### Honours

**BSc (Hons) **

Admission requirements: BSc degree with Applied Mathematics as a major subject, and a minimum mark of 60% in Applied Mathematics at the third-year level.

Courses: An Honours degree comprises 10 semester courses chosen from the following list, in consultation with the Chairman of the Department. Prospective students should note that, from time to time, some of these courses will not be offered.

**Computer algebra**

This is the field where computers are applied to the problems of mathematics which are not necessarily numerical. This includes symbolic computation which manipulates symbolic expressions according to mathematical rules, as well as applications such as modelling groups and operators. The techniques are extremely useful for doing tedious derivations and solving certain classes of equations. In the Computer Algebra course we describe the basic properties of a symbolic computation system. Important mathematical properties and techniques necessary for symbolic computation are introduced. A simple system is described in LISP. We proceed to build a symbolic system (SymbolicC++) using C++; the system is described in detail and the source code is freely available.

The prescribed book is Computer Algebra with SymbolicC++, Y. Hardy, Tan K.S. and W.-H. Steeb (World Scientific, Singapore, 2008, ISBN: 978-981-283-360-0).

**Differential equations**

In the first semester the application of Lie groups of transformations to ordinary differential equations (ODEs) is studied. Some of the points addressed are the basic theory of invariance, Lie point symmetries of ODEs, reducing the order of an ODE using a Lie point symmetry, the use of a two-dimensional Lie algebra to solve second-order ODEs, solvable Lie algebras and their use in reducing the order of ODEs, invariant solutions, separatrices and envelope solutions.

The second semester course builds on the first. The application of Lie groups of transformations to partial differential equations (PDEs) is studied. Some of the aspects covered are Lie point symmetries of PDEs, reducing the order of PDEs using Lie point symmetries, generalising point transformations and symmetries, contact symmetries, Lie-Backlund transformations, conservation laws and Noether's theorem.

**Dynamical systems**

Dynamical systems is an exciting new field in which the motion of a system is investigated in terms of generalised geometric concepts. In this course, the concepts of the discipline are developed with reference to one-dimensional and higher-dimensional maps. These concepts include periodic points of maps, stability conditions for these points, bifurcation theory, chaotic motion and symbolic dynamics. Students are regularly given demanding assignments which exercise their analytical, numerical and programming skills.

**Lie groups and algebras**

This course is an introduction to Lie groups and algebras, with applications in physics and engineering. The concepts of finite, continuous and Lie groups are introduced. The connection between Lie groups and Lie algebras is studied. The usage of Lie groups in solving problems in engineering and physics, particularly quantum theory, is covered.

The prescribed book is Continuous Symmetries, Lie Algebra, Differential Equations and Computer Algebra by W.H. Steeb (ISBN: 9810228910).

**Multilinear algebra**

A comprehensive introduction to multilinear algebra with applications to problems in science and engineering. The concepts of multilinear algebras and product spaces are described. The use of multilinear algebra and the Kronecker product in physics and quantum groups is studied. There is also a programming component.

The prescribed book is Matrix Calculus and Kronecker Product with Applications and C++ Programs by W.H. Steeb (ISBN: 98102324111).

**Neural networks and genetic algorithms**

This course is still under construction, but when presented will be based on the book The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and Symbolic C++ Programs, 2nd Ed. by W.H. Steeb (ISBN:9812382127).

**Numerical analysis**

Various concepts in the field of Numerical Analysis are studied. In the first semester these include solution of nonlinear equations, approximation theory using polynomials (interpolatory, least-squares, orthogonal), numerical differentiation, and numerical integration (Newton-Cotes, Gaussian, multiple integrals).

The second semester covers numerical methods for solving initial-value problems (IVPs) and boundary-value problems (BVPs) in ordinary differential equations. For IVPs multistep and Runge-Kutta methods are covered. For BVPs finite-difference methods for linear and nonlinear problems are studied. Numerical methods for partial differential equations based on finite-differences and the finite-element approach are presented.

The emphasis of the course is on the underlying mathematical derivation and properties of the methods, and their approximation error. An understanding of the approximation error is vitally important for implementing error control in these methods. A small programming component is included, designed to give students some experience in using the methods. The languages of preference are C++ or MatLab.

**Advanced scientific computation and programming**

Advanced problems in scientific computing are presented and solved using a combination of analytical and computational techniques. The subjects covered include numerical analysis, matrix theory, quantum mechanics, computer graphics and many other fields in scientific computing.

**Quantum field theory**

The central theme of condensed matter quantum field theory is the so-called formalism of second quantisation. This is the mathematical description which enables one to treat a system of particles either in terms of particle operators or fields. This mathematical formalism is first developed with reference to some simple physical systems and then is applied to many-body systems consisting of fermions or bosons, or both.

**Relativity**

The theory of relativity is one of the crowning intellectual achievements of the 20th century. This course covers a comprehensive treatment of the theory. It begins with a derivation of the Lorentz transformation and its application to uniform motion and the electromagnetic field. Then the Equivalence Principle is studied, and the necessary techniques from tensor analysis needed for the formulation of differential geometry are developed. The Einstein field equations are derived. The course concludes with the most important experimental verifications of General Relativity.

**Project**

The project is intended to be a substantial self-study component of the Honours program. The student is assigned a project by one of the staff members, although students are encouraged to propose their own topics. The work is submitted in the form of a typed document at the end of the second semester, and has the weight of one semester course (10% of the final Honours mark). Students are expected to give an oral presentation of their projects.

**Approved courses from other departments**

Subject to the approval of the Head of the Department, students may choose one or two (possibly more) courses from other departments. Of course, this does require that the student has sufficient background in those particular fields, and such choices will also require the approval of the Heads of those particular departments. Students who do exercise this option generally take courses from the Department of Mathematics and Statistics.