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Mathematical Sciences

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What you'll need

As part of the application process, you'll need to submit:

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  • Degree certificates
  • Personal statement outlining your learning ambitions

Our application process

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  • We'll email you when a decision has been made
  • If you've been made an offer, you can then accept or decline it using the Postgraduate Application Tracker.

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Master of Science

A Master of Science (MSc) is a master’s degree awarded for a postgraduate programme in the sciences.

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Course overview

This programme offers you the opportunity to specialise in a broad range of areas across pure and applied mathematics and theoretical physics.


Mathematical Sciences at Liverpool provides a centre for world class research and teaching across many areas and at the leading edge of the subject. Our teaching was evaluated as ‘excellent’ by the Quality Assurance Agency in its most recent survey and awarded 23 points out of a possible 24.

We are proud of our record on teaching quality, with five members of the department having received the prestigious Sir Alastair Pilkington Award for Teaching. We care about each student and you will find the staff friendly and approachable. We provide high quality supervision, teaching and IT support and you will benefit from the friendly and supportive atmosphere in the department.

This programme is available to both full-and part-time students, and offers a very wide range of options.

Who is this course for?

The programme is suitable for mathematics graduates who are seeking the opportunity to specialise and develop in a broad range of areas across pure and applied mathematics and theoretical physics.

What you'll learn

  • Advanced mathematical methods
  • Algebraic geometry
  • Dynamical systems
  • Fundamental particle physics
  • Mathematical biology
  • Singularity theory
  • Solid mechanics
  • Stochastic analysis
  • String theory
  • Wave propagation and scattering.

Course content

Discover what you'll learn, what you'll study, and how you'll be taught and assessed.

Studying this course part-time

International students may be able to study this course on a part-time basis but this is dependent on visa regulations. Please visit the Government website for more information about student visas.

If you're able to study part-time, you'll study the same modules as the full-time master's degree over a longer period, usually 24 months. You can make studying work for you by arranging your personal schedule around lectures and seminars which take place during the day. After you complete all the taught modules, you will complete your final dissertation or project and will celebrate your achievements at graduation the following term.

Studying part-time means you can study alongside work or any other life commitments. You will study the same modules as the full-time master's degree over a longer period, usually 24 months. You can make studying work for you by arranging your personal schedule around lectures and seminars which take place during the day. After you complete all the taught modules, you will complete your final dissertation or project and will celebrate your achievements at graduation the following term.

Semester one

Compulsory modules


Credits: 15 / Semester: semester 1

In this module you will learn to use the typesetting system LaTeX and a common mathematical software package (such as Maple or Matlab).  In the second half of the semester you will undertake a project where you will learn a new topic in Mathematics and will use you skills in mathematical programming and typesetting to investigate the topic and produce your project report.

Optional modules


Credits: 15 / Semester: semester 1

An introduction to the topology of manifolds, emphasising the role of homology as an invariant and the role of Morse theory as a visualising and calculational tool.


Credits: 15 / Semester: semester 1

​This module is concerned with linear partial differential equations (PDEs) that arise in mathematical physics, and advanced methods for solving them. There is a particular focus on methods that use singular solutions, which satisfy the PDE at all but a finite number of points. We will study three canonical PDEs: Laplace’s equation, the heat equation and the wave equation. In each case we will see how the solution to complicated problems can be built up from solutions to simpler problems, typically in the form of an infinite series or an integral.​


Credits: 15 / Semester: semester 1

Quantum Field Theory provides the mathematical language of modern theoretical particle and condensed matter physics. Historically Quantum Field Theory was developed to be the consistent theory of quantum mechanics and special relativity. The mathematical techniques developed in this course form the theoretical basis for varied fields such as high energy particle physics or superconductivity.

Singularity Theory of Differentiable Mappings (MATH455)

Credits: 15 / Semester: semester 1

​This module is an introduction to the calculus side of Singularity Theory.  Theory of singularities of differentiable maps is a far-reaching generalisation of the study of functions at maxima and minima. It has numerous applications in mathematics, the natural sciences and technology (as in the so-called theory of bifurcations and catastrophes). This module concentrates on the theory and stability of smooth maps, and classification techniques for critical points of smooth functions. Although not pre-requisites, any of MATH244 (Linear algebra and geometry), MATH248 (Geometry of curves), MATH343 (Group theory), MATH349 (Differential geometry) and MATH443 (Curves and singularities) would be helpful. MATH410 (Manifolds, homology and Morse theory) is a follow-up module but may be taken simultaneously.​

Riemann Surfaces (MATH445)

Credits: 15 / Semester: semester 1

This module will introduce students to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.


