Expert List Nonlinear Dynamics Universal PDE Solution Space
1. Partial differential equations, their numerical analysis and mathematical modelling
This focus area is subdivided as follows:
1.1 Partial Differential Equations, Ordinary Differential Equations and Stochastic Partial Differential Equation models in science and engineering
The research covers:
 Function spaces (distributions, Sobolev spaces, etc)
 Existence, uniqueness, regularity and singular properties of solutions
 Singularly perturbed problems
 Numerical treatment by finite element, finite difference and boundary element methods
 Dynamical systems
 Interval methods
 Modification of mathematical models
 Nonlinear theories of generalized functions
1.2 Study of dense singularities of solutions of nonlinear PDEs
The emphasis is placed on:
 Lie semigroups of noninvertible transformations of solutions
 Abstract differential geometry of algebras of generalised functions and de Rham cohomology
 Spacetime foam structures with dense singularities
1.3 Homogenization of elliptic and evolution problems
1.4 Geometric partial differential equations
The focus is on:
 Harmonic and wave maps on RiemannFinsler manifolds
 Ricci flow on Finsler manifolds
1.5 Mathematical biology with emphasis on epidemiology
This is a new direction of work on which the Department is embarking, with the aim of engaging into multidisciplinary research with the cluster of Biological Sciences at the University of Pretoria.
Associated staff: Prof M Abbas, Prof R Anguelov, Prof MK Banda, Prof M Chapwanya, Mr NKK Dukuza, Dr SM Garba, Prof NF Janse van Rensburg, Dr M Labuschagne, Dr C le Roux, Prof JMS Lubuma, Prof R Ouifki, Prof M Sango, Mr PE Shabangu, Dr Q van der Hoff, Prof JH van der Walt
2. Abstract analysis, topology and applications
This focus area is subdivided as follows:
2.1 Banach space analysis and measure theory
The following topics are covered:
 Tensor products and operator ideals
 Geometry of Banach spaces
 Interplay between Banach space theory and measure theory
 Amenability of Banach Algebra
2.2 Operator algebras
The focus is on noncommutative analysis on C*dynamical systems, with emphasis on the recurrence properties of such systems, and applications to quantum statistical mechanics.
2.3 Fixed point theory and its applications
2.4 Approximation theory and orthogonal polynomials
2.5 Stochastic analysis and applications to the mathematics of finance
2.6 Theory of double families of evolution operators, their spectral theory and applications
Applications are directed to:
 Dynamic boundary condition problems
 The NavierStokes equations (existence and uniqueness results)
 Problems in nonlinear elasticity theory
 NonNewtonian fluid mechanics
2.7 Convergence spaces and applications to analysis, including applications to:
 Riesz spaces and ordered topological vector spaces
 Generalized functions and solutions of nonlinear PDEs
Associated staff: Prof M Abbas, Dr AS Jooste, Dr E Kikianty, Dr R Kufakunesu, Ms L Mabitsela, Dr MD Mabula, Dr SM Maepa, Prof E Maré, Dr HJM Messerschmidt, Dr DV Moubandjo, Dr SA Mutangadura, Prof J Swart, Prof JH van der Walt and Dr AJ van Zyl
3. Discrete Mathematics
The following areas are currently under investigation:
 Graph theory, in particular hereditary properties of countable graphs
 Applications of group theory in the study of Cayley graphs with prescribed degree and diameter
 Covering designs
4. Mathematics Education
The issues being addressed are the following aspects of diversity at university level:
 The diversity of blending in teaching methods
 The diversity of assessment mechanisms in the classroom and online environment
 The diversity of students' conceptions of mathematics and its applications in their proposed professions.
 The diversity of thinking styles in undergraduate mathematics
 The transition of secondary to undergraduate mathematics
 The development and success of access programmes
 Undergraduate mathematics topics for enrichment
Associated staff: Mrs K Bothma, Prof AF Harding, Dr HZ Wiggins and Mrs B Yani
5. Algebra and Logic
The following areas are currently under investigation:
 Finite Group theory
 Ring and Module theory (emphasizing torsiontheoretic analysis of rings)
 Computational algebra
 Abstract differential geometry and commutative algebra (emphasizing sheaftheoretic context of classical differential geometry and quadratic morphisms on sheaves of modules)
 Model theory (emphasising the firstorder theories, axiomatisations, and classical model theory, of structures)
 Universal algebraic methods in nonclassical logics (emphasizing residuated structures and substructural logics)
 Abstract algebraic logic
Associated staff: Dr R Biggs, Dr R Kellerman, Dr SY Madanha, Dr MS Marais, Prof PP Ntumba, Prof JG Raftery and Prof JE van den Berg
Expert List
Prof M ABBAS
PhD (National College for Business Administration and Economics, Pakistan) Associate Professor
(Fixedpoint theory, Vector spaces, Best approximations, Fuzzy logic)

Prof R ANGUELOV
PhD (Unisa) Professor/HOD
(Differential Equations and their Numerical Analysis, Biomathematics, Image Analysis)

Prof M K BANDA
PhD (Darmstadt University of Technology, Germany) Professor
(Stochastic Partial Differential Equations, Optimisation, Networked flows)

Dr R BIGGS
PhD (Rhodes University) Lecturer
(Differential geometry, geometric control theory, Lie groups)

Dr M CHAPWANYA
PhD (Univ of Limerick, Ireland) Associate Professor
(Mathematical Modeling, Industrial Mathematics, Scientific Computing)

Dr M GANDHI
PhD (Indian Institute of Science) Senior Lecturer
(Deterministic and stochastic dynamical systems, especially topological dynamics, dynamical decomposition, nonautonomous attractors, recurrence, stability and bifurcation. Network of dynamical systems.)

