Mathematics in high school is almost exclusively concerned with calculating and computing. At university level (and beyond), however, mathematics – more precisely called pure mathematics, is increasingly abstract. Numbers, which are central to high school maths, play at best a peripheral role in several branches of pure mathematics. In their place we find a beautifully complex web of theorems, proofs, axioms, and definitions. The Department is home to a large number of internationally renowned mathematicians who conduct original research in many modern areas of pure mathematics, including algebra, analysis, category theory, discrete mathematics, and topology.
The Department of Mathematics and Applied Mathematics offers a major in (Pure) Mathematics, consisting of the following courses:

MAM1031F+MAM1032S or MAM1033F+MAM1034S  First year mathematics
Differential and integral calculus, their theory and applications, methods of proof, function approximation methods, complex numbers, linear algebra, differential equations. MAM1033F and MAM1034S is the same as MAM1031F and MAM1032S with the addition of twice weekly mathematical thinking workshops to aid with the transition from high school to university thinking.
Course convenor: Prof Jonathan Shock 
MAM1019H: Fundamentals of modern pure mathematics, including logic, set theory, relations and functions, basic mathematical structures
The aim of this course is to familiarise students with the most fundamental concepts and tools of modern mathematics at an elementary level. These include: fundamentals of logic and set theory, concepts of a function, of relations, of equivalence and order relations as well as some basic mathematical structures and the fundamental number systems.
Course convenor: Prof Anneliese Schauerte (1st sem) and Ms Morgan Vandeyar (2nd sem) 
MAM2000W  Second year mathematics
The course MAM2000W consists of five modules. Students must take four of these. In the first semester students take 2LA and 2AC, and in the second semester they take two of 2RA, 2IA and 2DE.
MAM2010F (MAM2000W 2AC) : ADVANCED CALCULUS Multivariable calculus. Curves and surfaces in three dimensions, change of coordinates. Line integrals, surface integrals. Stokes'. Green's and divergence theorems.
Module leader: Prof Francois Ebobisse Bille
MAM2011F (MAM2000W 2LA): LINEAR ALGEBRA Vector spaces, linear independence, spans, bases, row space, column space, null space. Linear maps. Eigenvectors and eigenvalues. Inner product spaces, orthogonality.
Module leader: Dr Holger Spakowski
MAM2012S (MAM2000W 2DE): DIFFERENTIAL EQUATIONS
Course is for Actuarial and Business Science students. Topics from: First and secondorder difference equations. Linear differential equations, constant coefficients. Laplace transforms. Nonlinear equations, phase plane analysis. Parabolic partial differential equations, separation of variables, boundary value problems. BlackScholes equation. Stochastic differential equations
Module leader: Mr Thomas van Heerden
MAM2013S (MAM2000W 2IA): INTRODUCTORY ALGEBRA Introduction to abstract algebra and number theory. Topics include: induction, strong induction and WellOrdering axiom. Divisibility and prime factorization. Modular arithmetic. Permutations. Groups. Subgroups. Cyclic groups. Isomorphisms. Simple groups. Factor groups. Lagrange's Theorem. The First Isomorphism Theorem.
Module leader: Prof Anneliese Schauerte
MAM2014S (MAM2000W 2RA): REAL ANALYSIS Axioms of the real numbers, supremum and infimum. Countable sets. Sequences and series. Open and closed sets, compactness. Limits, continuity, differentiability. Sequences and series of functions, uniform convergence, power series. Integration. Calculus of several variables, linear algebra, introductory analysis and abstract algebra.
Module leader: Prof Elena Berdysheva 
MAM3000W  Third year mathematics
The course MAM3000W consists of six modules. Students must take four of these, including at least one of 3AL and 3MS. 3AL: MODERN ABSTRACT ALGEBRA Group Theory (Isomorphism Theorems, pGroups, Sylow Theory, Direct Products and finitely generated Abelian Groups). Further Linear Algebra (Primary decomposition, Jordan normal forms, Bilinear forms).
Module leader: Ms Morgan Vandeyar 3CA: COMPLEX ANALYSIS Field of complex numbers. Power series. Analytic functions. Complex integration. Liouville’s theorem, Fundamental Theorem of Algebra. Maximum Modulus Theorem. Index of a closed curve. Cauchy’s Integral Formula. Counting Zeros and Open Mapping Theorems. Goursat’s Theorem. Singularities. Laurent series. Residues.
Module leader: Prof Elena Berdysheva
3DM: DISCRETE MATHEMATICS Graph theory, combinatorial counting, discrete probability theory, recurrences, algorithms, applications.
Module leader: Prof David Erwin
3MS: METRIC SPACES Metric spaces and topology; applications
Module leader: Dr Simon Chili 3TA: TOPICS IN ALGEBRA A selection from lattices and order, congruences, Boolean algebra, representation theory, naive set theory, universal algebra.
Module leader: Prof Anneliese Schauerte 3TN: TOPICS IN ANALYSIS Compactness in metric spaces, normed spaces, linear continuous mappings between normed spaces, Hilbert spaces, orthogonal projection, differential calculus on normed spaces, review of the Riemann integral and its limitations.
Module leader: Prof Francois Ebobisse Bille
Further details can be found in the Science Faculty Handbook.
Students who complete a degree with a major in Mathematics with a sufficiently strong academic record may apply for entry to the Department’s Honours Program.