School of Mathematics
MT3011/3022/3000 and MT4011/4022/4000
PROJECT OPTION FOR 3RD AND 4TH YEAR STUDENTS
2006-2007
The aims of this option is to give third and fourth year students an opportunity of personally
researching a chosen mathematical topic and of improving their personal transferable skills by
producing a short thesis that describes it. The School strongly urges students to select this
option. The experience of working independently provides useful practice for later life, and the
fact that you have chosen such an option gives you an advantage when you apply for jobs. Please
note that M.Math students must take this option, normally as a double project or two
(consecutive) single projects in their fourth year.
Warnings
 You are not allowed to do a project and also take a course that covers the same (as
opposed to related) material. Check with the supervisor, and plan ahead to make sure
there are no conflicts.

Many supervisors can only accept a limited number of students to take their project.
1. PROJECT TOPICS
Supervisor [default second examiner]: Topic
Brief Description
Projects available to third and fourth year students
Abrahams [MJS]: Simple models of the ear
The ear is an extremely complex and sensitive device for detecting sound over a wide range of
frequencies and amplitudes. It is anticipated that the student will examine simple models of the
mammalian inner ear which can help to explain qualitatively why it performs its function so well.
Such models will focus on the role of the basilar membrane inside the cochlear duct. Other
aspects, such as impedance matching across the middle ear, dynamics of the microscopic hair
cells, physiology and perception models could also be included in the study.
Abrahams [MJS]: Sonoluminescence
In the mid-1920s the remarkable discovery was made that sound at moderate intensities,
irradiating a flask of water containing microscopic bubbles, could generate light. At the time little
was understood of the processes involved, and so not-surprisingly the subject lay dormant for
many decades. In 1998 the field was resurrected when scientists created light from sound using
water containing just one bubble, so called single-bubble sonoluminescence (SSL). This enabled
precise measurements to be made and also allowed a more detailed explanation of this curious
phenomenon.
It is envisaged that the project will take the form of a review essay, examining the many articles
on the subject in the literature. However, there is also likely to be scope for some mathematical
analysis of the acoustic cavity modes excited in the experiment, the bubble dynamics (collapse,
expansion and possible instability) or some basic shock wave dynamics. The supervisor will
attempt to make arrangements during the project for a visit to a laboratory in the UK to see a
working demonstration of sonoluminescence.
Abrahams [MJS]: Applications of wavelets in applied mathematics
Wavelets offer an important new approach to determining detailed information about the
behaviour of solutions of model problems in geophysics and mechanics (fluids, solids etc.). Such
models may prove difficult to investigate by traditional direct or Fourier transform approaches.
The project will investigate wavelets in a non-rigorous fashion and apply them to several model
problems in applied mathematics. The usefulness of wavelets in such problems will be
ascertained, as will be the effect of changing the wavelets employed etc.
Coghlan [RCL]: Diophantine equations
Diophantine equations (named after Diophantus c.250A.D.) are equations (usually polynomial
equations) with integer coefficients where one tries to describe the integer solutions. Some
diophantine equations (e.g. x3  y 3  z 3 ) can be studied using knowledge from ring theory about
certain rings (e.g. Z[ w], w 
1
(1  3) , respectively). The project would concern itself with
2
this and some other equations and related matters in ring theory.
Coghlan [RJP]: p-adic numbers
The p-adic Numbers (for each prime p) is a field which, in some sense, combines all the rings
Z pn (n  1, 2,3 etc) in a ‘useful’ way. This field is the analogy of the real numbers where, instead
of using the ordinary absolute value (on Q ) to give you a metric (a notion of ‘distance’), one
uses a different ‘valuation’.
The project would concern itself with the construction of this field, a description of its algebraic
properties and of its number-theoretic applications.
Duck [MH]: Boundary layer separation
To compute the ‘boundary layer’ flow over a semi-infinite flat plate, with a retarding freestream
flow, using finite-differences. The aim is to determine the nature of the flow at a point on the flat
plate where the flow separates, i.e. the wall shear reduces to zero.
