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Dissertation in mathematics

This module enables you to carry out a sustained, guided, independent study of a topic in mathematics. Currently there are six topics to choose from: advances in approximation theory; algebraic graph theory; dynamical functional equations and applications; history of modern geometry; mathematical modelling of interfacial flows and microfluidics and variational methods applied to eigenvalue problems. This list is subject to change. You will be guided by study notes, books, research articles and original sources (or English translations where necessary), which are provided. You'll need to master the appropriate mathematics and ultimately present your work in the form of a final dissertation.


M840 is a compulsory module in our:


Module code


  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
Study level
Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU postgraduate modules correspond to these frameworks.
OU Postgraduate
Study method
Distance learning
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Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

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What you will study

Currently you can choose from six topics for your dissertation. We expect – but can’t at this stage guarantee – to offer an additional topic to students starting this October. We’ll provide further information to all postgraduate mathematics students as soon as possible.

Algebraic graph theory
Algebraic graph theory is a branch of mathematics that studies graphs and other models of discrete structures by a combined power of spectral methods of linear algebra (with basics treated in M208); group theory (covered in part in Further pure mathematics (M303)); and algebra over finite fields (as encountered in Further pure mathematics and Coding theory (M836)). You will need to get acquainted with appropriate mathematical tools by reading selected chapters from the book Algebraic Graph Theory by C. Godsil and G. Royle (Graduate Texts in Mathematics, Springer, 2001). About halfway through the module you will have the opportunity to choose a particular topic of your interest within algebraic graph theory which you will then develop into a dissertation.

Advances in approximation theory
This topic extends the material in Approximation theory (M832) to the study of splines and piecewise polynomials, and their possible application to the approximate solution of differential equations. You will need to master the appropriate mathematics which includes topics such as orthogonal polynomials, Green’s functions and the linearisation of non-linear differential equations. You will also need to carry out computational studies using the algebraic computing software, Maple, as introduced in Advanced mathematical methods (M833).

Dynamical functional equations and applications
Dynamical functional equations arise in the study of critical phenomena in the sciences and in complex social systems such as financial markets. They have been used to model geophysical phenomena (such as volcanic eruptions and earthquakes), financial crashes, stress in materials leading to rupture, and critical behaviour in physical systems, particularly in solid state physics. In this M840 topic you will study the basic theory of linear dynamical functional equations and then study in detail one or two applications, reading the original literature and, if desired, conducting your own explorations theoretically and/or numerically.

History of modern geometry
This topic covers the history of geometry in the nineteenth century. It follows the history of projective geometry and the discovery of non-Euclidean geometry from the 1820s and 1830s. It concentrates on algebraic developments in projective geometry and the work on abstract axiomatic geometry. Differential geometric aspects of non-Euclidean geometry are discussed, as is their influence on Einstein. The module ends with a discussion of geometry and physics, formal geometry and geometry and truth. The module is about geometry, specifically the history of geometry, but it is not a geometry module. ;What will be discussed is the production and reception of ideas, and how this was affected by the social context. The ideas are those of mathematics and the practices those of mathematicians. All the necessary mathematics will be presented but the ideas are to be understood as a historian would treat them and a good standard of English is required

Mathematical modelling of interfacial flows and microfluidics
Many natural and technological processes involve the understanding and modelling of systems in which a viscous liquid is in contact with other phases (e.g. gas and/or solid). Examples of applications include the coating of a substrate by a liquid, transport processes in falling liquid films, fluid flow in porous media, and many problems in the fields of nano- and micro-fluidics, such as inkjet printing or lab-on-a-chip devices. In this topic you will learn the mathematical modelling of interfacial phenomena. Some problems of current interest will be considered, such as for example, the motion of thin liquid films, droplets evaporating on solid surfaces, or fluid flow in confined systems. Basic knowledge of fluid mechanics (e.g. Mathematical Methods and Fluid Mechanics (MST326)) is desirable but not necessary.

Variational methods applied to eigenvalue problems
This topic extends the theory developed in Calculus of variations and advanced calculus (M820) to deal with some types of linear partial differential equations. After establishing functionals for various types of partial differential equations, the Rayleigh-Ritz method will be extended to obtain upper bounds on the smallest frequency of the oscillations of various shaped membranes. The most important part of your dissertation will involve deriving asymptotic estimates of the number, N(λ), of eigenvalues smaller that a given (large number), λ, for a various types of boundary value problems with arbitrary shaped boundaries. A number of theorems that establish increasingly better estimates of N(λ) will be investigated.

