conjecture is minimal. We prove that both are geometric, meaning they are built of positively oriented convex hyperbolic tetrahedra. We use the conjecturally minimal triangulation to present eight equations cutting out the A-polynomial of these knots.
On 9 December 2024 I gave a talk in the Topology, Geometry and Dynamics seminar at the National University of Singapore.
Title: Spinors and lambda lengths
Abstract: Work of Penrose and Rindler in the 1980s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors as flags in Minkowski space. We present some extensions of this work involving hyperbolic geometry. In particular, we give a correspondence between between spinors and horospheres with certain decorations, and show how spinors can be used to calculate a complex-valued, 3-dimensional generalisation of Penner’s lambda lengths. Time permitting, we may discuss some further applications, including inter-cusp distances in knot complements, higher-dimensional generalisations, and circle packings.
Slides from the talk are below.
Spinors_and_lambda_lengths_24-12-09_for_webOn 27 March 2024 I gave a talk in the Topology seminar at Monash.
Title: Contact geometry, Heegaard Floer homology, and skein theory
Abstract: I’ll give an introduction to contact geometry, Heegaard Floer homology, and some aspects of these subjects that are very combinatorial and skein-theoretic.
Slides from the talk are below.
24-03-27_Monash_talkJoint with Orion Zymaris – (17 pages) – on the arXiv
Abstract: 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.
(24 pages) – on the arXiv
Abstract: 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.
23-08-18_for_arxivOn 26 April 2023 I gave a talk in the Monash topology seminar.
Title: Spinors and horospheres
Abstract: Work of Penrose and Rindler in the 1980s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors in terms of Minkowski space. We present some extensions of this work, involving 3-dimensional hyperbolic geometry. In particular, we give a correspondence between between nonzero pairs of complex numbers, called spin vectors, and horospheres in 3-dimensional hyperbolic space decorated with certain spinorial directions. We show that the natural inner product on spin vectors describes a certain complex-valued distance between decorated horospheres, generalising Penner’s lambda lengths, and giving various applications.
Slides from the talk are below.
23-04-26_Monash_talk_slides
On 7 February I gave a talk at the 2023 ANZAMP meeting. Held in Hobart, Tasmania, this was the annual conference of ANZAMP, the Australian and New Zealand Association of Mathematical Physics.
Title: The geometry of spinors in Minkowski space
Abstract: Work of Penrose and Rindler in the 1908s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors in terms of Minkowski space. We present some extensions of this work, involving 3-dimensional hyperbolic geometry.
Slides from the talk are below.
Mathews_ANZAMP_talk_geometry_spinors_MinkowskiOn 15 November 2022 I gave a talk as part of the STELR (Science and Technology Education Leveraging Relevance) program. They run STEM webinars to inspire school students. This was a short webinar for school students, talking about my career path and work, followed by discussion with a facilitator.
Here’s the blurb for the talk:
“Hi, I’m Dan. I’m a mathematician. I work at Monash Uni.
I’ve been into maths since I was very young. At school I enjoyed learning lots of things, including mathematics and science and history and everything. I’ve always wanted to know how the world works.
At school I found out I enjoyed maths, and I got involved in the Maths Olympiad. I actually represented the country at the International Maths Olympiad. I’ve been doing maths ever since. I’ve done maths in California, in Boston, in France, before coming home to Australia and doing it here too. I’ve also worked in IT and as a lawyer and a writer. Nowadays I teach mathematics and do mathematical research at Monash University.”
The slides from my talk are available here (11mb pptx and 2mb pdf).
STELR_talk