Duck [MH]: Flow in a driven cavity
To compute the flow in a rectangular cavity, using the full (‘Navier-Stokes’) equations, by means
of finite-difference approximations. Particular emphasis may be placed on ‘very long’ or ‘very
deep’ cavities to determine the nature of the flow in these two limits.
Foster [EKK]: Normal linear model checking
After fitting a selected statistical model to a set of data an examination of its fit should be carried
out to help prevent the drawing of erroneous conclusions. A formal approach involves
embedding the current model in a wider class and checking for an improvement in fit while more
informal methods are generally graphical in nature. It is envisaged that this project will involve
the study of a number of both formal and informal techniques. Also, they will be applied to
certain data sets using already prepared GLIM programs.
Gajjar [PJE] Modelling of traffic flow.
In the last 50 years there have been several attempts to model traffic flow. Initial attempts
focussed on simple one-dimensional continuum models in which traffic flow was described by
the kinematic wave equation. More sophisticated models lead to highly complicated nonlinear
partial differential equations similar to the Navier-Stokes equations. Recently there have been
attempts to model traffic flow using discrete ideas. The aim of this project is to survey the
different models used and discuss the relative success and failures of the various approaches.
Gray [AJ]: Shock waves in granular media
Granular flows are abundant in geophysics, industrial processes and our day to day lives. This
project will review the derivation of some recent models for free-surface flows of granular
materials and derive a series of shock-wave solutions for the flow. If time permits, it may be
possible to realise these flows in an existing laboratory experiment and make comparisons with
the exact solutions.
Hazel [TS]: Practical Cryptography (for Maths + CIT students)
How can sensitive information be transmitted on an open communication channel? What makes
an Internet transaction secure? The answer is that the information must be encrypted or encoded
in such a way that only the desired recipient can decode the message. This project will involve the
study and implementation of modern cryptographic algorithms in the Java programming language.
There will be a very strong programming component to the project, but the student must also
understand the mathematics underlying the algorithms. Algorithms studied could include RSA,
AES, 3DES and Blowfish.
Heil [AFJ]: Wave propagation and blood flow in the arteries
The regular contractions of the heart pump blood around the body’s circulatory system which
consists of a highly branched network of elastic tubes. This project will examine analytical
models for the pulse wave propagation and the flow fields in the large arteries.
Hewitt [AH]: Shuffling 101
An internet company (NexTag.com) sets the following challenge for job applicants. Given a deck
of N unique cards, cut the deck M cards from top and perform a 'perfect shuffle'. A perfect shuffle
begins by putting down the bottom card from the top portion of the (cut) deck followed by the
bottom card from the bottom portion of the (cut) deck followed by the next card from the top
portion etc., interleaving the cards until one portion is used up. The remaining cards then go on
top of the pile. The problem is to find the number of 'perfect shuffles' required to return the deck
to its original order. Your task is to write a small object-oriented code (in C++/Java/Python) to
solve this task for N=1002 and M=101, discussing some of the (mathematical) ideas behind the
process. Key to the problem will be the efficiency of the method(s) described.
(If you have ideas for other interesting programming projects you would prefer to tackle, let me
know)
Higham [FT]: Powers of matrices
Many numerical processes are governed by the behaviour of powers of a matrix. For example,
iterative processes often rely on the fact that the powers of a matrix converge to zero if all the
eigenvalues of the matrix have modulus less than 1. The aim of this project is to survey bounds
for the powers of a matrix, and to gain further insight by conducting numerical experiments in
MATLAB.
Higham [FT]: 2 x 2 matrices
Many theorems and algorithms in numerical linear algebra can be illustrated well with a 2 x 2
matrix. The purpose of this project is to explore 2 x 2 matrices.
Juel [RH]: Standing pendulums upside-down: not quite the Indian Rope Trick?
A pendulum can be be stabilized in an upside-down position by rapid vertical oscillations of its
pivot. Chains of linked pendulums can equally be made to defy gravity by vibrating the system at
higher frequencies the larger the number of links in the chain. The aim of this project is to review
the mathematical analysis of this curiosity of classical mechanics and discuss its relevance to the
understanding of the stabilization of a length of floppy wire, also known as the Indian Rope Trick.