Dr D J FRESEN
PhD (University of Missouri) Lecturer
(Probability, functional analysis, and convex geometry)

Dr S M GARBA
PhD (Putra University, Malaysia) Senior Lecturer
(Mathematical Biology)

Prof A F HARDING
DSc (Pretoria) Professor
(Mathematics Teaching)

Dr A S JOOSTE
PhD (Witwatersrand) Associate Professor
(Special functions, Orthogonal Polynomials)

Dr R KELLERMAN
PhD (Witwatersrand) Lecturer
(Mathematical Logic)

Dr E KIKIANTY
PhD (Victoria University, Australia) Senior Lecturer
(Geometry of Banach spaces, Inequality theory, Integral approximations, Bioinformatics)

Dr A A KOTZE
PhD (WITS) ExtraOridinary Lecturer
(Independent derivatives pricing, Valuation, Model validation, Risk expert, Financial engineer)

Dr R KUFAKUNESU
DPhil (Univ of Zimbabwe) Senior Lecturer
(Mathematics of Finance, Financial Engineering)

Dr M LABUSCHAGNE
PhD (University of Pretoria) Lecturer
(Partial Differential Equations, Numerical Analysis)

Dr T LÉ
PhD (Wayne State University) Senior Lecturer
(Finite group theory, Design theory)

Dr C LE ROUX
PhD (Pretoria) Senior Lecturer
(Partial Differential Equations, Fluid Mechanics)

Prof J MS LUBUMA
PhD (Louvain) Professor/Dean
(Partial Differential Equations, Numerical Analysis, Function Spaces, Integral Equations)

Dr SY MADANHA
PhD (Pretoria) Lecturer
(Representation theory of finite groups, products of finite groups)

Dr S M MAEPA
PhD (Pretoria) Senior Lecturer
(Geometry of Banach Spaces)

Dr G MALULEKE
PhD (WITS) ExtraOrdinary Lecturer
(Cryptography, Differential equations)

Dr M S MARAIS
PhD (Stellenbosch) Lecturer
(Computational Algebra, Algebraic Geometry, Ring Theory)

Prof E MARÉ
PhD (UOFS) Associate Professor
(Financial Engineering)

Dr M MESSERSCHMIDT
PhD (Leiden) Lecturer
Functional analysis. Ordered Banach spaces. General Banach space theory. Operator theory. Ergodic theory. Ergodic Ramsey theory. History of mathematics. Computational geometry. Cryptocurrency.

Dr D V MOUBANDJO
PhD (Stellenbosch) Lecturer
(Approximation Theory, Wavelets Analysis)

Prof P P NTUMBA
PhD (UCT) Associate Professor
(Algebraic Topology, Differential Geometry)

Dr R OUIFKI
PhD (University of Cadi Ayyad) Senior Lecturer
Mathematical Modeling: HIV, Tuberculosis, Malaria, Trypanosomiasis, Cancer; Stability and Bifurcation: Ordinary and Delayed Differential Equations; Optimal Control cost benefit and Costeffectiveness Analyses

Dr E PINDZA
PhD (UWC) ExtraOridinary Lecturer
(Business processes, Numerical analysis)

Prof J G RAFTERY
PhD (Natal), Professor
(Logic, Algebra and Foundations)

Dr PA RAZAFIMANDIMBY
PhD (UP) Lecturer
Stochastic Analysis, Applied Probability, Theory of Stochastic/Deteministic Partial Differential Equations (PDEs) and their Applications; Numerical treatment of Stochastic PDEs; Stochastic control and its application to the dynamics of nematic Liquid Crystal with external fluctuating control, Mathematical Finance.

Prof M SANGO
PhD (Valenciennes) Professor
(Partial Differential Equations)

Prof N SAUER
PhD (Unisa) Emeritus Professor
(Equations of Evolution, Fluid Dynamics)

Prof A STRÖH
PhD (Pretoria) Professor and Dean of Faculty of Natural and Agricultural Sciences
(Operator Theory, Von Neumann Algebras, Noncommutative Integration Theory, Ergodic Theory)

Prof J SWART
Dr Phil (Zürich) Professor
(Abstract Analysis: Geometry of Banach Spaces, Measure and Integration Theory, Mathematics of Finance)

Prof JE VAN DEN BERG
PhD (Natal) Associate Professor
(Algebra, Analysis)

Dr J H VAN DER WALT
PhD (Pretoria) Associate Professor
(Partial Differential Equations, Convergence Spaces, Topology)

Prof N F J VAN RENSBURG
DSc (Pretoria) Associate Professor
(Partial Differential Equations, Numerical Analysis)

Dr A J VAN ZYL
PhD (Pretoria) Senior Lecturer
(Functional Analysis, Stochastics)

Dr H Z WIGGINS
PhD (Pretoria) Lecturer
(Mathematics Teaching)

Dr M WORTEL
PhD (Leiden University) Lecturer
Functional analysis, Ordered vector spaces, Operator algebras, Metric geometry

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