Credits: 15 / Semester: semester 1

Ordinary and partial differential equations (ODEs and PDEs) are crucial to many areas of science, engineering and finance. This module addresses methods for, or related to, their solution. It starts with a section on inhomogeneous linear second-order ODEs which are often required for the solution of higher-level problems. We then generalize basic calculus by considering the optimization of functionals, e.g., integrals involving an unknown function and its derivatives, which leads to a wide variety of ODEs and PDEs. After those systems of two linear first-order PDEs and second-order PDES are classified and reduced to ODEs where possible. In certain cases, e.g., `elliptic’ PDEs like the Laplace equation, such a reduction is impossible. The last third of the module is devoted to two approaches, conformal mappings and Fourier transforms, which can be used to obtain solutions of the Laplace equation and other irreducible PDEs.


Credits: 15 / Semester: semester 1

​This module provides an introduction to basic concepts and principles of continuum mechanics. Cartesian tensors are introduced at the beginning of the module, bringing simplicity and versatility to the analysis. The module places emphasis on the importance of conservation laws in integral form, and on the fundamental role integral conservation laws play in the derivation of partial differential equations used to model different physical phenomena in problems of solid and fluid mechanics.


Credits: 15 / Semester: semester 1

The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time. The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

Relativity (MATH326)

Credits: 15 / Semester: semester 1

Einstein’s theories of special and general relativity have introduced a new concept of space and time, which underlies modern particle physics, astrophysics and cosmology. It makes use of, and has stimulated the development of modern differential geometry. This module develops the required mathematics (tensors, differential geometry) together with applications of the theory to particle physics, black holes and cosmology. It is an essential part of a programme in theoretical physics.


Credits: 15 / Semester: semester 1

The module provides an introduction to the modern theory of finite non-commutative groups. Group Theory is one of the central areas of Pure Mathematics. Being part of Algebra, it has innumerable applications in Geometry, Number Theory, Combinatorics and Analysis, but also plays a very important role in Theoretical Physics, Mechanics and Chemistry. The module starts with basic definitions and some well-known examples (the symmetric group of permutations and the groups of congruence classes modulo an integer) and builds up to some very interesting and non-trivial constructions, such as the semi-direct product, which makes it possible to construct more complicated groups from simpler ones. In the final part of the course, the Sylow theory and its applications to the classification of groups are considered.


Credits: 15 / Semester: semester 1

This module studies discrete-time Markov chains, as well as introducing the most basic continuous-time processes. The basic theory for these stochastic processes is considered in detail. This includes the Chapman Kolmogorov equation, communication of states, periodicity, recurrence and transience properties, asymptotic behaviour, limiting and stationary distributions, and an introduction to Poisson processes. Applications in different areas, in particular in insurance, are considered.

Linear Statistical Models (MATH363)

Credits: 15 / Semester: semester 1

This module extends earlier work on linear regression and analysis of variance, and then goes beyond these to generalised linear models. The module emphasises applications of statistical methods. Statistical software is used throughout as familiarity with its use is a valuable skill for those interested in a career in a statistical field.


Credits: 15 / Semester: semester 1

Number theory begins with, and is mainly concerned with, the study of the integers. Because of the fundamental role which integers play in mathematics, many of the greatest mathematicians, from antiquity to the modern day, have made contributions to number theory. In this module you will study results due to Euclid, Euler, Gauss, Riemann, and other greats: you will also see many results from the last 10 or 20 years.Several of the topics you will study will be familiar from MATH142 (Numbers, groups, and codes). We will go into them in greater depth, and the module will be self-contained from the point of view of number theory. However, some background in group theory (no more than is in MATH142) will be assumed.


Credits: 15 / Semester: semester 1

Differential geometry studies distances and curvatures on manifolds through differentiation and integration. This module introduces the methods of differential geometry on the concrete examples of curves and surfaces in 3-dimensional Euclidean space. The module MATH248 (Geometry of curves) develops methods of differential geometry on examples of plane curves. This material will be discussed in the first weeks of the course, but previous familiarity with these methods is helpful. Students following a pathway in theoretical physics might find this module interesting as it discusses a different aspect of differential geometry, and might take it together with MATH326 (Relativity). MATH410 (Manifolds, homology and Morse theory) and MATH446 (Lie groups and Lie algebras).​


Credits: 15 / Semester: semester 1

​Stochastic processes are ways of quantifying the dynamic relationships of sequences of random events. Stochastic models play an important role in elucidating many areas of the natural and engineering sciences. They can be used to analyse the variability inherent in biological and medical processes, to deal with uncertainties affecting managerial decisions and with the complexities of psychological and social interactions, and to provide new perspectives, methodology, models and intuition to aid in other mathematical and statistical studies. This module is intended as a beginning course in introducing continuous-time stochastic processes for students familiar with elementary probability.  The objectives are: (1) to introduce students to the standard concepts and methods of stochastic modelling; (2) to illustrate the rich diversity of applications of stochastic processes in the science; and (3) to provide exercises in the applications of simple stochastic analysis to appropriate problems.