Pan [EKK]: Statistical modelling of longitudinal data
In randomized controlled trials patients are assigned to different treatment groups and are then
followed up for some period. Repeated measurements taken from the same patient over time are
likely to be correlated although responses between patients may be independent. One of the aims
of such studies is to measure the difference between treatments over time and to quantify the
dependence of responses on certain baseline covariates of interest, by taking into account the
within-patient correlation.
Data such as these are called longitudinal data and are very common in practice, e.g., in medicine,
epidemiology, economics, etc. This project aims to use some advanced statistical methodology
such as growth curve models and linear mixed models to analyze longitudinal data. Some
practical data arising in randomized controlled trials will be studied.
Paris [GMW]: The History of the Propositional Calculus
Who should be credited with discovering the Propositional Calculus as we understand it today?
Who first formalized the notions of logical consequence? formal proof ? Who was the first
person to prove a Correctness Theorem? a Completeness Theorem? for this calculus.
The project would involve tracing these concepts and results back to their, sometimes multiple,
origins and writing a coherent account of this history.
MT2151 (Propositional Calculus) is surely a prerequisite, and it might be a help, though it is
certainly not essential, to have a smattering of German for consulting some of the original papers.
Premet [GW]: Root Systems and Weyl Groups
A crystallographic root system is a finite subset  of the Euclidean space E with scalar product
( | ) which satisfies the following three conditions:
1.      for any   ;
2. s ()   for any   , where s denotes the orthogonal reflection of E associated with
;
3. 2( |  ) /( |  )  Z for all  ,   .
The Weyl group of  is the subgroup of the group of all orthogonal linear transformations of E
generated by all reflections s with   . This group is finite and crystallographic, i.e.,
preserves a lattice in E .
Root systems and their Weyl groups were first introduced in the course of classifying the finite
dimensional simple Lie algebras over
. They play a prominent role in Lie theory,
representation theory of algebras, geometry, and combinatorics.
This area provides a lot of material for 3/4 year projects. One possibility could be to study
maximal subsystems of an irreducible root system in E .
Books James E. Humphreys, Reflection Groups and Coxeter groups, Cambridge University Press,
1990
Prerequisites: Group Theory, Further Linear Algebra
Prest [AAP]: Rings and modules
For this you should have met rings in second year algebra or elsewhere. Modules are structures
on which rings act linearly (vector spaces over fields are examples). Here are some of many
possible projects: in each case the balance between theory and examples/computation can be
shifted either way.
a. Structure of ideals (prime decomposition in particular) in commutative rings, with possible
emphasis on rings of integers in number fields.
b. Weyl algebras - these are important non-commutative rings which consist of polynomial
differential operators. Groebner basis methods have recently been developed in this context so
this could be brought in if desired.
c. Representations of quivers. A quiver is a directed graph and a ring can be formed from the
paths through the quiver. A representation of this quiver is essentially the same thing as a module
over this ‘‘path algebra’’. This is a nice mix of theory and computation.
d. Localisation: The process by which non-zero integers are inverted to obtain the field of
rational numbers can be extended to many situations including the non-commutative setting (Ore
localisation) and rings of matrices (Cohn’s localisation) and is a recurring theme in many
mathematical situations.
Prest [GW]: Algebraic geometry
a. The foundation of algebraic geometry is the connection between ideals in polynomial rings
and algebraic varieties (zero sets of polynomials). After some basics this could take a number of
directions, including the next topic.
b. Computational algebraic geometry. Use of Groebner bases (on the computer) to investigate
algebraic curves, surfaces and higher-dimensional varieties.
c. Toric varieties. These are algebraic varieties defined in terms of linear combinatorial data.
They have very nice properties and include many important examples. Again Groebner bases
would fit in naturally (if desired).
Prest [PHA]: Categories, functors and natural transformations
These are simple and powerful ideas which connect and unify much of pure mathematics. A onesemester project could reasonably aim for the adjoint functor theorem. Not recommended unless
you’re happy with abstract notions and, preferably, have seen some algebra or algebraic topology
(so that you have some examples to refer to).
