Main research focus areas

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
Space-time 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 Riemann-Finsler 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 JM-S 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 Navier-Stokes equations (existence and uniqueness results)
Problems in nonlinear elasticity theory
Non-Newtonian 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 torsion-theoretic analysis of rings)
Computational algebra
Abstract differential geometry and commutative algebra (emphasizing sheaf-theoretic context of classical differential geometry and quadratic morphisms on sheaves of modules)
Model theory (emphasising the first-order theories, axiomatisations, and classical model theory, of structures)
Universal algebraic methods in non-classical 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 (Fixed-point 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) Extra-Oridinary 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 M-S 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) Extra-Ordinary 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)
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 Cost-effectiveness Analyses	Dr E PINDZA PhD (UWC) Extra-Oridinary 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, Non-commutative 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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Published by Annel Smit