- UCAS course code
- GG13
- UCAS institution code
- M20
Master of Mathematics (MMath)
MMath Mathematics and Statistics
Duration: 4 years
Year of entry: 2025
UCAS course code: GG13 / Institution code: M20
- Key features:
Scholarships available
Accredited course
- Typical A-level offer: A*AA including specific subjects
- Typical contextual A-level offer: A*AB including specific subjects
- Refugee/care-experienced offer: A*BB including specific subjects
- Typical International Baccalaureate offer: 37 points overall with 7,6,6 at HL, including specific requirements
Fees and funding
Fees
Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £34,500 per annum. For general information please see the undergraduate finance pages.
Policy on additional costs
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).
Scholarships/sponsorships
The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.
For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages
Course unit details:
Measure Theory and Ergodic Theory
| Unit code | MATH41021 |
|---|---|
| Credit rating | 15 |
| Unit level | Level 4 |
| Teaching period(s) | Semester 1 |
| Available as a free choice unit? | No |
Overview
In this course we will learn about the abstract theory of measures and the theory of integration that sits on top of it. Then we will cover examples and properties of measure-preserving transformations and the pointwise ergodic theorem before applying ergodic theory to other parts of mathematics.
Pre/co-requisites
| Unit title | Unit code | Requirement type | Description |
|---|---|---|---|
| Metric Spaces | MATH21111 | Pre-Requisite | Compulsory |
| Real Analysis | MATH11112 | Pre-Requisite | Compulsory |
| Real Analysis A | MATH20101 | Pre-Requisite | Compulsory |
| Real Analysis B | MATH20111 | Pre-Requisite | Compulsory |
Aims
The unit aims to: Introduce the abstract theory of integration with respect to a measure, introduce measure-preserving transformations, and apply ergodic theory to other parts of mathematics.
Learning outcomes
- Recognise, deduce and apply properties of sigma-algebras and measures.
- Construct measures using Caratheodory’s extension theorem and the Riesz representation theorem.
- Compute integrals of measurable functions.
- Define Lebesgue spaces and deduce whether a given function belongs to a specific Lebesgue space.
- Determine whether transformations are measure-preserving or ergodic.
- Interpret applications of the pointwise ergodic theorem to measure-preserving transformations.
- Distinguish measure-preserving transformations via their dynamical properties.
- Describe applications of ergodic theory to other areas of mathematics.
Syllabus
- Measures and sigma-algebras [4 lectures]
- Integration and Lebesgue spaces [4 lectures]
- Irrational rotations and Bernoulli shifts [4 lectures]
- Measure-preserving transformations and ergodicity [2 lectures]
- The pointwise ergodic theorem [2 lectures]
- Spectral properties and entropy [4 lectures]
- Applications of ergodic theory [2 lectures]
Teaching and learning methods
In addition to delivery of content in the two lectures per week, feedback will be given on the weekly problem sheet assignments. Tutorials will provide an opportunity for students' work to be discussed and for feedback on their understanding to be given. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
Assessment methods
| Method | Weight |
|---|---|
| Written exam | 50% |
| Set exercise | 50% |
Feedback methods
For weekly problem sheets, feedback will be returned scripts, within a week of submission
Recommended reading
Bartle, R. G. The Elements Of Integration And Lebesgue Measure Wiley 1995
Folland, G. B. Real Analysis: Modern Techniques and Their Applications Wiley 1999
Walters, P. An Introduction to Ergodic Theory Graduate Texts in Mathematics, Springer 1982
Einsiedler, M. and Ward, T. Ergodic Theory with a view towards Number Theory Graduate Texts in Mathematics, Springer 2011
Study hours
| Scheduled activity hours | |
|---|---|
| Lectures | 22 |
| Tutorials | 11 |
| Independent study hours | |
|---|---|
| Independent study | 117 |
Teaching staff
| Staff member | Role |
|---|---|
| Donald Robertson | Unit coordinator |