Shardlow [NJH]: Numerical solution of Hamiltonian systems
Hamiltonian systems arise as models of conservative physical systems, systems which do not
loose energy. The simplest examples are those in classical mechanics, for example planetary
motion. Hamiltonian systems have a rich mathematical theory. This project focuses on how
Hamiltonian systems are solved with a computer, particularly on the derivation and analysis of
symplectic methods. Some computer work in Matlab will be necessary.
Shardlow [AH] Scientific Computing for students taking the Math with C+IT degree
We will explore the use of Java and Matlab to understand and visualize a mathematical model.
There will be a strong programming aspect to this project, though the student will need to
understand the mathematics behind the model. Here are two suggested problems: (i) rainbows:
study the physics of rainbows and write a program to illustrate the creation of rainbows from
sunlight and rain drops; (ii) images: mathematics is used to encode images for transmission on the
web. Images are often compressed to reduce the file size and speed up transmission on the web.
We will look at how these techniques change the image.
Sharp [CW]: Fractals and their dimension
In the theory of chaotic dynamical systems there are natural invariant sets (typically, attractors or
repellers) which may have very beautiful and complicated “self-similar” features. For example,
this occurs with Iterated Function Schemes and Julia sets.
A key quantity for fractal sets is their dimension (for example, either the Hausdorff dimension or
box dimension). For linear dynamical systems, the dimension is often explicitly known (e.g., for
the middle third Cantor set it is log 2/log 3). For nonlinear dynamical systems one has to calculate
the dimension.
This project deals with different types of fractals, their construction and methods for estimating
their dimensions.
Simon [IDA]: Energy from ocean waves
There has been substantial interest in Britain in renewable energy sources, and ocean waveenergy in particular, since the oil crisis twenty years ago. Ocean waves can be considered as small
amplitude waves, for which the equations are linear and can be described by the use of a scalar
potential. The project is to review the general theoretical aspects of harnessing the energy of such
waves, including looking at the different types of structures proposed for wave-energy devices.
The project should also touch on any relevant considerations of engineering, biology, energy
supply and politics.
Simon [IDA]: Loudspeaker design
Hi-fi enthusiasts often design and build their own loudspeaker systems. Although there is a
certain art to this, there are also quantitative aspects, such as room and cabinet acoustics, and
directional sound fields. Ref. J. Borwick (ed.), Loudspeaker and Headphone Handbook (534.868,
B17).
Tisseur [NJH]: Solving Systems of Second-order Differential Equations Arising in Practical
Problems1
Problems such as the analysis of vibration of structures (to avoid bridges collapsing for instance),
and simulation of the transient current of electrical circuits give rise to a system of linear secondorder differential equations of the form Mq(t )  Dq(t )  Kq(t )  f (t ) where M , D and K are
n x n matrices and f (t ), q (t ) are nth order vector. The solution of this system is intimately related
to the solution of the quadratic eigenvalue problem (  2 M   D  K ) x  0.
The aims of the project are to study the analytical solution of such systems and to explore
numerical methods of solution.
1 This project is available in the second semester only
Walkden [RS] The Riemann Zeta Function
The Riemann zeta function plays an important role in the study of prime numbers. Despite being
easy to define, there are still many basic properties that are unknown. (For example: what are its
zeros? The infamous Riemann hypothesis - one of the most important unsolved problems in
mathematics - states that the (non-trivial) zeros of the zeta functions are complex numbers s with
( s)  1/ 2. ) Other, easier, properties of the Riemann zeta functions have attractive and
occasionally surprising connections with prime numbers. The aim of this project would be to
develop some of the basic properties, and to use them to study prime numbers.
The project requires a good knowledge of MT2101 Calculus with Complex Numbers.
References S.J. Patterson, An introduction to the theory of the Riemann zeta-function, C.U.P.,
(1988). E.C. Titchmarsh, The theory of the Riemann zeta-function, O.U.P,, (1986). T.M. Apostol,
Introduction to analytic number theory, Springer-Verlag, (1976).
Most books on analytic number theory contain material on the zeta function. Some books on
complex analysis (eg: Ahlfors’ book) do too.