Aperiodic order and dynamics
Aperiodic tilings are of interest not only for their aesthetic appeal, but also due to their applications in mathematical crystallography, where they serve as structure models of quasicrystalline materials. In this M840 topic you will explore some concepts of symbolic dynamics, in particular substitution rules on finite alphabets, and the dynamical systems they generate. Their geometric counterparts give rise to inflation tilings. The topic involves reading original literature in the field and offers the option of constructing and exploring tilings.

You will learn

Successful study of this module should enhance your skills in understanding complex mathematical texts, working on open-ended problems and communicating mathematical ideas clearly.

Teaching and assessment

Support from your tutor

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

  • Marking your assignments (TMAs) and providing detailed feedback for you to improve.
  • Guiding you to additional learning resources.
  • Providing individual guidance, whether that’s for general study skills or specific module content.
  • Facilitating online discussions between your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.


The assessment details can be found in the facts box above.

You will be expected to submit your TMAs online using the eTMA system, unless there are some difficulties which prevent you from doing so. In these circumstances, you must negotiate with your tutor to get their agreement to submit your assignment on paper. We strongly recommend that you submit these TMAs electronically.

There is no examination. The final assignment, the dissertation (or EMA), should be typeset and must be submitted electronically.

Course work includes

3 Tutor-marked assignments (TMAs)
End-of-module assessment
No residential school

Future availability

Dissertation in mathematics (M840) starts once a year – in October.

This page describes the module that will start in October 2021.

We expect it to start for the last time in October 2025.


As a student of The Open University, you should be aware of the content of the academic regulations which are available on our Student Policies and Regulations website.

    Entry requirements

    This module is a dissertation and assumes a high level of mathematical maturity.

    To study this module you must:

    • declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention
    • normally have successfully completed at least four other modules in the MSc in Mathematics (F04).

    Provided you have successfully completed at least three other modules in the MSc in Mathematics (F04) programme you may be given permission to take M840 alongside other modules.

    The ‘Advances in approximation theory’ topic builds on the material in Advanced mathematical methods (M833) and Approximation theory (M832). Normally, you should have completed both these modules to be accepted to study this topic.

    To study the 'Variational methods applied to eigenvalue problems' topic, normally, you should have completed Calculus of variations and advanced calculus (M820)

    The number of students on each topic may be limited so you are advised to register early, noting that you may not be offered your first choice of topic.

    All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.

    If you have any doubt about the suitability of the module, please speak to an adviser.


    Start End Fee Register
    02 Oct 2021 Jun 2022 Not yet available

    Registration closes 09/09/21 (places subject to availability)

    October 2021 is the final start date for this course. For more information, see Future availability.

    Future availability

    Dissertation in mathematics (M840) starts once a year – in October.

    This page describes the module that will start in October 2021.

    We expect it to start for the last time in October 2025.

    Additional costs

    Study costs

    There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

    Study weekend

    This module normally includes an optional study weekend. For each day you choose to attend, you must pay an additional charge of around £60 to cover tuition and refreshments during the day. You’ll pay this charge when you book, after you’ve registered on the module. You must also pay for your travel to and from the venue and your accommodation if you need it.

    Ways to pay for this module

    We know there’s a lot to think about when choosing to study, not least how much it’s going to cost and how you can pay.

    That’s why we keep our fees as low as possible and offer a range of flexible payment and funding options, including a postgraduate loan, if you study this module as part of an eligible qualification. To find out more, see Fees and funding.

    Study materials

    What's included

    You'll have access to a module website, which includes:

    • a week-by-week study planner
    • course-specific module materials
    • audio and video content
    • assessment details and submission section
    • online tutorial access.

    You will need

    You will need to buy the set book relevant for your topic (see 'Materials to buy' section).

    Computing requirements

    A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

    Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

    • A desktop or laptop computer with an up-to-date version of Windows
    • The screen must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

    To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).

    Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

    Materials to buy

    Set books

    • de Boor, C. A Practical Guide to Splines (Revised edn) Springer £69.99 - ISBN 9780387953663 Advances in approximation theory option set book.
    • Godsil, C. & Royle, G. Algebraic Graph Theory Springer £36.99 - ISBN 9780387952208 Algebraic graph theory option set book.
    • Gray, J. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century Springer £29.99 - ISBN 9780857290595 History of modern geometry option set book. Please ensure you purchase the December 2010 edition.

    Note: All of these books are Print on Demand, please allow at least 2 weeks for receipt following order.

    If you have a disability

    The material contains small print and diagrams, which may cause problems if you find reading text difficult. Adobe Portable Document Format (PDF) versions of printed material are available. Some Adobe PDF components may not be available or fully accessible using a screen reader and mathematical  materials may be particularly difficult to read in this way. Alternative formats of the study materials may be available in the future. 

    If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Visit our Disability support website to find more about what we offer.