Research interests
Research interests
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I am currently a Professora Auxiliar (Assistant Professor) in Mathematics at the University of Minho, Portugal, and an Exceptional Associate Professor at Stellenbosch University, South Africa. Before joining the University of Minho in January 2026, I was a full-time faculty member at Stellenbosch University, where I served first as Senior Lecturer and subsequently as Associate Professor. Prior to Stellenbosch, I held a postdoctoral position at the University of Cape Town from 2017 to 2019, and from 2013 to 2017 I was at the Courant Institute of Mathematical Sciences, New York University, where I held positions as Visiting Assistant Professor and Clinical Assistant Professor.
My research lies broadly at the intersection of algebraic number theory, algebraic geometry, algebra, and category theory, with connections to invariant theory, algebraic combinatorics, and representation theory. My work includes the study of ramification and field extensions, Galois module structures, permutation invariants and Cohen–Macaulay rings, near-vector spaces and near-linear algebra, categorical approaches to descent and algebraic structures, and more recently questions involving radicial extensions, Frobenius morphisms, and explicit descent problems for algebraic extensions.
One strand of my research concerns holomorphic differentials on soluble coverings of the projective line. In this setting, I have constructed explicit bases for spaces of holomorphic differentials and described their associated Galois module structures. This work contributes more generally to the modular representation theory of non-cyclic groups and to the understanding of Galois actions in positive characteristic.
Another part of my research focuses on invariant theory and Cohen–Macaulayness. In particular, my work on permutation invariants characterizes when polynomial invariant rings of permutation groups are Cohen–Macaulay independently of the choice of coefficient field. The characterization shows that this occurs precisely when the group is generated by transpositions, double transpositions, and 3-cycles. Related questions have led me to study Stanley–Reisner rings, face rings, simplicial and Boolean complexes, and the interaction between combinatorial structures and invariant theory.
A further direction of my research concerns field extensions, Galois theory, and ramification. I have worked on explicit descriptions and classifications of cubic and biquadratic extensions, including algorithms for determining their structure over fields of arbitrary characteristic, as well as on the ramification of separable cubic extensions of global function fields. More recent work develops explicit methods for understanding quartic extensions through resolvents and descent to biquadratic extensions, and investigates finite radicial extensions in positive characteristic through Frobenius factorization and related algebraic structures.
I also work on near-vector spaces and near-linear algebra, which generalize classical vector spaces by weakening the distributivity properties of scalar multiplication. My research in this area includes structural and decomposition results for near-vector spaces, isomorphism theorems, the study of scalar bases and quasi-kernels, and classifications arising from the near-field structures induced by vectors. More recent work investigates products of near-vector spaces and the extent to which classical linear-algebraic phenomena survive in this substantially more flexible setting. These results contribute to the development of a systematic theory of quasi-linear algebra and provide tools for studying algebraic structures lying beyond the classical framework of modules and vector spaces.
Finally, a substantial part of my current research uses category theory as a structural framework for algebra and geometry. I am particularly interested in monads, descent theory, internal categories, and categorical approaches to algebraic structures. This includes work on monadic and lax descent, actions of internal categories, and the use of categorical methods to clarify constructions arising naturally in algebraic geometry and Galois theory.
A recurring theme throughout my research is the combination of explicit algebraic constructions with structural and categorical methods. I am particularly interested in problems where concrete computations—such as bases, generators, resolvents, ramification data, or decomposition formulas, can be connected with broader conceptual structures.
Published research papers
Published research papers
Tame actions of affine group schemes and tame stacks, C. R. Acad. Sci. Paris. Ser. I, 350 (3-4):125-128, https://doi.org/10.1016/j.crma.2012.01.020 (2012).
Existence of slices in a tame context, European Journal of Mathematics 1(1):54-77, https://doi.org/10.1007/s40879-014-0022-4 (2015).
Generic polynomials for cyclic function field extensions over certain finite fields. European Journal of Mathematics, 4(2):585-602, https://doi.org/10.1007/s40879-017-0188-7 (2017).