Walkden [RS]: Continued fractions
A continued fraction is an expression of the form
a0 
1
a1 
1
a2 
1
1
a3  
a4
where the a j are integers. Any real number can be expanded as a continued fraction, and this
expansion often tells us a great deal about that number. For example, continued fractions tell us
why 22/7 is a good approximation to  (and why 355/113 is even better). Continued fractions
also have many applications ranging from number theory (finding integer solutions to
x 2  dy 2  1 , for example) to designing models of the solar system and explaining why every 4th
year is a leap year (unless the year is divisible by 100 but not by 400).
References:
There are lots of books on continued fractions. See for example:
H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, CUP.
G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, OUP.
A.M. Rockett, P. Sz z, Continued Fractions, World Scientific.
Wilmers [JBP]: Proving theories decidable - The method of elimination of quantifiers
A celebrated consequence of Godel’s work is the undecidability of consistent mathematical
theories ‘containing’ a version of both addition and multiplication on the natural numbers, (i.e.the
impossibility for such theories of the existence of an algorithm to determine which sentences are
or are not consequences of the axioms). In contrast to such results however there are some
interesting mathematical theories which ARE decidable, two of the best known examples being
the theory of the real numbers with addition and multiplication, or the theory of the complex
numbers with addition and multiplication. The project would investigate the principle method
used to prove such theories decidable, and would survey some of the results in the literature.
Prerequisites: students will need some knowledge of the predicate calculus before taking this
project. The internal (School of Maths) course on this is MT4191.
Zhang [RAD]: Option Pricing
Consider a financial market consisting of d risky assets (e.g. d stocks) and one riskless asset (e.g.
a bank account). An option is a financial instrument (agreement) giving one the right but not
obligation to make a specified transaction at (or by) a specified date at a specified price. For
example, a European call option gives one the right to buy on the specified date, the expiry date,
at a specified price K, which is called the strike price. The project aims to use probabilistic
methods to establish a formula for the selling prices of options and to calculate the prices
explicitly in some concrete models.
This is available as a third option (10 credit) option only and is independent of MT3672
(Mathematical Modelling of Finance).
Projects available to fourth year students only
Aczel [GMW]: Frege’s construction of the natural numbers
At the end of the last century Frege claimed to have demonstrated that the arithmetic of the
natural numbers could be developed in purely logical terms. Unfortunately Russell found a
paradox (Russell’s Paradox) that showed that Frege’s ‘logic’ was inconsistent, thereby
invalidating Frege’s claim. In recent years it has been discovered that Frege’s development of
arithmetic can be reconstructed within a consistent part of Frege’s ‘logic’.
The aim of this project is to present a coherent account of the historical and mathematical issues
concerning Frege’s failed claim and its modern reconstruction. A student engaged on this project
will need to be familiar with first order logic (which effectively makes MT4191, Predicate
Calculus a prerequisite for this project) and will need to learn about second order logic and the
additional ingredients of Frege’s ‘logic’. The student will also need to read about the recent work
and should have an interest in the history and philosophy of mathematics.
Aczel [GMW]: Substitution and variable binding
To study predicate logic and other formal languages it is necessary to study syntax, particularly
the substitution operation. This has usually been considered a tedious subject, best got over as
quickly as possible, even though there are awkward problems when substituting inside variable
binding operators such as the universal and existential quantifiers of predicate logic. The
standard treatments of this topic are usually somewhat vague and imprecise. This has been
unsatisfactory, particularly when formal languages are implemented on computers, and this has
led to a variety of approaches to try to develop a standard generally accepted mathematical theory
of syntax with variable binding operations.
The aim of the project is to read some of the interesting literature on the topic, some of it very
recent, and write a survey of the present state of the issue.
Duck [RH]: The mathematical modelling of finance
This project (for which MT3672 is a necessary prerequisite) is based on extensions to the BlackScholes equation (e.g.transaction costs, dividend payments, etc.), and on various aspects of
American (and other) options, not considered in MT 3672. Depending on the particular interests
of the student, the project could develop in various directions, including a consideration of
numerical techniques for treating such problems, or on other (but related aspects) such as the
modelling of interest rates, volatility, etc.
Heil [AFJ]: Flow in a collapsible channel
Many fluid conveying vessels in the human body are elastic and deform substantially in response
to the traction that the fluid exerts on them. The resulting fluid-structure interaction can lead to
interesting phenomena such as the flow limitation frequently experienced during forced
expiration.
