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My research field is the overlap of Topology and Category Theory.  My principal research activity is to use category theory to organise and express new and existing mathematics, particularly in general topology. I have a historical expertise in closure operators with more recent interest in neighbourhood operators, interior operators and uniform structures in categories. I also work in frame theory - so called "point free topology".

Within UWC, I have collaborative research projects with Ando Razafindrakoto and our students.

Other active research projects:  Prof J Slapal (Brno University of Technology), Prof M Sioen (Free University Brussels), Dr J Masuret (Stellenbosch).
Research collaboration: There is a strong group working in category theory and topology around SA.  I keep particularly close links (seminars, sharing students) with UNISA and UCT.  The main professors there are Prof T Dube (UNISA), Prof H-P Kunzi (UCT) and Prof G Janelidze (UCT).

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My current research interest lies in mathematical modelling by using both deterministic and stochastic models of infectious disease dynamics in a host population. Mathematical models offer possibilities to investigate the infectious disease dynamics over time and may help in informing policy-making on disease control by public and global institutions. Much of my work is based on the dynamics of TB in a crowded environment such as South African prisons, mines and camps as these provides favourable conditions for TB to flourish.

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My research focus includes the application of mathematical modelling in finance. More specifically, for commercial banks, I am interested in finding the optimal investment policies for maximising expected terminal time utilities of the banks’ asset portfolios and capital. By embedding formulae for the optimal policies into the banks’ asset portfolio and capital models, the effects of the optimal policies on the bank’s capital adequacy ratios and deposit insurance premium can be observed.

My principal research interests lie in the numerical methods for differential and integro-differential equations, scientific computing and applications of mathematics in biology. Current projects include:

1. higher order numerical methods for singularly perturbed systems of differential equations,
2. design, analysis and implementation of numerical methods for integro-differential equations and
3. mathematical epidemiology and analysis of malaria enhancer and reducer factors. I am also interested in the understanding of factors that influence the teaching and learning of first year undergraduate mathematics.

I am currently working on projects of common interest with KC Patidar (UWC), AYA Mukhtar (UWC), R Ouifki (UP), RA Appadu (NMU), M Khumalo (UNISA), and M Bani-Yaghoub (U. Missouri).

My area of research is:
a)  the study and determination of symmetries abstract discrete structures;
b)  representation and granulation of graphs as abstract structures;
c)  determination of parameters and symmetries of codes from abstract discrete structures;
d)  the determination of organising principles of biological networks as discrete abstract structures.

My main research focus is in modelling with ordinary differential equations as well as stochastic differential equations. The current research is devoted to stochastic modelling of HIV/AIDS with treatment, the use of oral pre-exposure prophylaxis (PrEP) and post-exposure prophylaxis (PEP). HIV has become one of greatest public health disasters in history despite some new significant advances recorded in the research. Antiretroviral treatment, PrEP and PEP have been used efficiently in management of HIV infection. I also have research interest in computational finance, biostatistics and econometrics.

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My research is focused on numerical methods and scientific computing for application problems that arise from the interactions between natural and life sciences as well as those from the engineering domain. This research involves analytical investigations and numerical experiments using finite difference, finite element, spline approximation and spectral methods for parameter sensitive differential models. I have published more than 90 research articles (in journals of international standing), six invited book chapters, graduated 17 PhD and 16 MSc students under my supervision and am busy supervising several students at present. Some of my accomplished and ongoing well-funded research projects include the development of reliable methods for the mathematical models arising in population biology as well as computational finance and numerical singular perturbations. Currently, I am working on some collaborative projects with Prof N Vaidya (San Diego State University, USA), Prof X Li (Arizona State University, USA), Prof F Gideon (University of Namibia, Namibia) and JB Munyakazi at UWC.

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My research lies between the fields of general topology and category theory. It studies different ways of looking at spaces and constructions involving them. The main recurrent topics are: closure operators and neighbourhood operators. It also looks at some intuitionist viewpoint of topology, hence the study of frames or locales in some cases (pointfree topology).

My area of research is:?

• Probabilistic methods in group theory. I am mainly interested in the theory of the commuting elements in algebraic structures, and corresponding generalisations. In particular, I work on the theory of the random walks, because it is possible to look at problems of commutativity from the perspective of the dynamical systems, and this gives many relations with measure theory and mathematical physics.
• Nilpotent groups and related lie algebras. I like to investigate the classical theory of finite $p$-groups ($p$ prime) and the corresponding infinite version in the context of profinite groups. One of the main interests I have is the classification via the size of the Schur multiplier. This object can be treated both with homological machineries and with appropriate generators and relations, so its theory fits very well many categories and that of lie algebras shows interesting analogies which I explore systematically.
• Locally compact groups. The general theory of locally compact groups is fundamental in abstract harmonic analysis, topology and group theory. Beginning from some problems regarding profinite groups, it came quite naturally to generalise several notions to the case of locally compact groups, so that I was involved in many aspects of the theory of these important structures.
• Isoperimetric inequalities in algebraic structures. The idea is to adapt some technical notions of differential geometry, like the notion of isoperimetric dimension, or the notion of Cheeger constant, to the context of metric spaces. Graphs fit very well this context.
• Geometric Group Theory. I have mainly studied two problems. The first is the so called qsf property, due to seminal papers of V. Poenaru, A. Casson and J. Stallings. These works involve an application of certain techniques of differential geometry to those finitely presented groups which can be naturally associated to a manifold via fundamental groups. The main point is to show a conjecture of Stallings for which there are no non-qsf finitely presented groups.
• Applications of topology to dynamical systems. I found surprising to justify a sophisticated mathematical model, which involves the dynamics of neutron stars, via pure considerations of topology. In fact, I am recently interested in applications of the topology in complex dynamical systems.

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I have performed research in gravitation and cosmology, as well as biotechnology, applying analytical and numerical techniques to solve partial differential equations in fluid mechanics. My current research interests are differential geometry, gravitation, and cosmology. Specific interests include the role of vacuum energy in physics and cosmology, and the application of exterior calculus in gravitation. I am currently working on a project that investigates the equivalence principle in modified theories of gravity. I also have research interest in teaching and learning of mathematics at institutions of higher education.

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I received a PhD degree at the University of the Western Cape in 2019. In recent years, I have been working on mathematical modelling. My current research interests include financial mathematics and epidemiology.

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Mathematical modelling with a focus in Biology and Health Science. This includes modelling, with stochastic differential equations (SDEs), of processes in banking, working collaboratively with G.E. Muller and G.J. van Schalkwyk. In biology, the main activity is on compartmental modelling with ODEs and SDEs of population dynamics of infectious diseases. Work on Malaria is done in collaboration with the iDEWS (Infectious Diseases Early Warning Systems) joint project between RSA and Japan. Bioinformatical TB research is done with A. Christoffels (UWC, SANBI) and IHI Ahmed (UWC, Computer Science). Current and past student co-workers include also GJ Abiodun (CPUT), S. Maku Vyambwera, M. Nsuami, M. Ongansie. Current and past student co-workers include also Dr. GJ Abiodun (CPUT), S. Maku Vyambwera, M. Nsuami, M. Ongansie, Dr. I.H.I. Ahmed, Dr. G.E. Muller, Dr. G.J. van Schalkwyk and Mr. Brandon Beukes (BCB).