Research interests
Research interests
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I am currently serving as an Associate professor at Stellenbosch University, following 2-year Postdoctoral position at UCT and 4 years at the Courant Institute, NYU, where I held positions as a Visiting Assistant Professor and Clinical Assistant Professor.
My research focuses primarily on the following areas: algebraic number theory, algebraic geometry, algebra, and category theory. It covers topics such as holomorphic differentials in soluble coverings, Cohen-Macaulayness of permutation invariants, classification of cubic and biquadratic extensions, and the study of ramification.
My research on the study of holomorphic differentials in soluble coverings of the projective line has led to the construction of explicit bases and description of the associated Galois module structure. This work also contributes to the theory of representation of non-cyclic groups in positive characteristic.
My research explores the Cohen-Macaulayness of permutation invariants on polynomial rings, demonstrating that polynomial invariants of a permutation group are Cohen-Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles.
Another contribution concerns the classification of cubic and biquadratic extensions, with explicit algorithms to determine their structure over fields of arbitrary characteristic. Furthermore, we have characterized the ramification of separable cubic extensions over global function fields.
My research also explores the domain of near-vector spaces, extending classical linear algebra to quasi-linear algebra. Fundamental results have been established, such as the isomorphism theorem for near-vector spaces and the characterization of these spaces based on the presence of a scalar base. These works open new perspectives in the field of linear algebra and offer valuable tools for addressing nonlinear problems.
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.
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)
Distributive decomposition of near-vector spaces. (with Daniella Moore, PhD student and Leandro Boonzaaier) Journal of Algebra and Its Applications. Available at https://doi.org/10.1142/S021949882650088X (2024)
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).
A note on the transport of (near-)field structures. (with Leandro Boonzaaier). PDF Accepted for publication in Afrika Mathematika. (2025)
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)
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)
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)
Master Students (On going):
Master Students (On going):
Fabian Palmer, A study of Elliptic curves and their applications. (2024-now)
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)
Ph.D. Students (On going):
Ph.D. Students (On going):
Joseph Atalaye, (co-supervision with Dr. Liam Baker) Near-field theory. (2023-Now)
Daniella More (co-supervision with Prof. Zurab Janelidze) Category of near-modules. (2023-Now).
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
Number theory conference
Stellenbosch University
Stellenbosch University
(Federico Bambozzi, Stephane Vinatier, Bruno Chiarellotto)
(2025)
(2025)
MSRI visit with Ben Blum-Smith (2025)
MSRI visit with Ben Blum-Smith (2025)
MSRI visit with Ben Blum-Smith (2025)
MSRI visit with Ben Blum-Smith (2025)
Elena Caviglia (collaborator), Luca Mesiti (postdoctoral student), Frank Neumann (collaborator)
LMS-SAMS-SAMSA MEETING
Elena Caviglia (collaborator), Luca Mesiti (postdoctoral student), Frank Neumann (collaborator)
LMS-SAMS-SAMSA MEETING
2025
2025
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