Holomorphic differentials of solvable Galois towers of curves over a perfect field (joint with Kenneth Ward), Mathematische Nachrichten, https://doi.org/10.1002/mana.201500347 (2018).
Cubic fields: a primer (with Kenneth Ward), European Journal of Mathematics, https://doi.org/10.1007/s40879-018-0258-5 (2018).
An explicit triangular integral basis for any separable cubic extension of a function field (with Kenneth Ward), European Journal of Mathematics, https://doi.org/10.1007/ s40879-018-0243-z., (2018).
When are permutation invariants Cohen-Macaulay over all fields? (with Ben Blum Smith), in Algebra and Number Theory, Vol. 12, No. 7, 1787–1821 https://doi.org/10.2140/ant.2018.12.1787 (2018)
A complete study of the ramification for any separable cubic global function field, (with Kenneth Ward) Research in Number Theory 5, 36, https://link.springer.com/article/10.1007/s40993-019-0173-y (2019).
The Geometry Of The Moduli Space Of non-cyclic Biquadratic Field Extensions. (with Mpendulo Cele, master's student). Quaestiones Mathematica, https://doi.org/10.2989/16073606.2021.1951391 (2021)
Cubic function fields with prescribed ramification. (with Valentijn Karemaker and Jeroen Sijsling). International Journal of Number Theory. Vol. 17, No. 09, pp. 2019-2053, https://doi.org/10.1142/S1793042121500755 (2021)
Toward an intuitive understanding of the structure of near-vector spaces. (with Karin-Therese Howell). Comm. Algebra 50 (2022), no. 9, 3664–3677. Available at https://doi.org/10.1080/00927872.2022.2042543 (2022)
Near-linear algebra. (with Daniella Moore, master's student). Journal of Algebra and Its Applications. Available at https://doi.org/10.1142/S0219498825501257 (2023)
A characterization of ramification groups via Taylor morphism. (joint with Luigi Pagano). European Journal of Mathematics, https://doi.org/10.1007/s40879-024-00768-8, (2024).
Distributive decomposition of near-vector spaces. (with Daniella Moore, PhD student and Leandro Boonzaaier) Journal of Algebra and Its Applications. https://doi.org/10.1142/S021949882650088X (2025)
A note on the transport of (near-)field structures. (with Leandro Boonzaaier). Afrika Matematika 36, 57 (2025). https://doi.org/10.1007/s13370-025-01276-y
Why is the Category of Near-Vector Spaces Abelian? (with Zurab Janelidze and Daniella Moore), Applied Categorical Structures 34, article 3 (2026). https://doi.org/10.1007/s10485-025-09837-9
Galois descent theorem in general categories. Research in the Mathematical Sciences 13, 49 (2026). https://doi.org/10.1007/s40687-026-00629-8.
Published books
Published books
Linear Algebra I Frederick P. Greenleaf and Sophie Marques, AMS and Courant lectures. ISBN-10: 1-4704-4871-8, ISBN-13: 978-1-4704-4871-4 (2019)
Linear Algebra II Frederick P. Greenleaf and Sophie Marques, AMS and Courant lectures. ISBN-10: 1-4704-5425-4, ISBN-13: 978-1-4704-5425-8 (2020)
Ph.D. Students (Complete):
Ph.D. Students (Complete):
Damas Mgani, On lifting of finite group schemes and Lagrange theorem. (completed in 3 years, lecturer (tenure) at the University of Dar es Salaam, Tanzania) (2021-2024)
Elizabeth Mrema, Study of the existence of a radical closure, radical descent for quintic extensions and applications. (completed in 3 years, lecturer (tenure) at the University of Dar es Salaam, Tanzania) (2021-2024)
Joseph Atalaye, (co-supervision with Dr. Liam Baker) Exploring the multiplicative structure of Z/nZ. (2023-2026)
Daniella More (co-supervision with Prof. Zurab Janelidze) An exploration of near-vector space theory. (2023-2026).