This project will investigate a simplified model (viscous flow in a channel with elastic walls)
which shows many characteristic features of such fluid-structure interaction problems.
Kyprianou [PJF]: Survival analysis
Survival analysis is the term used to describe the analysis of data that correspond to the time from
a well-defined origin until the occurrence of some particular event, or endpoint. In medical
research, the time origin will often correspond to the recruitment of an individual into an
experimental study such as a clinical trial to compare two or more treatments, or to investigate the
factors which affect the progression of a disease. This in turn may coincide with the diagnosis of
a particular condition, the commencement of a treatment regime or the occurrence of some
adverse event. If the end point is the death of the patient, the resulting data are literally survival
times. However, data of a similar form can be obtained when the end point is not fatal, such as
the relief of pain, or the recurrence of symptoms. The methodology can also be applied to data
from other application areas such as times to re-offend in criminology, times to perform a task in
psychological experiments, or lifetimes of industrial or electronic components.
The reasons why survival data are not amenable to standard statistical procedures are firstly
because survival data are generally highly skewed with a fair proportion of cases having
unusually long survival times, and secondly because survival times are frequently censored. The
survival time of an individual is said to be censored when the end point of interest has not been
observed for that individual. This may be because the data from a study are to be analyzed at a
point when some individuals are still alive. Alternatively, the survival status of an individual at
the time of the analysis might not be known because that individual has, at some point in time,
withdrawn from the study. An actual survival time can also be regarded as censored when death
is from a cause unrelated to the treatment.
This project involves reviewing and applying modern statistical methods suitable for analyzing
specific survival data sets.
Kyprianou [PJF]: Factorial experiments, response surface methodology and Taguchi
designs
Quality control of industrial products or a management system requires that sources of variation
be identified. This can be achieved by performing factorial experiments. Once these sources of
variation are identified the levels at which they should be controlled for optimal and robust
product/system performance should be identified. This can be achieved using response surface
methodology. So, in what way do Taguchi designs contribute in building quality into a product?
McCrudden [???]: Haar measure on locally compact groups
Lebesgue measure on n is invariant under translation, and is the unique such measure on n .
In this project we look at the natural generalization of this result to locally compact groups, and at
various consequences. The intention is to investigate some elementary properties of locally
compact groups and then to show the existence and uniqueness of Haar measure on such groups.
Prerequisites: MT2222, MT2262. Co-requisites: MT 3101, MT3131. (Can be taken as a single or
as a double project in year 3 or 4.)
Pre-requisites: 2nd year Group Theory, 3rd year Metric Spaces; Co-requisite: 4th year Measure
and Dimension. (This can be taken as a single project by M.Math students in either semester, and
could naturally extend to a double project.)
Walkden [RS] Hyperbolic dynamical systems
Dynamical systems can be thought of as the study of iterating a single map T : X  X .
Hyperbolic dynamics form one of the most important, and best understood, classes of dynamical
system and are still a major source of current research. The aim of this project is to gain an
understanding of hyperbolic dynamics in terms of both specific examples (hyperbolic dynamics is
very ‘hands-on’!) and general techniques.
There are many directions a project could take. You could explore the notion of structural
stability: a dynamical system is structurally stable if its dynamics are ‘the same’ (in some
appropriate sense) as nearby dynamical systems. Alternatively, you could explore how to
represent a hyperbolic system in terms of a space of sequences (‘symbolic dynamics’)---a
technique that has proved to be extremely useful and powerful.
This project ties in very naturally with MT4512 Dynamical Systems and Ergodic Theory,
although this is not a formal co-requisite.
Reference A. Katok and B.~Hasselblatt, Introduction to the Modern Theory of Dynamical
Systems, C.U.P., (1996) contains lots of material and a large number of additional references.
