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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 four topics to choose from: history of modern geometry; advances in approximation theory; variational methods applied to eigenvalue problems; algebraic graph theory. 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.

Qualifications

M840 is a compulsory module in our:

MSc in Mathematics (F04)

Some postgraduate qualifications allow study to be chosen from other subject areas. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.

Postgraduate Loans

If you study this module as part of an eligible qualification, you may also be eligible for a Postgraduate Loan available from Student Finance England. For more information, see Fees and funding.

Module

Module code

M840

Credits

Study level

OU	Postgraduate
SCQF	11
FHEQ	7

Study method

Distance learning
Find out more in Why the OU?

Module cost

See Module registration

Entry requirements

Find out more about entry requirements.

Module details Entry requirements Module registration Study materials

What you will study

Currently you can choose from four topics for your dissertation.

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

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).

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.

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.

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

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance.

Assessment

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 but you will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of material in your TMAs would take you an unacceptably long time.

There is no examination. The final assignment, the dissertation, must be submitted on paper.

Course work includes

3 Tutor-marked assignments (TMAs)

End-of-module assessment

No residential school

Course satisfaction survey

See the satisfaction survey results for this course.

Future availability

The details given here are for the module that starts in October 2017. It starts once a year – in October.

Regulations

As a student of The Open University, you should be aware of the content of the academic regulations which are available on our Essential Documents 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
have successfully completed at least four other modules in the MSc in Mathematics (F04).

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.

Register

Start	End	England fee	Register
07 Oct 2017	Jun 2018	£1000.00	New to OU? Registration closed but there may be places available. Ring +44 (0)300 303 5303. Already an OU Student? You have until 21/09/17 to register. Register
This module is expected to start for the last time in October 2021.

Future availability

The details given here are for the module that starts in October 2017. It starts once a year – in October.

Additional costs

Study costs

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

Study weekend

This module includes an optional study weekend. The cost of this is around £50 per day and includes tuition and refreshments during the day. This cost is payable after your registration as part of confirming attendance of the study weekend. You will be responsible for the cost of travel to the venue and for securing appropriate accommodation near the venue if required.

Ways to pay

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. To find out more see Fees and funding.

Study materials

What's included

Module notes, other printed materials.

You will need

The module requires you to carry out research, and study materials are delivered via the web, so a computer or access to one is essential. It is also possible that the University may require you to submit your dissertation in electronic and not manuscript form.

Computing requirements

You will need a device with internet access to study this module as a web browser is used to access learning materials and activities. Any other computer-based activities you will need to carry out, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment, are specified in the module materials. If any additional software is needed for these tasks it will either be provided or is freely available. This module requires the installation of software from a hardware device e.g. DVD drive or USB stick. You may need administrative privileges to install software required for this module. Windows 10 S is not suitable as it restricts software installation to software available in the Windows Application Store.

A Windows desktop or laptop running Windows 7 or later operating system is suitable for this module. This module requires installation of Microsoft Windows specific software.

Some software will not run on Mac OS X, Linux, iOS or Android devices.

A netbook, tablet, phone, Mac or Linux computer that supports one of the browsers below may be suitable for some activities. However, these devices may not be suitable for one or more activities within this module. If you intend to use one of these devices please ensure you have access to another suitable desktop or laptop device that uses the Windows operating system so that you can carry out all activities on your mobile device. You will need a broadband internet connection to complete this module. Better video performance is available with higher connection speeds.

Recent versions of the following browsers for carrying out web-based activities:

Safari
Chrome
Firefox
Edge

Or Internet Explorer 9 and above.
Using a browser upgraded to the latest version will maximise security when accessing the internet.

Using company or library computers may prevent you accessing some internet materials or installing additional software.

To be able to talk and listen in our online discussions you will need both a microphone and speakers/headphones.

Devices with small screens may make it difficult to view the material provided and carry out the activities. However, a device that has a resolution of at least 1024 pixels horizontally and also at least 768 pixels vertically should be adequate.

See our Skills for OU study website for further information about computing skills for study and educational deals for buying Microsoft Office software.

Materials to buy

Set books

de Boor, C. A Practical Guide to Splines (Revised edn) Springer £66.99 - ISBN 9780387953663 Advances in approximation theory option set book. This book is Print on Demand, please allow at least 2 weeks for receipt following order.
Godsil, C. & Royle, G. Algebraic Graph Theory Springer £36.99 - ISBN 9780387952208 Algebraic graph theory option set book. This book is Print on Demand, please allow at least 2 weeks for receipt following order.
Gray, J. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century Springer £24.95 - ISBN 9780857290595 History of modern geometry option set book. Please ensure you purchase the December 2010 edition. This book is 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. Written transcripts of any audio components and 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. Find out more about our services for disabled students.

Dissertation in mathematics

Qualifications

Postgraduate Loans

Module

What you will study

History of modern geometry

Advances in approximation theory

Variational methods applied to eigenvalue problems

Algebraic graph theory

You will learn

Teaching and assessment

Support from your tutor

Assessment

Course work includes

Course satisfaction survey

Future availability

Regulations

Entry requirements

Register

Future availability

Additional costs

Study costs

Study weekend

Ways to pay

Study materials

What's included

You will need

Computing requirements

Materials to buy

Set books

If you have a disability

The Open University

OU Community

OU Students Community

Support

Study

Policy

What you will study

History of modern geometry

Advances in approximation theory

Variational methods applied to eigenvalue problems

Algebraic graph theory

You will learn

Teaching and assessment

Support from your tutor

Assessment

Course work includes

Course satisfaction survey

Future availability

Regulations

Entry requirements

Register

Future availability

Additional costs

Study costs

Study weekend

Ways to pay

Study materials

What's included

You will need

Computing requirements

Materials to buy

Set books

If you have a disability

Where do you live?