Papers:

1. A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs
   (Submitted)
   [arXiv]
2. Dynamic nowcast of the New Zealand greenhouse gas inventory.
   (with Hannah Chorley, Flynn Ownen, Tamsyn Hilder, Holly Trowland and Paul Bracewell)
   Environmental Modelling & Software 167 (2023): 105745.
   [doi, arXiv]
3. Filtering germs: groupoids associated to inverse semigroups.
   (with Becky Armstrong, Lisa Orloff Clark, Astrid an Huef and Ying-Fen Lin)
   Expositiones Mathematicae 40.1 (2022): 127-139.
   [doi, arXiv]

Invited talks:

• York Semigroup Seminar 2 April 2025

  Title and abstract

  Title: Noncommutative Stone duality and nonfinitely aligned categories of paths

  Abstract: The power set of a given set is a Boolean algebra, and Boolean algebras identify with Boolean spaces by Stone duality. This is only the commutative case in that intersection of sets is commutative. The collection of partial bijections of the given set is an inverse semigroup. This is the noncommutative case in that composition of partial bijections is not commutative. Lawson (2010) showed there is a noncommutative Stone duality identifying some inverse semigroups with some groupoids. In this talk, we will discuss classical and noncommutative Stone duality. We will then suppose the given set has the extra structure of a category of paths (a broad generalisation of the finite-path category of a directed graph). In this example, the groupoids coming from noncommutative Stone duality have a nice description, provided the category of paths is finitely aligned. The groupoids of nonfinitely aligned categories of paths, such as those of Spielberg (2020), are hard to describe. We will discuss results from my thesis that attempt to simplify the treatment of nonfinitely aligned categories of paths. In doing so, we uncover examples of constellations as introduced by Gould and Hollings (2010).

• Leeds Algebra Seminar 1 April 2025

  Title and abstract

  Title: Reconstructing algebras from filters with a twist

  Abstract: A large class of algebras (those admitting quasi-Cartan subalgebras) can be reconstructed from naturally occurring inverse semigroups and groupoids. I will describe my work with collaborators comparing constructions of groupoids from inverse semigroups that supported this reconstruction. I will not assume any background on inverse semigroups or groupoids.

• Wollongong Operator Algebras and Noncommutative Geometry Seminar December 2024

  Title and abstract

  Title: Path groupoids of nonfinitely aligned \(P\)-graphs

  Abstract: The directed graph approach to C*-algebras was generalised to \(P\)-graphs by Brownlowe–Sims–Vittadello in 2013. The unit space of the usual path groupoid of a \(P\)-graph can fail to be locally compact if the \(P\)-graph is not finitely aligned. In 2020, Spielberg developed a groupoid for any left cancellative small category (for example, any \(P\)-graph) that is ample and not necessarily Hausdorff. In 2023, Neshveyev–Schwartz gave a nonfinitely aligned left cancellative small category whose reduced C*-algebra is not modeled by Spielberg's groupoid. Moreover, Spielberg's groupoid is complicated to construct. In this talk, we associate path groupoids to (not necessarily finitely aligned) \(P\)-graphs that are ample, Hausdorff and relatively easy to construct. Our path groupoid coincides with the path groupoid of finitely aligned \(P\)-graphs and is very different from Spielberg's groupoid for nonfinitely aligned \(P\)-graphs. These results appear in my thesis, which I successfully defended in October this year, supervised by Lisa Orloff Clark and Astrid an Huef.

• Queen's University Belfast Mathematical Sciences Research Centre June 2023

• Algebra, Geometry and C*-algebras , ICMS, Bayes Centre, Edinburgh June 2023

• WWU Münster Kleines Seminar May 2023

• Wollongong OANCG Seminar April 2023

  Title and abstract

  Title: Ample groupoid of a (finitely aligned) \(P\)-graph where \(P\) embeds in a group