Ph.D. Students (on-going):
Ph.D. Students (on-going):
Omega Zimba (co-supervision with Prof. Florian Luca) Biquadratic descent. (2026-now)
Wessel Esterhuyse (cosupervision with Prof. Naina Ralaivaosaona) Radical moduli spaces. (2026-now)
Salash Nabaala (cosupervision with Prof. Jonathan Weinberger) Galois category theory. (2026-now)
Postgraduate/research students
Postgraduate/research students
Undergraduate research
Undergraduate research
Junichi Koganemaru, On the study of vanishing sum of roots of unity. (2014)
Paul Vladimir Kulik, Group theory via Caley graphs. (2015)
Aims Master Students (Complete):
Aims Master Students (Complete):
Ali Taore (co-supervised), On Serre’s homological criterion for regularity of Noetherian local rings, supervised by Professor Neil Strickland, University of Sheffield. PDF (received the Stephen Hawking Scholarship) (2020)
Tamara Tembo, From Classic Cryptography to Elliptic Curve Cryptography. PDF (2020)
Master Students (Complete):
Master Students (Complete):
Timmy Shun, An Introduction to the Nonnegative Inverse Eigenvalue Problem with an Algorithm for the RNIEP. PDF (2015-2016)
Mpendulo Cele, The Geometry Of The Moduli Space Of Biquadratic Field Extensions. PDF (2019-2020) Won the S2A3 Award, best Master Student at Stellenbosch University. Completed Master with Distinction.
Lesley Scoot (co-supervisor) Perceptions and Lived Experience: Exploring Factors That Affect Student Retention in Mathematics. Supervised by Mariam Salie (Interdisciplinary collaboration with the psychology department). (2020-2021)
Daniella More (co-supervision with Prof. Zurab Janelidze) On the abelianess of the category of near-vector spaces. (2022-2023). Completed Master with Distinction.
David Smith (co-supervision with Prof. Gareth Boxall) Zilber’s Conjecture on Intersections with Tori. (2022-2024)
Wisaarkhu
Wisaarkhu
In 2019, I sowed the seeds of the Wisaarkhu project, a collaborative effort with a remarkable team of individuals and the generous support of the Rubbi Fund and Nithecs. Our primary goal was to unveil the human face of mathematics, challenging its seemingly abstract and impersonal facade. We aimed to inspire people to discover how mathematics can empower them to achieve their aspirations, rather than viewing it solely as a subject reserved for mathematicians.
In March 2020, we officially launched our magazine, which has since become a vibrant platform where disciplines such as psychology, history, neurobiology, economics, and more converge to contribute to thoughtful discussions centered around carefully chosen themes. Our contributors span the spectrum, from students to educators and scholars, and each piece undergoes thorough review. We take pride in presenting these contributions alongside thoughtfully curated artwork, a testament to the artistic talents within our team.
The impact of our project has garnered attention, with numerous articles written about us and recognition as finalists in the science engagement category at the 2020 Falling Walls event. If you're curious to learn more about our journey, we have a video that provides further insight.
On my personal journey, I am deeply engaged in a diverse array of roles and initiatives:
- African Women Conferences and Seminars: I play a role in the organization of the African Women Conferences and Seminars at Stellenbosch University, collaborating closely with Professor Karin-Therese Howell the main organizer.
- Algant Program Representative: Proudly representing Stellenbosch University as its Algant Program Representative, I actively contribute to the program's objectives and activities.
- NITcheCS Associate: As an associate of NITcheCS, I wholeheartedly participate in their mission and collaborative endeavors.
- Ubuntu Mathematics Institute (UMI): I serve as an integral member of the UMI's driving team, actively shaping its vision and guiding its initiatives.
- External Editor at Person Global Edition: In the realm of editorial responsibilities, I take on the role of an external editor for Person Global Edition, where I contribute to the enhancement of content quality.
- University Exchange Program Coordinator: Fostering academic connections and cultural exchanges, I undertake the coordination of a vibrant exchange program between the University of Bordeaux and Stellenbosch University, generously supported by the French Embassy.
Preprints
Preprints
Cubic fields: A primer (joint with Kenneth Ward). Full version, available at https://arxiv.org/pdf/1703.06219.pdf. (2017).
A complete classification of cubic function fields over any field (joint with Kenneth Ward). Available at https://arxiv.org/pdf/1612.03534.pdf (2017).
