Keele University Mathematics Undergraduate Programmes 2020 2020 Entry | Page 14

14 | Mathematics
Year 3 Modules
In the third year, all students have a wide choice of modules. The availability of modules, and their content,
may change from time to time subject to student uptake, staff availability and timetabling restrictions.
Graph Theory Complex Variable II Partial Differential Equations
This module introduces the concept
of a graph as a pictorial representation
of a symmetric relation. A variety of topics
are investigated and, for each one,
at least one of the major theorems
is proved. The emphasis is on pure
graph theory, although some applications
are explored. This module begins with applications
of contour integration and the Residue
Theorem. It then provides an introduction
to conformal mappings, together with
some of their applications, in particular
their application in determining harmonic
functions in two-dimensional regions.
The module also covers certain analytical
aspects of complex functions. A derivative of a function of two variables,
with respect to one of these variables
whilst the other variable is held fixed,
is called a partial derivative. Partial
differential equations (PDEs) are
equations that contain partial derivatives,
and many physical processes, such as
wave propagation and heat conduction,
are governed by PDEs. This module
discusses basic solution techniques
for PDEs. A typical question that can
be answered by the theory of PDEs
is “Where should we place the radiators
in a room such that the temperature
is as uniform as possible?”
Group Theory
Building on the concept of a group
introduced in Abstract Algebra,
this module develops some of the
mathematics underlying the classification
of finite groups, culminating in a proof
of Sylow’s First Theorem. The module
also develops some applications of
group theory.
Linear Algebra
This module builds on the concept of an
abstract vector space. Concepts such
as linear independence, span and scalar
products of vectors are generalised from
Euclidean space to other vector spaces,
such as function spaces, you will then
see that seemingly disparate results from
different branches of mathematics are
sometimes just different specialisations
of the same general concept.
Waves
This module aims to give an account
of the underlying mathematical theory
that describes the behaviour of waves
in diverse physical settings. These
include waves on beams, membranes
and stretched strings, sound waves and
waves in liquids with a free surface.
keele.ac.uk/scm
Metric Spaces and Topology
A metric space is a set of mathematical
objects of any kind, in which it is possible
to define the ‘distance’ between two
objects. Concepts such as convergence
of infinite sequences and continuous
functions, which arose from the study
of real numbers, can be generalised
to metric spaces, making them more
powerful and versatile. Topology is a
further generalisation, in which there
is no numerical measure of distance
but simply a qualitative notion of
‘arbitrary closeness’.
Nonlinear Differential
Equations
The great variety of behaviour in
physical systems is reflected in the
solutions of the differential equations
used to model them. The majority of
these equations are nonlinear, and very
few have exact solutions. The module
focuses on methods for obtaining
approximate solutions and phase-plane
representations of dynamics.
Fluid Mechanics
Fluids are everywhere – your blood is a
fluid, as is the air in your lungs, the water
in the oceans and the burning gas in
the sun. Despite this diversity, most fluid
motion can be explained by a tiny number
of core principles. The module derives
these core principles, and illustrates them
in some simple examples of fluid flow.
Number Theory
and Cryptography
Number Theory studies the properties of
the natural numbers and the integers. It is
one of the oldest and most beautiful areas
of Pure Mathematics, and is famous for a
large number of problems that are easy to
state, but turn out to be difficult to resolve.
Indeed, many of these problems remain
unsolved, and so Number Theory is also
one of the most active areas of modern
research. Recently, ideas from Number
Theory have been applied to problems
in Cryptography, such as the design of
ciphers and secret sharing schemes.
This module will trace the development
of the subject from ancient problems to
these modern applications.