Honours Complex Analysis is an advanced course in complex analysis which presents properties of analytic functions, in particular relating to zeros and poles of analytic functions.

The results emphasize the rich structure of analytic functions. The course content includes: Möbius transformations; Montel's theorem; Riemann mapping theorem; infinite products of analytic functions; approximation of analytic functions; analytic continuation; harmonic functions; entire functions of finite order; the range of analytic functions.

This module deals with mainstream and advanced concepts and trends in Elementary, Analytic and Algebraic theory of Numbers.

These will include a selection from the following topics:

1. Infinitude of primes, primes numbers of different kinds, solution of Diophantine equations and congruences, arithmetic functions, Euler function, quadratic residues, irrational numbers and continued fractions, decimal expansions of real numbers.
2. Algebraic properties of arithmetical functions, pseudoconvergence, average values, densities, the zeta function, the nth prime, Prime Number Theorem, Dirichlet characters, Ramanujan expansions, orders of magnitude.
3. Ring localizations, integral elements, prime and maximal ideals, Dedekind domains, unique factorization of ideals, algebraic number fields, integral bases, discriminants, norms, class number.

Starting at an elementary level, the course introduces students to a selection from the following topics:

1. Permutations and combinations.
2. Binomial coefficients, Stirling numbers and combinatorial identities. 
3. The principle of inclusion and exclusion.
4. Recurrence relations
5. Ordinary and exponential generating functions
6. The exponential formula and trees
7. Lagrange inversion
8. The symbolic method of enumeration
9. Discrete probability.
10. Bivariate generating functions and combinatorial parameters
11. Polya’s Theory of Counting.

The course deals with the Invariance approach to the analysis of variational differential equations as introduced by Sophus Lie.

The course content is as follows:

1. Differential Geometric Preliminaries (Manifolds, Groups, Lie Groups, Lie group transformations).
2. Lie point symmetries of ordinary differential equations (methods and applications).
3. Calculus of Variations (Introduction and definitions, Euler Lagrange equations, Inverse problems, conservation laws).
4. Noether symmetries
5. Noether's theorem (conservation laws).
6. Association between symmetries and first integrals.
7. Symmetries of PDEs.
8. Conservation laws of variational PDEs.

This is a continuation of the Honours Topic Combinatorics. It includes a selection from General principles of enumeration, Symbolic computer algebra with Mathematica Methods of asymptotic enumeration including Asymptotics of sums, asymptotics of recurrence relations, Mellin transforms, Rice’s method, singularity analysis, saddle point method, limiting distributions.

References:

Flajolet and Sedgewick, Analytics Combinatorics. (Available for free download); Sedgewick and Flajolet, an introduction to the analysis of algorithms.

The course will introduce the fundamental concepts of Graph theory. Elements of topological graph theory, graph polynomials, connectivity, and embedding will be introduced.

 The Major part of the course will be devoted to some or all the following parts:

PART I: Basic Graph Theory. Basic concepts and results in graph theory and introduction to open problems. Traversals (Eulerian graphs, Hamiltonian graphs), connectivity and planarity. Research in graph theory on these topics.

References: R. Diestel, Graph Theory, 4th edition, Springer-Verlag 2010 (corrected edition in 2012); G.F. Royle and C. Godsil, Algebraic Graph Theory, Springer-Verlag 2001.

PART II: Topological Graph Theory. Fundamental concepts of the relationship between graph theory and Knot theory. Some knot invariants calculated via the corresponding graphs: pathwidth, component number, the Kauffman polynomial, the Jones polynomial and the Alexander polynomial. Reference: C. Adams, The Knot Book.

PART III: Graph Polynomials. Fundamental concepts of graph colouring and graph operations. Graph polynomials namely chromatic polynomial, the Tutte polynomial, the Martin polynomial and Penrose polynomial.

Reference: F. M. Dong, K.M. Koh and K.L. Teo, Chromatic polynomials and Chromatic graphs.

The course is an introduction to algebraic topology with geometric applications.

It is aimed at honours students who have some knowledge of basic topology and differential geometry, such as what is provided by the third-year courses MATH 3010 and MATH 3031 offered at Wits.

The main content follows:

Introduction to the basic concepts and tools of algebraic topology, such as fundamental group, covering spaces, homology and cohomology. Introduction to certain applications of algebraic topology to geometry, such as de Rham cohomology, the classification of closed surfaces, and the Jordan Curve Theorem.

Mathematical logic is a fundamental mathematical field. In the early twentieth century, the discovery of Russell's and other foundational paradoxes shook the mathematical community to its core and it became a matter of urgency to re-establish mathematics on firm foundations. In the ensuing quest, mathematical logic began to emerge in its modern form and to produce some of the most sensational and shocking results of the last hundred years, including Gödel's incompleteness theorems, and the independence of the continuum hypothesis from the axioms of Zermelo-Fraenkel set theory.

Besides its central role in mathematics, mathematical logic has wide-ranging applications, especially in theoretical computer science, artificial intelligence and linguistics.

This course introduces the field of Mathematical Logic. It consists of propositional logic, first-order logic and non-classical logic. The course explores the syntax, semantics and proof systems for each logic considered, pursuing these themes up to and including soundness and completeness theorems and the characterisation of expressivity in terms of model-theoretic invariance results.

For honours students, there are the following prizes:

• IBM Gold Medal Awarded for outstanding performance in the Honours course of study in the Mathematical Sciences.
• UNICO Chemical Company Gold Medal Awarded annually to the most distinguished Honours graduand in the Faculty of Science.
• Bronze Medal of the South African Mathematical Society Awarded annually by the South African Mathematical Society to the top student in Mathematics or Applied Mathematics Honours at each South African University.