### Coordinator: Dr E. Andriantiana

Here is a list of the 9 courses currently offered inthe department at honours, with a short description of each of them. Please contact the designated lecturers and/or the course coordinator for more details.

• ### Algebraic Graph Theory

Lecturer: Dr E. Andriantiana

The AGT Hon course introduces graph theory by studying basic properties of graphs. For this it
covers the notion of homomorphism, transitivity, and spectra of graphs. The main part of the course
consists of studying adjacency matrices of graphs. This involves looking at characteristic
polynomial and spectrum of the adjacency matrices and using them for combinatoric purpose, to
extract information about the graph. Laplacian matrices of graph will be considered as well. As for
adjacency matrices, relations between their characteristic polynomials and spectra to properties of
graphs will also be analysed. The last part of the course is on applications of graph theory to
chemistry, physics and computer science.

• ### Artificial Neural Networks:

Lecturer: Prof M. Burton

Syllabus: Neural Computing, perceptrons, performance Surfaces, general feed-forward networks, neural networks with MATLAB, applications to prediction and classification, radial basis function networks.

• ### General relativity:

Lecturer: Prof N. Bishop

The mathematical tools of general relativity: vectors and tensors in general coordinates, tensor calculus, the covariant derivative, Riemann curvature tensor, Ricci tensor, Einstein tensor.
Special relativity: equivalence of inertial observers, constancy of the speed of light, Lorentz transformation, Minkowski spacetime.
Physical foundations of general relativity: equivalence principle, spacetime is not flat, Einstein’s equations (“geometry = matter”), linearized static approximation, geodesic motion.
Schwarzschild vacuum metric: solution of Einstein equations under spherical symmetry, Birkhoff’s theorem, classical tests (bending of light, perihelion precession, spectral red-shift).

• ### Manifolds:

Lecturer: Prof Larena

The aim of this course is to provide an introduction to differentiable manifolds with an eye on their relevance in modern theoretical physics. Nevertheless, no physics background is required. The course could be beneficial to any student interested in maths or applied maths.

Syllabus: Multilinear and tensor algebra; topological spaces; differentiable manifolds; calculus on manifolds (vectors, one-forms, tensors); flows and Lie derivatives; Riemannian and pseudo-Riemannian manifolds; parallel transport, connection and covariant derivative; curvature and torsion; Levi-Civita connection; isometries; Killing vector fields; non-coordinate bases. If time left, one of the two following topics, depending on the interest of the students: differential forms and integration of differential forms OR Lie groups and Lie algebras and action of Lie groups on manifolds.

Prerequisites: Advanced Calculus and Linear Algebra

• ### Matrix Groups:

Lecturer: Dr C. Remsing

Syllabus: Concrete matrix groups; the matrix exponential function; matrix Lie groups; Lie algebras. (optional: group actions, symmetry and invariance; Klein geometries/Klein's Erlangen Program).

• ### Measure theory:

Lecturer: Dr A. Pinchuck

Measure theory is the study of measure and integration. In this course we
begin by looking at the shortcomings of Riemann integration as a motivation
for a better theory of integration. This leads to the definition of a
measure and a measure space. Measure theory can essentially be divided into
the study of integration and the related study of measurable functions and
has applications in probability theory, ergodic theory, functional analysis
and other areas. Important topics include the Lebesgue measure on the real
numbers, the Lebesgue integral, properties of measurable functions, Lp
spaces and a comparison between the theories of Riemann integration and
Lebesgue integration.

Pre-requisites:  3rd year real analysis or equivalent.

• ### Numerical Modelling:

Lecturer: Prof Pollney

This course takes students through the construction of a numerical model of a time-evolution PDE problem. Topics include: Grid construction, finite differences, method-of-lines, convergence testing, Riemann problems, finite volume methods, and high-resolution schemes.

Prerequisites: Numerical Analysis, PDEs

• ### Stochastic Processes:

Lecturer: Dr M. Gandhi

Measure spaces and Measurable functions, Lebesgue Integration, Lebesgue Measure, Distribution and densities of Random Variables, Random vectors and Joint densities, Moments, Independence, Examples of Stochastic Processes,  Stationarity, Law of Large numbers, Convergence of a Stochastic Process.

• ### Topology:

Lecturer: Dr V. Naicker

Topology combines and generalizes various ideas from analysis, algebra and geometry. This course presents a broad introduction to point set topology, differential topology and algebraic topology. It is aimed at those students interested in how mathematics comes together as a unifed subject - the fundamental question: ''How is a coffee cup, the same as a doughnut?"