Tame action on affine group schemes and tame stacks, Ph.D. Thesis. PDF (2013).
Multiplicative automorphisms of certain fields. (joint with Leandro Boonzaaier). PDF (2022)
Gluing index category and generalized gluing data functor. (with Damas Mgani, P.h.D student). PDF 2022
A Categorical Perspective on Gluing. (with Damas Mgani, P.h.D student). PDF 2023
A note on quadratic cyclotomic extensions. (with Elizabeth Mrema, P.h.D student). PDF (2024)
2-Cyclotomic field extensions. (with Elizabeth Mrema, P.h.D student). PDF 2024
A note on the multiplicative structure of Z/peZ. (joint with Joseph Atalaye (PhD student), Liam Baker). PDF (2024).
A study of a recursive sequence of polynomials revealing weighted Catalan Numbers (with Elizabeth Mrema). PDF. (2025)
Why is the category of near-vector spaces abelian? (joint with Zurab Janelidze, Daniella Moore). PDF. (2025)
Joseph Atalaye, PhD student (2024)
David Smith, Master student
(2024)
Damas Mgani, PhD Student
(2024)
Elizabeth Mrema, PhD Student
(2024)
Joseph Atalaye, PhD student
Daniella Moore, PhD Student
Leandro Boonzaaier, collaborator
(2025)
Joseph Atalaye, PhD student
Liam Baker, cosupervisor
(2025)
Pedagocical publications
Pedagocical publications
What is mathematics? Mathemafrica. Available at http://www.mathemafrica.org/?p=15981 (2019)
Reasoning and making sense: a pillar of mathematics? Mathemafrica. Available at http://www.mathemafrica.org/?p=15985 (2019)
Could any type of competition be pushing us away from our “best self”? Wisaarkhu. Available at https://wisaarkhu.co.za/article/could-any-type-of-competition-be-pushing-us-away-from-our-best-self/ (2020)
The human side of mathematics. Wisaarkhu. Available at https://wisaarkhu.co.za/article/the-human-side-of-mathematics/ (2021).
The art of distinguishing. Wisaarkhu. Available at https://wisaarkhu.co.za/article/the-art-of-distinguishing/ (2021).
Building a community empowered with 21st Century skills within an inclusive and innovative vision through mathematics. (written with John Gilmour & John Volmink). Proceedings of the 26th Annual National Congress of the Association for Mathematics Education of South Africa Developing Equitable Mathematical Teaching and Learning Practices that Empower Teachers and Learners in the 4IR Era (Volume 2). http://www.amesa.org.za/AMESA2021/Volume2.pdf (2021)
Empowering learners with critical life skills in our current world by exploring the human side of mathematics. Proceedings of the 26th Annual National Congress of the Association for Mathematics Education of South Africa Developing Equitable Mathematical Teaching and Learning Practices that Empower Teachers and Learners in the 4IR Era (Volume 1). http://www.amesa.org.za/AMESA2021/Volume1.pdf (2021)
MSRI visit with Ben Blum-Smith (2024)
MSRI visit with Ben Blum-Smith (2024)
MSRI visit with Ben Blum-Smith (2024)
MSRI visit with Ben Blum-Smith (2024)
Number theory conference Stellenbosch University (Federico Bambozzi, Stephane Vinatier, Bruno Chiarellotto)
2025
Number theory conference Stellenbosch University (Federico Bambozzi, Stephane Vinatier, Bruno Chiarellotto)
2025
MSRI visit with Ben Blum-Smith (2025)
MSRI visit with Ben Blum-Smith (2025)
Joseph Atalaye and Daniella Moore PhD student graduation (2026)
Joseph Atalaye and Daniella Moore PhD student graduation (2026)
Elena Caviglia (collaborator), Luca Mesiti (postdoctoral student),
Frank Neumann (collaborator)
LMS-SAMS-SAMSA MEETING 2025
Elena Caviglia (collaborator), Luca Mesiti (postdoctoral student),
Frank Neumann (collaborator)
LMS-SAMS-SAMSA MEETING 2025
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