2. DETAILED ARRANGEMENTS
(a) Topics. A list of topics is given above. Each project title is given together with the name of
the member of staff willing to supervise that project. General advice on the projects can be
obtained from your Personal Tutor or the Project Organizer (Mike Prest). Once you have
identified a project that interests you, you should discuss the project with its supervisor. If more
than one project interests you, approach all the relevant supervisors. It is recommended that you
consult with the supervisor in the academic year before you plan to do the project. This allows
you to do some reading about the project over the summer vacation which in turn, should assist
you in assessing whether the project is suited to your personal tastes. It should also gives you a
head start on the work involved, and can lead to more relaxed semester later.
Supervisors may limit the number of students on a particular project, and will normally operate
on a first come, first served basis. If you are doing the same project as another student you may
wish to discuss your work; this is allowable but you may lose credit if the essays overlap
substantially. You are not allowed to take a course that substantially duplicates the material in
the project so check with the supervisor to ensure that no conflicts arise.
If you wish to do a project on a subject that is not included on the list, you need to find a member
of staff who is willing to supervise you. Lecturers are normally helpful in this regard providing
they have at least some experience in the project field and so are able to offer proper supervision.
If you are unsure about whom you should approach, begin by discussing the matter with your
Personal Tutor or the Project Organizer.
(b) Work and Credit Obtainable. Six project options are possible. First semester projects,
which have the ending -11, and second semester projects, which have the ending -22, are single
projects. Third level single projects carry a credit rating of 10, while fourth level projects carry a
credit rating of 15. Double projects, which carry the ending -00, have respective credit ratings of
20 and 30.
The single project replaces and carries the same credit as a typical single-semester 3rd/4th-year
lecture course and the amount of work put into it should correspond to that involved in offering
such a course for examination. As a rough guide, the material for a 10 credit project should
correspond approximately to the material that would normally be covered in seven lectures, and
the amount of time spent on the project should be 80-100 hours. These figures should be
increased proportionately for higher credit projects.
You may find that certain projects are only offered at limited levels of credit. You should check
with the supervisor that the credit level that you require is available for any given project. The
classification of the projects above is not always relevant here. The projects described as ‘Fourth
Year Only’ are principally so classified because they require level three courses as pre-requisites.
However, the amount of work required for a ‘Third Year’ project can often be varied so that the
project is suitable for either level 3 or level 4.
Many topics are offered as single projects, but with the option of being converted to double
projects at a later stage. This has relevance if you have embarked on a single project in the first
semester. If you feel that it is going well, and wish to take the subject further, then you may
convert to a double project. Evidence needs to be given that you are progressing satisfactorily,
and the change is subject to the agreement of the supervisor and your Personal Tutor. Changeover
is effected by completing a standard change-of-course unit form, and this must be done before the
first day of the January examination period.
(c) Registration. Any third/fourth year student may do a project (or more than one) subject to
acceptance by the supervisor(s) and the agreement of his/her Personal Tutor, and M.Math
students must do thirty credits of project work, normally in their fourth year. Registration for a
project is done in much the same way as registration for a lecture course, namely by entering the
relevant course code on the registration form. However, you must have the explicit agreement of
the project supervisor, and to ensure this you must also get your registration form countersigned
by the supervisor.
(d) Supervision. As the project is essentially your own work, the amount of supervision is kept to
a minimum. The recommendation is about five hours overall. The supervisor’s job is to explain
the meaning and scope of the title, to put you in touch with the relevant literature, and to make
sure that you understand what you are expected to do. You should also check at regular intervals,
either fortnightly or as agreed with your supervisor, that you are making satisfactory progress. A
progress form should be completed half way through the semester as confirmation that progress
has been checked by your supervisor.
Supervisors are asked to read and comment on an early chapter of the report, and also to give
instruction on the overall structure, but not to correct the whole of the report. In the main, this
should be your own effort.
(e) The Essay. The length of a 10 credit project should typically be about 25--30 A4 pages with
12pt type, although reports of an essay-type nature are usually longer, as are reports that contain
numerical output or many diagrams. These figures are intended as guidelines only. Your
supervisor will be aware of the amount of work s/he is expecting, and if the work is described
with fewer pages you would not be penalized. Projects with higher credit allocations are
proportionately longer. The report must be typed, but equations may be written in by hand. (Note
that if your equations are illegible this may prejudice the examiners, who can require you to resubmit the essay). A list of sources that you have used must be given.
