MSc Pure Mathematics

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).

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.

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 the 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 .

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

The basic method of Algebraic Topology is to associate an algebraic object to each
topological space so that homeomorphic spaces have isomorphic algebraic objects. These
algebraic objects are then topological invariants and so can be used to distinguish topological
spaces. The fundamental group introduced in MATH31051/2 is such an algebraic object. This
course unit introduces higher homotopy groups and homology groups.
In order to provide motivation for homology theory, the first part of the course unit discusses
the classification theorem of closed surfaces using a combinatorial method of proof via
simplicial surfaces. Roughly speaking the theorem states that every closed surface can be
built up out of the sphere, the torus and the projective plane. Non-homeomorphic surfaces
are distinguished using two key topological invariants: the Euler characteristic and
orientability.
The second part then generalizes the idea of a simplicial surface to the idea of a simplicial
complex. This enables us to introduce a wide class of topological spaces called polyhedra which can be represented by a simplicial complex. This representation can then be used to define the homology groups of the polyhedron. The Euler characteristic and orientability type of a surface are determined by the homology groups. These powerful invariants have many attractive applications.
The supplementary reading introduces higher homotopy groups, categories and functions
and the axiomatic approach to homology theory.

 

Students are not permitted to take MATH41071 and MATH61071 for credit in an undergraduate programme and then a postgraduate programme.

The unit aims to:
develop further the concepts introduced in the Level 3 introductory course on Topology, emphasizing topics coming from Algebraic Topology. The course will introduce students to the basic concepts of homotopy and homology theory and explain the need for different algebraic invariants of topological spaces focussing on the classification theorem for closed surfaces

 

On successful completion of this course unit students will be able to:

• recognize when a collection of triangles in a Euclidean space forms a simplicial surface
   
• identify the underlying space of a simplicial surface by reducing a representing symbol to standard form
   
• state the basic definitions (including homotopy, homology groups, simplicial complexes) and main theorems from the course, and apply them in typical examples
   
• recognize whether two topological spaces are homotopic in simple cases
   
• calculate the homology groups of naturally occurring topological spaces
   
• recall proofs of basic properties of homology groups and apply these properties as topological invariants in examples
   
• determine the Euler characteristic and orientability of a simplicial surface from its homology groups
   
• calculate the homology groups of standard spaces using a variety of methods
   

Syllabus:
• Topological surfaces: definition and basic examples; the connected sum of two
surfaces; the classification theorem for compact surfaces; handles and cross-caps. [3]
• Simplicial surfaces: definition, triangulation of a topological surface and the statement
of the triangulation theorem for compact surfaces; representing the underlying space of
a simplicial surface by a symbol; equivalent symbols and the statement and proof of the
classification theorem for surface symbols; geometrical interpretation of the
classification theorem. [3]
• Topological invariants of surfaces: definition of the Euler characteristic and
orientability of a simplicial surface; statement of topological invariance; using these
invariants to recognize the underlying space of a simplicial surface. [2]
• Homotopy and homotopy equivalence: definition and simple examples, deformation retractions. [2]
• Simplicial complexes: definitions and examples; the underlying space,
polyhedral. [3]
• Simplicial homology groups: definitions and examples. [3]
• Simplicial approximation and topological invariance: statement of the Simplicial
Approximation Theorem and how this may be used to prove the topological invariance of
homology groups. [3]
• Applications: Brouwer Fixed Point Theorem, Borsuk-Ulam Theorem, Lefschetz Fixed
Point Theorem. [2]
 

22 hours of lectures plus 11 hours of tutorials. 
Feedback is available at the weekly tutorials as well as the mid-term coursework. 
 

Mid-semester coursework: weighting 20%,

End of semester examination: three hours weighting 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 following books contains most of the material in the course and much more.

• M.A. Armstrong, Basic Topology, Springer 1997 (classification of surfaces, simplicial complexes, homology)
• A. Hatcher, Algebraic Topology. (free download) (simplicial complexes, delta-complexes, homology, higher homotopy groups, categories and functors)
• W.S. Massey, Algebraic Topology: an Introduction, Springer 1990 (classification of surfaces)