MSc Pure Mathematics and Mathematical Logic

For entry in the academic year beginning September 2027, the tuition fees are as follows:

• MSc (full-time)
  UK students (per annum): £16,500
  International, including EU, students (per annum): £39,200
• MSc (part-time)
  UK students (per annum): £8,250
  International, including EU, students (per annum): £19,600

The fees quoted above are fully inclusive of tuition, administration and computational costs.

Fees for entry are subject to yearly review. The University reserves the right to increase your tuition fee by up to 7% each year for courses lasting more than one year, including to reflect rising costs associated with delivering our educational and wider student experience. Postgraduate fees information .

Always contact the admissions team if you are unsure which fee applies to your qualification award and method of attendance.

International student CAS deposit

Self-funded international applicants are required to pay a deposit of £2500 towards their tuition fees before a confirmation of acceptance for studies (CAS) is issued. Some applicants will be required to pay a higher deposit. More information on tuition fee deposits .

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

Our Manchester Master's Bursaries are aimed at widening access to master's courses by removing barriers to postgraduate education for students from underrepresented groups.

We also welcome the best and brightest international students each year and reward excellence with a number of merit-based scholarships. See our range of master’s scholarships for international students .

If you have completed or are in your final year of an undergraduate degree at The University of Manchester, you may be eligible for a discount of 10% on tuition fees for postgraduate study. Find out if you're eligible and how to apply .

The Martingale Foundation helps provide access to postgraduate mathematics study for UK students facing financial barriers by offering fully-funded MSc and PhD programmes. Find more information on the Martingale Foundation website . Applications for 2027 scholarships open Wednesday 12 August 2026.

For more information on master's tuition fees and studying costs, visit the University of Manchester funding for master's courses website.

Lie algebras are abstract algebraic structures, like groups, but they are inherently linear. One way to interpret this is that any Lie algebra can be realised as a vector space of matrices with the algebra structure inherited from matrix multiplication. This means that we can utilise concrete tools from linear algebra when studying Lie algebras, as well as utilising abstract ideas like those applied to groups. Our focus will be on describing the basic structure theory of Lie algebras.

 

The language we use is the language of weights, which generalises the usual theory of eigenvalues from linear algebra. This is used to introduce the root space decomposition of a Lie algebra. We will compute several explicit examples of this decomposition. To enable you to compute more interesting examples you will learn to use the computer algebra system GAP, which can efficiently perform the necessary multiplication of matrices.

 

To develop the basic properties of the root space decomposition we will need to utilise two key tools: representation theory and the Killing form. Of upmost importance is the representation theory of the Lie algebra sl2, which will be treated in detail.

 

Students are not permitted to take more than one of MATH42112 or MATH62112 for credit or in an undergraduate programme and then a postgraduate programme, as the contents of the courses overlap significantly.

Lie algebras are a fundamental algebraic object arising in mathematics and physics. This unit aims to introduce students to the basic structure of Lie algebras and the key techniques involved in their study.

On successful completion of the course students will be able to: 

1. Provide and identify examples of Lie algebras such as abelian, solvable and semisimple Lie algebras.
2. Analyse the structure of a Lie algebra using the adjoint representation.
3. Construct weight space decompositions of representations and show how weight spaces are used in the representation theory of the Lie algebra sl2.
4. Apply the representation theory of sl2 to the root space decomposition of a Lie algebra.
5. Compute the Killing form and apply it to produce ideals of a Lie algebra.
6. Use the computer algebra system GAP to produce examples of Lie algebras and to demonstrate and explore the theory covered in course.

The course will be taught through three in-person contact hours consisting of lectures, tutorials, and computer lab sessions. In some weeks, as part of their independent study, students will engage with additional material provided through written notes and videos. Students will be encouraged to collaboratively solve problems online through the message board system Piazza.

Coursework: weighted 20%

Examination: weighted 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

The course notes will be self-contained. However, the following two books provide good background reading on the subject.

 

• Karin Erdmann and Mark J. Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathematics Series, Springer-Verlag London Limited, 2006.
• J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer, 1972.

 

For further reading on linear algebra one can consult the following texts:

 

• Sheldon Axler, Linear algebra done right (third edition), Undergraduate Texts in Mathematics, Springer, Cham, 2015.
• Thomas S. Blyth and Edmund F. Robertson, Further linear algebra, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2002.
• Thomas S. Blyth and Edmund F. Robertson, Basic linear algebra, Springer Undergraduate Mathematics Series,Springer-Verlag London, Ltd., London, 1998.