(f) Submission. Completed projects (2 copies) must be handed in to the Student Support Office
(Lamb Building) for the attention of the Examinations Officer (Dr F. Coghlan) by the first day of
the relevant examination period (January period for first semester projects, May/June period for
second semester and double projects). It must be accompanied by a completed Project
Supervision Questionnaire (available from the Student Support Office). You must also specify
on a cover sheet whether the project is a single or double project, and give your own name and
the name of your supervisor. No projects will be accepted after that date unless, by virtue of
special circumstances, permission has been obtained from the Year Tutor (Dr. M. McCrudden for
3rd year and Dr. P. Eccles for 4th year). You should also keep a copy of the project for your own
use at the oral examination.
(g) Assessment. Each essay is read by the supervisor and one other examiner. Marking guidelines
are provided by the School for the examiners, and these are reproduced at the end of this
document. The description given there of the five skills categories should be carefully noted since,
effectively, they define the course objectives. Since the examiners award marks for achievement
in each of these specific areas you should make sure that your thesis exhibits suitable competence
in each specific skill. Over-achievement in one category, for instance in Initiative, would not
allow compensation for under-performance in a different category, e.g. in Presentation. It is also
important to realize that the essay should be your own interpretation of the subject; a verbatim
transcription of the sources would attract little credit (and, if the source(s) were not stated, would
amount to plagiarism – see the programme handbook for guidance on this).
After the examiners have read the essay, and during the relevant examination period or the first
week after that period, an oral examination, which typically lasts between 30 -- 60 minutes, is
held at which both examiners and the student are present. This has two objectives: (i) to ensure
that the project is the student’s own work, (ii) to clarify whether the student understands the essay.
The examiners should not ask questions outside the scope of the written work presented. The
examiners then compare their marks and agree on a joint mark.
(h) Completed Projects are retained by the School until after the final exam meeting (i.e.the
meeting that awards the final degree classification; in the longest scenario a first semester third
year project of an M.Math student would be retained for a year and a half). If requested, a
photocopy would be supplied at a nominal charge.
3. ADVERTISEMENT
A comprehensive, detailed guide to mathematical writing (which is well worth reading even if
you are not contemplating a project) is the excellent Handbook of writing for the mathematical
sciences’, 2nd Ed 1998, by Nicholas J. Higham, published by the Society for Industrial and
Applied Mathematics (SIAM), ISBN 0-89871-420-6.
It is available in bookshops and in the JRUL.
Mike Prest (Project Organizer)
Room: Newman 1.21; Tel: 55875; email: mprest@maths.man.ac.uk
Appendix: Guidelines for the Examiners
The following guidelines are provided to the examiners. You may wish to bear these in mind
when preparing and constructing your report.
The different categories are comprised of the following factors:STRUCTURE [10%]
Clear definition of aims and objectives
Selection of material
Organization and ordering of material
PRESENTATION [25%]
Precise and effective communication
Clarity of writing and exposition
ACCURACY [20%]
Correctness of the arguments
Factual accuracy
Mathematical precision
INITIATIVE [20%]
Originality
Independent thinking and individual expression
Critical writing (as opposed to routine copying)
Independent use of library and external sources
UNDERSTANDING [25%]
Appreciation of the meaning, context and significance of the work
The mark distribution indicated is the standard School mark scheme for 3rd and 4th year projects.
If the project is such that this would not reflect the effort required, then an alternative mark
scheme should be assigned. Note, however, that this mark scheme must have been provided to
the student at the start of the project, and cannot be assigned a posteriori. In no case should any
category carry a maximum mark of more than 30%.
When awarding marks, be aware that achievement is required in both quality and quantity. An
insubstantial report should earn reduced marks in all categories.
The purpose of the oral exam is to verify that the project is the student’s own work, and to verify
the student’s understanding of the thesis. The student should not be questioned on topics outside
the scope of the presented document. In the normal course of events the oral exam should only
affect the mark awarded in the ‘understanding’ category. Only in cases of plagiarism should
marks in other categories be affected.
After reading the thesis and conducting the oral examination, the supervisor and co-examiner
should confer together before agreeing on the final marks to be awarded. After completion,
please return this form to the Examinations Officer.