Hello! I’m Josh (he/him), a Lecturer in Mathematics within the Department of Mathematical Sciences at Sol Plaatje University (SPU), and a pure mathematics PhD candidate at the University of the Western Cape (UWC). I joined SPU on 01 October 2026.
My research is focused mainly on pointfree topology, which I like to view as a lattice-based approach to general topology that also draws on ideas from category theory. My broader interests include topology, category theory, algebra, and logic. I completed my Master’s in 2025; you can find my thesis online.
I am very passionate about higher education in the South African context, particularly in the mathematical sciences; I believe that education can empower individuals and communities, and strengthen unity in our society. I enjoy sharing ideas with, and learning from, others – through teaching and collaborative research.
On this page: Education · Research · Publications and Theses · Work history · Teaching · Presentations · Leadership and Service · Awards · Posts
Education
- February 2026 – present: PhD in progress, UWC. Research topic: $C$-ideals, freely generated frames, and compactness. (Supervised by David Holgate and Mark Sioen.)
- 2023 – 2025: MSc in Mathematical Science (Summa Cum Laude), UWC.
- 2022: BSc (Hons) in Mathematical Science, UWC.
- 2018 – 2021: BSc in Mathematics and Physics, University of Cape Town (UCT).
Research Projects
- 2026 – present: Revisiting the localic Kuratowski-Mrówka theorem. TACL 2026 slides (PDF).
- 2026 – present: $C$-ideals, freely generated frames, and compactness.
- 2023 – 2025: The foundations of locale theory, MSc research supervised by Prof David Holgate.
- 2022: Frames and ideals in pointfree topology, Honours research.
- 2020: Gravitation and general relativity, third-year physics project.
Research Groups
- 2023 – present: Member of the Topology for Tomorrow research seminar at UWC.
- February – July 2026: Member of the AIMS Research Centre, in collaboration with the Algebra and Topology research group at AIMS South Africa.
As of present, I do not practice constructive mathematics, although I rarely make use of the axiom of choice. Whenever possible, I avoid the use of excluded middle, but if all else fails, I will resort to proof by contradiction!
Publications and Theses
- 2026 - 2027: Revisiting the localic Kuratowski-Mrówka theorem — to appear.
- 2025: The foundations of locale theory (PDF), MSc thesis, UWC. Also available through the UWC repository.
- 2022: Frames and ideals in pointfree topology (PDF), Honours thesis, UWC.
- 2020: Exploring the shores of the general relativistic ocean (PDF), third-year physics project, UCT.
Work History
- October 2026 – present: Lecturer in Mathematics, Sol Plaatje University.
- May 2025 – September 2026: Full-time Teaching Assistant (TA) for the structured Master’s programme in Mathematical Sciences at AIMS South Africa.
- 2023: Lecturing Assistant for MAM737 (Topology Honours), UWC.
- 2019 – 2023: Tutor in mathematics and physics for high school and university students.
- 2019: Part-time receptionist at Blue Cross Veterinary Hospital, Newlands, Cape Town.
Teaching Experience
I am of the opinion that mathematics undergraduate students (including applied mathematics students) should have a course on set theory and logic as early as possible in their curricula. Set theory and propositional logic should be introduced together, ideally in the first year of one’s studies, with emphasis on intuition rather than axioms — the axioms can always come later.
At AIMS SA, I worked as a TA on the following courses:
- Model Theory: May 2026 and May 2025. First-order logic, set theory, languages, structures, formulas, sentences, models, and the compactness theorem.
- Algebraic Systems Biology: April–May 2026. Biochemical reactions, Boolean and algebraic models of molecular networks, finite dynamical systems, and the dynamics of Boolean models.
- Discrete Mathematics: March–April 2026. Graph theory, the travelling salesperson and minimum weight spanning tree problems, and graphic and nongraphic matroids.
- Mathematics and Art in Africa: March 2026. Mathematical ideas in Ndebele house decorations, Shwe-Shwe fabric designs, Mozambican basketry, and Ghanaian Adinkra symbols; symmetries, dihedral groups, graph theory, Euler cycles, and elementary number theory.
