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, Dr AR Appadu, Prof MK Banda, Prof M Chapwanya, Prof J Djoko Kamdem, Mr NKK Dukuza, Dr SM Garba, Prof NF Janse van Rensburg, Dr M Labuschagne, Dr C le Roux, Prof JM-S Lubuma, Dr 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, Dr WS Lee, 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

Associated staff:  Prof I Broere
 
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 MS Marais, Prof PP Ntumba, Prof JG Raftery and Prof JE van den Berg