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

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.

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 and carry out computational studies using the algebraic computing software, Maple.

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, 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.

Algebraic graph theory

This builds on and extends selected themes covered in Graphs, networks and design (MT365), the discontinued module 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 speak to an adviser.

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 as the study materials and activities are accessible via a web browser. You may also be required to perform other tasks, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment. The additional software for these tasks will either be provided or is freely available. For this module you will also need to install software provided by the OU on a disk or USB stick.

A Windows desktop or laptop computer running Windows 7 or later operating system is suitable for this module. You will be required to install Microsoft Windows specific software.

A netbook, tablet, smartphone or Linux computer that supports one of the browsers listed below may be suitable. The screen size should be at least 1024 (H) x 768 (W) pixels. If you intend to use one of these devices please ensure you have access to a suitable desktop or laptop computer in case you are unable to carry out all the module activities on your mobile device.

We recommend a minimum 1 Mbps internet connection and any of the following browsers:

• Internet Explorer 9 and above
• Apple Safari 7 and above
• Google Chrome 31 and above
• Mozilla Firefox 31 and above.

Note: using the latest version for your browser will maximise security when accessing the internet. Using company or library computers may prevent you accessing some internet materials or installing additional software.

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.
• 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 £24.95 - ISBN 9780857290595 History of modern geometry option set book. Please ensure you purchase the December 2010 edition.

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 us 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.

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.

There is no examination. 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
• Fractal geometry
• Analytic number theory I
• Nonlinear ordinary differential equations
• Approximation theory

Future availability

The details given here are for the module that starts in October 2015. 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.