Honours

The course consists of a selection from the following topics.

PART I: RINGS AND MODULES

This course is an introduction to the theory of associative rings and their modules. Review the fundamental concepts of algebras.

Topics covered include the complete ring of quotients of commutative rings, prime ideas and prime ideal spaces, primitive rings and radicals and completely reducible rings and modules. Artinian and Noetherian rings and idempotents. Injective and projective modules. Introduction to homological algebra.

PART II: AUTOMORPHISMS GALOIS THEORY

This course introduces the study of field extensions. The main idea of Galois Theory is to consider the relation of the group of permutations of the roots of a polynomial to the algebraic structure of its splitting field. The course includes the Fundamental Theorem of Galois Theory, composite extensions and simple extensions, the Galois group of polynomials, solvability and radical extensions leading to the result on the insolvability of the quintic.

PART III: FINITE DIMENSIONAL VECTOR SPACES

The purpose of this course is to treat linear transformations on finite dimensional vector spaces by simple geometric notions common to many parts of mathematics and in a language that is used in the theory of integral equations and Hilbert Theory. The course builds on the elementary notions of vectors spaces over fields and introduces dual spaces, quotient spaces and the direct sum of vector spaces. Central to the development is the introduction of bilinear forms and inner products and the Riesz Representation Theorem with the ideas of adjoint and self-adjoint linear transformations. The course includes a survey of orthogonal projections, eigenvalues and the Spectral Decomposition Theorem.

PART I: ASYMPTOTICS

This is a continuation of the Honours Topic Combinatorics. It includes a selection from 1) general principles of enumeration; 2) Symbolic computer algebra with Mathematica; 3) 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

PART II: APPROXIMATION THEORY

1) Qualitative approximation: (Does a closest element exist, is it unique, can we characterize it?) Existence and Uniqueness of best approximations. Theorems of Chebyshev, H r, Borel, the alternation/ equioscillation theorem. 2) Possibility of Approximation: (Can we approximate arbitrarily well?) Weierstrass Theorem via Bernstein polynomials; Bohman-Korovkin theorem. Lagrange and Hermite interpolation. 3) Quantitative approximation: (How fast can we approximate?) Jackson s and Bernstein s theorems, Bernstein s and Markov s inequalities. 4) Additional topics: M? Theorem, rational approximation, Pad?pproximation, continued fractions, wavelets.

References: E.W. Cheney, Introduction to Approximation Theory, Chelsea, New York, 1982; R. De Vore and G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.

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.

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.

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: M?bius 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 is a foundational course in functional analysis and as such it requires as a prerequisite only a knowledge of classical real analysis. The topics covered in the course are: (1) Normed linear spaces, inner product spaces, Banach spaces & Hilbert spaces. (2) Properties and characterization of bounded linear operators on normed linear spaces. (3) The principle of uniform boundedness, the open mapping theorem, the Hahn-Banach theorem or the Hilbert space analogues thereof in terms of orthogonality, depending on the focus for that year (4) The Riesz-Fischer Theorem, duality and reflexivity. (5) Spectral theory of compact operators. (6) Bounded selfadjoint operators.

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.

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 of 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.

This is a foundational course in measure theory and as such it requires as a prerequisite only a knowledge of classical real analysis.The topics covered in the course are: (1) Algebras and sigma algebras of sets. (2) Definition and properties of measures (3) Completions of measures. (4) The monotone class theorem and the Caratheodory construction of measures. (5) Properties of measurable functions. (6) Construction of the Lebesgue integral. (7) Fatou's Lemma, the Lebesgue monotone convergence and dominate convergence theorem. (8) The space of Lebesgue integrable functions. (9) Signed measures and the Hahn-Jodan decomposition. (10) The Radon-Nikodym-Lebesgue decomposition.

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:

I. 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;
II. 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;
III. Ring localizations, integral elements, prime and maximal ideals, Dedekind domains, unique factorization of ideals, algebraic number fields, integral bases, discriminants, norms, class number.

The course will cover a selection from the following topics:

i) General Topology: Axiom of Choice, cardinal arithmetic, a topological space, a continuous mapping, cardinal functions, separation axioms, Urysohn's lemma, compact spaces, Tychonoff theorem.
ii) Algebraic Topology: Homology and Cohomology, Winding Numbers, Covering spaces, topology of surfaces, de Rham cohomology of surfaces, the Mayer-Vietoris sequence, classification of compact surfaces, Riemannian surfaces.
iii) Differential Geometry: Manifolds, vector bundles, differential forms, integration of differential forms, introduction to Stokes¡¯ theorem. Introduction to Riemannian geometry (connections, curvature, covariant differentiation).

References: R. Engelking, General topology, Springer-Verlag, 1989; K. Kunen and J. Vaughan (eds.), Handbook of set-theoretic topology, Elsevier, Amsterdam, 1984; W. Fulton, Algebraic Topology: A First Course; G. E. Brendon, Topology and Geometry.

The course consists of a research project on a pure mathematics topic which is carried out under standard exploratory, investigative and analytical principles. The stages consists of Topic selection, Proposal Construction, Approval of Proposal, Project Work, Project Report Writing and Report Submission. The report should not exceed 35 printed pages on A4 sized paper. The following items are recommended for the proposal: (i) Title (ii) Aim (iii) Problem Statement (iv) Research Questions (v) Methodology (vi) Contents (vii) Literature review (viii) Further Wor; (ix) References.

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 G?del'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. 

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.