Research Description

My primary research interest is in Near Vector Spaces, with the aim of developing a theory which parallels that of vector spaces. My key research questions were related to a better understanding of the structure of near-vector spaces and how their lack in linearity affects the theory linked to traditional vector spaces. These questions have led to me working in representation theory, model theory, incidence geometry, matroid theory and most recently, hypergroups.

As a secondary research interest, in collaboration with the late Professor Carl Maxson (Texas A & M University, USA), we investigated algebras of functions associated with a given algebraic structure. We were interested in when the set of endomorphisms of the universal algebra A will be a maximal structure in the set of all maps of A?

For an idea of the research involved, please refer to:

  • J. Andre, Algebra über Fastkörpern, Math.Z.136 (1974), 295-313.

  • K-T Howell, Contributions to the Theory of Near Vector Spaces, PhD thesis (2008).

  • A. Kreuzer and C.J. Maxson, E-locally Cyclic Abelian Groups and Maximal Near-rings of Mappings, Forum Math. 18 (2006), no.1, 107-114.

  • J.D.P. Meldrum and M. Samman, On Endomorphisms of Semilattices of Groups, Algebra Colloquim 12:1 (2005) 93-111.

  • A.P.J. van der Walt, Near-Linear Transformations of Near-Vector Spaces, Proceedings of a Conference held at Oberwolfach, 5-11 November 1989 (ed. G.Betsch, G.Pilz).

  • Near Rings: http://www.algebra.uni-linz.ac.at/Nearrings/.

Research Rating

In November 2020 I was awarded a C2 rating from the South African National Research Foundation (NRF).

Professional Societies

I am a member of the South African Mathematical Society and the American Mathematical Society. As of January 2020 I am the secretary of the South African Mathematical Society. I also serve on the Mathematics Committee of the National Graduate Academy (NGA) in South Africa and the Advisory Committee on Mathematics (ACM) of the South African Mathematics Foundation (SAMF).

Since November 2014 I am a reviewer for the American Mathematical society. I have refereed papers for Journal of Algebra, Results in Mathematics, UFA Mathematical Journal, Discussiones Mathematicae, General Algebra and Applications.

Promoting Women in Mathematics

I am the founder and convenor of a committee that organises the African Women in Mathematics (AWiM) Conferences and Virtual Seminars at Stellenbosch University since 2019. As part of our virtual seminars we have hosted Eugenic Chang, Federico Ardila and Nalini Joshi.

Students

Foundations Course Projects

  • M. Chimes – supervised a third-year project, “Near-vector spaces” for the Foundations Course (2013)

  • R. Ferguson – supervised a second-year Foundations Course project on Near-rings and one on Rings (2015)

  • P. Thompson – supervised a second-year Foundations Course project on Rings (2015)

AIMS (South Africa) Projects

  • S.P. Sanon – supervised a project “An introduction to Nearrings” (2016), completed with distinction

  • P. Djagba – supervised a project “Nearfields” (2017), completed with distinction

  • K. Tefoetsile – supervised a project “Near-vector spaces constructed over \(Z_p\) for \(p\) a prime” (2018)

  • J. Kalunga – supervised a project “The Decomposition Theorem for Near-vector spaces” (2018)

  • F. Makuo – co-supervised (with M.Marais and J.Boehm) a project “Near-vector spaces constructed over \(Z_p\) for \(p\) a prime” (2018), completed with distinction

  • A.Rakotondrafara – supervised a project “An introduction to Beidleman Near-vector Spaces” (2020)
  • I.Kyapwanyama – supervised a project “Matrix Nearrings” (2021)

Hons Projects

  • J.Rabie (2019)
  • D.Moore (2020)

MSc

  • S.P. Sagnon (2016-2017) – graduated cum laude.
  • D. Moore (2020-) (supervised with Prof Z. Janelidze)

PhD

  • P. Djagba (supervised with Dr G. Boxall) (2017-2019)
  • L. Wessels (supervised with Dr S. Dorfling) (2017-2020)
  • J. Rabie (2022-)(upgraded from a MSc to PhD)



“All wish to possess knowledge, but few, comparatively speaking, are willing to pay the price.”
Juvenal