  Abstract: Beginning in the 80s with the work of Enomoto, Fujii and Watatani, directed graphs and their generalisations (such as the \(P\)-graphs of Brownlowe, Sims and Vittadello) have been used to model C*-algebras. Associating an auxiliary groupoid to the graph has been a fruitful technique. This was demonstrated for directed graphs by Kumjian, Pask, Raeburn and Renault in the 90s, and demonstrated for finitely aligned higher-rank graphs by Kumjian, Pask, Farthing, Muhly and Yeend in the 2000s. In the present decade, Spielberg has given us a groupoid model for the non-finitely aligned setting. In fact, Spielberg's techniques apply to all left cancellative small categories, making the techniques as complex as the class of left cancellative small categories is broad. Moreover, the length functions that moderate the behaviour of the graph are omitted. We are interested in finding a more accessible groupoid for the non-finitely aligned setting. We reintroduce length functions to make this feasible. In this talk, we associate a topological groupoid to any \(P\)-graph, where P is only assumed to be a submonoid of a group. We describe a sufficient condition for the groupoid to be ample. When the \(P\)-graph is finitely aligned, the sufficient condition holds, so the groupoid is ample. We discuss how to modify our construction to include non-finitely aligned \(P\)-graphs. Lastly, we highlight that our construction applies to discrete groups, in which case the groupoid is isomorphic to the group itself. This work is being undertaken with my supervisors Lisa Orloff Clark and Astrid an Huef.

Awards:

• University of Münster, Visiting Doctoral Researcher with Becky Armstrong (2023)
• New Zealand Mathematics Research Institute (Inc.), Kalman Summer Scholarship (2022)
• Victoria University Postgraduate Students' Association, Research Excellence Award (2022)
• Victoria University of Wellington, Doctoral Scholarship (2021)
• University of Otago, Scholarship in Science (2018)
• University of Otago, Summer Research Scholarship (2017-18), supervised by David Bryant on diversity theory
• University of Otago, Staff Prize in Mathematics and Statistics (2016)

PhD Thesis:

MSc Thesis:

Teaching Assistant at the University of Leeds (2025)

• MATH1013 Computational Mathematics and Modelling (tutorials)
• MATH1110 Real Analysis (tutorials)
• MATH1700 Probability and Statistics (practicals)
• MATH2017 Real Analysis (workshops)

Teaching Fellow at Victoria University of Wellington (2022)

• MATH142 Calculus 1B

Tutor at Victoria University of Wellington (2019—2024)

• MATH311 Algebra
• MATH321 Introduction to Applied Mathematics
• MATH251 Linear Algebra
• MATH212 Introduction to Real Analysis
• MATH161 Discrete Mathematics and Logic
• DATA101 Introduction to Data Science

Tutor at the University of Otago, Department of Mathematics and Statistics (2018)

• COMO101 Modelling and Computation

Learning Lab Mentor at the University of Otago, Carrington College (2017—2018)

• STAT110 Statistical Methods
• STAT115 Introduction to Biostatistics

• Co-organiser, New Zealand Mathematics and Statistics Postgraduate Conference (2023)
• Co-organiser and founding member, Postgraduate Seminar, School of Mathematics and Statistics, Victoria University of Wellington (2020-2023)
• Organiser, Operator Algebra Group Seminar, School of Mathematics and Statistics, Victoria University of Wellington (2022)

• New Zealand Mathematical Society (2022—present)
• Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) division of the Australian Mathematical Society (2022—present)

YesCite is an online tool I created that tidies a .bib file by removing unused entries in a .bib, leaving only those that have been used according to a .bbl file (e.g. when using BibTeX).

Please contact me with any feedback on how to improve the tool (or if someone else has done it better). Send me an email or raise an issue on GitHub

Here is the code for this website. Feel free to download and customise.

• Detexify: Draw the LaTeX symbol you want the command for.
• Point and click editors for creating diagrams in LaTeX:
• tikzcd-editor (light-weight and can do loops on vertices)
• quiver (more powerful than the above, but can't do loops on vertices)
• RandomA11y: Generate accessible colour combinations.
• OpenDyslexic: Typeface designed against some common symptoms of dyslexia.
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• Journals:
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• Springer tool for finding a journal for your manuscript
• SCImago tool for comparing journals on various metrics