- Topology: February 2026. Metric spaces, point-set topology, compactness, connectedness, and filter convergence.
- Mathematical Problem Solving: January 2026 and October 2025. Problems in number theory, combinatorics, probability, and geometry, together with set theory and methods of proof.
- Verified Computation: November–December 2025. Floating-point and interval arithmetic, verified linear algebra, rigorous evaluation of elementary functions, and verified equation solving.
- Ordinary Differential Equations: November 2025. Separable equations, changes of variables, integrating factors, and qualitative behaviour through trajectories, orbits, flows, and phase portraits.
- Hilbert Spaces: June–July 2025. Orthonormal bases, generalised Fourier series, functionals, and the Riesz representation theorem.
I also assisted with the Topology Honours module at UWC in 2023 and tutored mathematics and physics from 2019 to 2023.
Presentations (conferences, workshops, seminars, etc.)
2026
- 28 July: Revisiting the localic Kuratowski-Mrówka theorem, Topology, Algebra, and Categories in Logic (TACL) Summer School and Conference, Kraków, Poland. Slides (PDF).
- 21 February: A Bourbaki-style proof of Tychonoff’s theorem, Topology for Tomorrow workshop, AIMS South Africa.
2025
- 09 September: The Knaster-Tarski fixpoint theorem and its applications in set theory, Departmental Seminar, UWC.
- 15 August: Ideals and coherent frames: how ideals can be used to generate frames, UJ Category Theory Seminar.
- 14 August: Ideals and coherent frames: how ideals can be used to generate frames, Wits Category Theory Seminar.
- 13 June: Ideals and ideal lattices, Topology for Tomorrow seminar.
- 21 February: Nuclei and sublocales, Topology for Tomorrow seminar.
- 14 February: Ideals and filters in lattices, Topology for Tomorrow seminar.
2024
- 12 December: The localic Kuratowski-Mrówka theorem, Brümmer 90 Conference, UCT.
- 06 December: The localic Kuratowski-Mrówka theorem, South African Mathematical Society (SAMS) Congress, University of Pretoria.
- 08 November: The Kuratowski-Mrówka theorem for frames and locales, Topology for Tomorrow seminar.
- 21 September: A first introduction to pointfree topology, Analysis and Differential Equations with Applications Conference, University of KwaZulu-Natal.
- 07 June: Adjoint functors and the adjunction between spaces and locales, Topology for Tomorrow seminar.
2023
- 06 December: Freely generated frames, compactness and the strong Hausdorff property for locales, SAMS Congress, University of Johannesburg.
- 20 October: Sites, $C$-ideals and freely generated frames, Topology for Tomorrow seminar.
- 30 August: Sublocales of open-set lattices, Topology for Tomorrow seminar.
- 16 June: From subspaces to sublocales, Topology for Tomorrow seminar.
- 24 February: The relation between topological spaces and frames, Topology for Tomorrow seminar.
Earlier presentations
- 26 October 2022: Frames and ideals in pointfree topology, Honours presentation, UWC.
- 09 November 2020: Exploring the shores of the general relativistic ocean, third-year physics project presentation, UCT.
Leadership and Service
- February - July 2026: Lead Teaching Assistant for the Biomathematics Honours Programme at AIMS South Africa.
- 23–27 March 2026: Organiser, Women in Mathematics Research Conference, AIMS South Africa.
- 19–21 February 2026: Organiser, Topology for Tomorrow workshop, AIMS South Africa.
- 2023 – 2025: Organiser, Topology for Tomorrow seminar series, UWC.
Awards and Bursaries
- 2023 – 2025: Topology for Tomorrow bursary for the MSc in Mathematical Science, UWC.
- 2023: Lifetime membership of the Golden Key International Honour Society.
- 2023: Faculty of Natural Sciences Dean’s Merit Award for outstanding performance during the 2022 academic year, UWC.
- 2022: Topology for Tomorrow bursary for the BSc (Hons) in Mathematical Science, UWC.
- 2022: Attraction Bursary from the office of the Deputy Vice-Chancellor for Research and Innovation, UWC.
Posts
- Joshua – a brief introduction — 5 June 2026