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.

What you will study

The topic ‘History of modern geometry’ 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 topic ‘Advances in approximation theory’ 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 and carry out computational studies using the algebraic computing software, Maple.

The topic ‘Variational methods applied to eigenvalue problems’ extends the theory developed in 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, A(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 A(λ) will be investigated.

The topic ‘Algebraic graph theory’ builds on and extends selected themes covered in Graphs, networks and design (MT365), Groups and geometry (M336) and Coding theory (M836) from an algebraic viewpoint; the selection will depend on your interest. You will learn methods of investigation of graphs and graphical models of other discrete structures by algebraic means, such as spectral analysis, group theory and integer and rational polynomials. This will be achieved by studying selected chapters of the set book Algebraic Graph Theory by C. Godsil and G. Royle (Graduate Texts in Mathematics 207, Springer, 2001) and the accompanying module notes. After acquaintance with the fundamentals of these three algebraic tools, you will select an approach and deal with its application in the study of discrete structures in greater detail. The dissertation will then consist of the application of the selected algebraic approach to analysis of concrete graphs and other structures outside the set book and the module notes.

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.

Entry

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 contact our Student Registration & Enquiry Service.

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.

Regulations

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

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. 

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 computer with internet access to study this module. It includes online activities – you can access using a web browser – and some module software provided on disk.

• If you have purchased a new desktop or laptop computer running Windows since 2008 you should have no problems completing the computer-based activities.
• A netbook, tablet or other mobile device is not suitable for this module – check our Technical requirements section.
• If you have an Apple Mac or Linux computer – please note that you can only use it for this module by running Windows on it using Boot Camp or a similar dual-boot system.

You can also visit the Technical requirements section for further computing information (including details of the support we provide).

Materials to buy

Set books

• 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. Note: Please purchase the 17 December 2010 edition
• Godsil, C & Royle, G Algebraic Graph Theory Springer £36.99 - ISBN 9780387952208 Algebraic Graph Theory option
• Carl de Boor A Practical Guide to Splines - Revised edition Springer £66.99 - ISBN 9780387953663 Advances in Approximation Theory Option

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.

Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

Assessment

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

There is no examination. The tutor-marked assignments take the form of two drafts. You will be expected to submit your drafts online using a special maths eTMA processor, which is used in place of the main 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 drafts electronically but you will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of material in your drafts would take you an unacceptably long time. The final assignment, the dissertation, must be submitted on paper.

Students also studied

Students who studied this course also studied at some time:

• Applied complex variables
• Coding theory
• Nonlinear ordinary differential equations
• Analytic number theory I
• Fractal geometry

Future availability

The details given here are for the module that starts in October 2014. We expect it to be available once a year.

How to register

To register a place on this course return to the top of the page and use the Click to register button.

Distance learning

The Open University is the world's leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you're at home, at work or on the move. As an OU student you'll be supported throughout your studies - your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

For more information about distance learning at the OU read Study explained.