Spinors and horospheres

(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 Pl├╝cker coordinates, illuminating the connection between Grassmannians, hyperbolic polygons, and type A cluster algebras.

23-08-18_for_arxiv

Oklahoma State topology seminar, November 2022

In November 2022 I gave a zoom talk in the Oklahoma State University Topology seminar. It was on 9 November, although it was the 8th at the time in Oklahoma.

Title: Symplectic structures in hyperbolic 3-manifold triangulations

Abstract: 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 an 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 for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This talk involves joint work with Jessica Purcell, Joshua Howie and Yi Huang.

Slides from the talk are below.

22-11-08_oklahoma_talk_slides

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 organised through the Australian Council of Deans of Science.

Karen and I (mostly Karen) created a poster based on my experiences of teaching online during the pandemic, compared to other mathematics classes taught at the same time. The poster is titled “Accidental experiment in mathematics classroom shows how to engage online students”. It is available at the ACSME 2022 poster website and below.

Karen and I (again, mostly Karen) gave a talk at the conference, also entitled “Accidental experiment in mathematics classroom shows how to engage online students”.

The abstract was published in the Proceedings of The Australian Conference on Science and Mathematics Education (2022). It can be found at the proceedings website and below.

Title: Accidental Experiment in Mathematics Classroom Shows how to Engage Online Students

Abstract: Melbourne COVID rules during semester 1, 2021, unintentionally created a large comparative study between students learning mathematics entirely online and those with some face-to-face classes. An analysis of student results for semester 1 found that students enrolled in online mathematics tutorials had both consistently lower participation and lower final marks than on-campus students. Except for one first-year mathematics subject where there was no difference between the two groups of students.

Class participation can be used as a measure of student engagement (Alrajeh & Shidel, 2020). In this particular first-year mathematics subject, the Unit coordinator made significant efforts to create an inclusive environment reducing barriers to participation faced by online students. Students were given multiple opportunities and incentives to stay engaged. The tutorials were highly structured and students were placed into formal cooperative learning groups, creating a learning environment both collaborative and collegiate (Johnson, Johnson & Smith, 2006), facilitating individual accountability, intrapersonal relationships and social support.

22-09_Hogeboom_Mathews_ACSME_Poster

22-09_Hogeboom_Mathews_ACSME_proceedings_abstract

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

On 20 September 2022 I gave a zoom talk at the Yau Mathematical Sciences Center at Tsinghua University in Beijing. The talk was in the YMSC Topology Seminar.

Title: A symplectic approach to 3-manifold triangulations and hyperbolic structures

Abstract: 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 an 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 for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This talk involves joint work with Jessica Purcell and Joshua Howie.

Video: The talk was recorded and a video of the talk is available through the YMSC website here.

Slides: from the talk are below.

22-09-20_Tsinghua_talk_slides

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.

Title: A symplectic approach to 3-manifold triangulations and hyperbolic structures

Abstract: 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 an 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 for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This is joint work with Jessica Purcell and Joshua Howie.

Slides from the talk are below

22-09-14_monash_talk

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

On 30 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.

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

Abstract: In July of this year, Maryna Viazovska became the second woman and second Ukrainian to be awarded a Fields medal. Among other amazing achievements, she gave a wonderful proof of the most efficient way to pack spheres in 8 dimensions, bringing together ideas from all over mathematics. In this talk I’ll give a brief biography of Prof Viazovska and attempt to explain some of her mathematics.

The slides from my talk are available here (15mb pptx and 5mb pdf).

viazovska_talk

A symplectic basis for 3-manifold triangulations

Joint with Jessica Purcell – (45 pages) – on the arxiv

Abstract: 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.

SymplecticBasis_202208

Talk at Knots in Washington 49.75

On 22 April 2022 I gave a (virtual) talk at the 49.75’th (!) Knots in Washington conference, an international conference on knot theory held regularly since 1995.

Title: A symplectic basis for 3-manifold triangulations

Abstract: Neumann and Zagier in the 1980s introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We will discuss an interpretation for this symplectic structure in terms of the topology of the 3-manifold. This joint work with Jessica Purcell involves train tracks, Heegaard splittings, and is related to Ptolemy varieties, geometric quantisation, and the A-polynomial.

Slides from the talk are below.

washington_talk

A Symplectic Basis for 3-manifold Triangulations, AustMS 2021

On 8 December 2021 I gave a (virtual) talk in the Topology session of the 2021 meeting of the Australian Mathematical Society.

Title: A symplectic basis for 3-manifold triangulations

Abstract: Walter Neumann and Don Zagier in the 1980s introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold. We will discuss an interpretation for this symplectic structure in terms of the topology of the 3-manifold. This work, joint with Jessica Purcell, involves train tracks, Heegaard splittings, and is related to Ptolemy varieties, geometric quantisation, and the
A-polynomial.

Slides from the talk are below.

austms_talk_v2