# Modules

### Modules for Third-Year Mathematics Courses

Semester |
Prerequisites |
||

3AL | Algebra | 1 | Module 2IA |

3CA | Complex Analysis | 2 | Module 2RA |

3LC | Logic and Computation | 1 | - |

3MS | Metric Spaces | 1 | Module 2RA |

3TA | Topics in Algebra | 2 | - |

3TN | Topics in Analysis | 2 | Module 3MS |

**Allowed combinations:**

All combinations of modules are subject to the restrictions imposed by the timetable and the approval of the course co-ordinator.

**Recommended modules for Mathematics Honours courses**

Five types of Honours programmes are available to students who have completed senior courses in Mathematics:

- BROAD COVERAGE OF MATHEMATICS: Intended for prospective researchers and mathematicians.
- TEACHING: Intended for prospective high school mathematics teachers.
- MATHEMATICS OF COMPUTER SCIENCE: A co-operative venture with the Department of Computer Science. Each Department offers one half of the degree.
- INDUSTRIAL MATHEMATICS: Designed to prepare a mathematician to enter industry, this programme is run jointly through the Department of Mathematics & Applied Mathematics and the Department of Statistical Sciences.
- FINANCIAL MATHEMATICS: A course run jointly with the Universtiy of Stellendbosch and the African Institute for Mathematical Sciences, for those interested in employment in the financial sector.

Students registering for MAM3000W and intending to take (a) are advised to take modules 3MS, 3CA and 3AL as part of their course, and those intending to do (c) are advised to take Modules 3LC and 3AL. Students intending to do (b) are also advised to do 3MS, 3CA and 3AL, but may also do one of these as part of their Honours course. For (c) and (d) please refer to MAM4007W and MAM4008W.

**Syllabuses:**

3AL ALGEBRA

An introductory course of modern abstract algebra involving the following concepts: algebraic operations; magmas and unitary magmas; semigroups; monoids; closure operators; equivalence relations; categories; isomorphism; initial and terminal objects; algebras, homomorphisms, isomorphisms; subalgebras; products; quotient algebras; canonical factorizations of homomorphisms; free algebras. Various classical-algebraic constructions for groups, rings, fields,

and vector spaces, seen as examples of these concepts, will be described in tutorials.

3CA COMPLEX ANALYSIS

An introduction to the theory of complex functions with applications.

3LC LOGIC AND COMPUTATION

The propositional and predicate calculi: their syntax, semantics and metatheory. Resolution theorem proving.

3MS METRIC SPACES

An introduction to metric spaces and their topology, with applications.

3TA TOPICS IN ALGEBRA

A selection from lattices and order, congruences, Boolean algebra, representation theory, naive set theory, universal algebra. (Please note that this module is not a prerequisite for entry to the Honours course in Algebra.)

3TN TOPICS IN ANALYSIS

A selection from the implicit function theorem and inverse mapping theorem, Lebesgue integral, Fourier analysis, Hilbert spaces, Lebesgue and Sobolev spaces, Fractals and approximation theory. (Please note that this module is not a prerequisite for entry to the Honours course in Functional Analysis.)