Mathematical Biology (MATH335)

Credits: 15 / Semester: semester 1

In the current age of big data, mathematics is becoming indispensable in order for us to make sense of experimental results and in order to gain a deeper understanding into mechanisms of complex biological systems. Mathematical models can provide insights that cannot be gained through experimental work alone. This module will focus on teaching students how to construct and analyse models for a wide range of biological systems. Mathematical approaches covered will be widely applicable.

Advanced topics in mathematical biology (MATH426)

Credits: 15 / Semester: semester 1

Mathematics can be applied to a wide range of biological problems, many of which involve studying how systems change in space and time. In this module, an example selection of mathematical applications will be presented chosen from staff research interests involving developmental biology, epidemic dynamics & biological fluid dynamics.


Credits: 15 / Semester: semester 1

Groups appear everywhere in mathematics where symmetry is involved. Representation Theory analyses groups by using tools from linear algebra. This course studies the representation theory of finite groups, a beautiful theory that starting with a few basic facts from group theory and linear algebra quickly leads to beautiful and powerful theorems unlocking major insight into the structure of finite groups and their representations. Representation theory also has spectacular applications to Chemistry, making it possible to calculate the colours of the world!

Programme details and modules listed are illustrative only and subject to change.

Our curriculum

The Liverpool Curriculum framework sets out our distinctive approach to education. Our teaching staff support our students to develop academic knowledge, skills, and understanding alongside our graduate attributes:

  • Digital fluency
  • Confidence
  • Global citizenship

Our curriculum is characterised by the three Liverpool Hallmarks:

  • Research-connected teaching
  • Active learning
  • Authentic assessment

All this is underpinned by our core value of inclusivity and commitment to providing a curriculum that is accessible to all students.

Course options

Studying with us means you can tailor your degree to suit you. Here's what is available on this course.

Postgraduate Certificate

If you enrol on a master’s degree and complete 60 credits you may be able to exit the programme early and earn a Postgraduate Certificate (PGCert) for the work you have completed. This usually amounts to one semester when completed full-time.

Postgraduate Diploma

If you enrol on a master’s degree and have completed 120 credits you may be able to exit the programme early and earn a Postgraduate Diploma (PGDip) for the work completed. This usually amounts to two full semesters when completed full-time, without having to complete the final dissertation module.

Your experience

Virtual tour

Supporting your learning

From arrival to alumni, we’re with you all the way:

Careers and employability

A mathematically-based degree opens up a wide range of career opportunities, including some of the most lucrative professions.

Career planning

Three career coaches standing outside the Careers Studio

Our campus Career Studio is a space for students and graduates to drop into and talk to a career coach. Career coaches are highly trained to help no matter what stage you are at in your career planning. You can access support to find and apply for full-time and part-time roles, placements, internships and graduate schemes. You will also find the help you need if you have a start-up idea or want to create a business plan. You can explore the world of work, prepare for job interviews, and access careers events and workshops. The Career Studio is open Monday to Friday from 10am-5pm, simply drop in at a time that works for you.

From education to employment

Two graduates in postgraduate robes.

We develop our programmes with employers in mind. You will be supported to enhance your long-term employment prospects as you learn. We do this by exposing you to professionals, a variety of sectors and supporting you to work collaboratively with others to develop transferable skills. You are equipped with a clearer view of what to focus on in your area of interest, and to reflect on your studies. Our digital employability tools give you a tech-enhanced curriculum experience and make it easy for you to prepare for the world of work. You can use tools like the Handshake platform to connect with employers and message the Career Studio 24/7.

Networking events

Postgraduate students hold a discussion while sat round a table in in the Liverpool Guild of Students.

You can start building good professional networks by attending events and employability activities. Our events are designed to develop your skills and expose you to many different employers, as well as to help you make contacts in your field. We help you improve your confidence when speaking to employers and give you access to unique opportunities. Our networking events also boost your understanding of the competencies and skills that employers are looking for in their recruitment process, giving you a competitive edge.

Your future

Recent graduates have moved into fast track teacher programmes, jobs in finance (actuarial, banking, insurance), software development, drugs testing and defence work.

The MSc programme is a natural route into doctoral study in mathematics and related fields. Some of our PhD students move on to postdoctoral positions, academic teaching jobs and roles in research institutes, both in the UK and elsewhere.

87.5% of mathematical sciences graduates go on to work or further study within 15 months of graduation.

Discover Uni, 2018-19.

Fees and funding

Your tuition fees, funding your studies, and other costs to consider.

Tuition fees

UK fees (applies to Channel Islands, Isle of Man and Republic of Ireland)
Full-time place, per year £12,400
Part-time place, per year £6,200
International fees
Full-time place, per year £26,400
Part-time place, per year £13,200
Fees stated are for the 2024-25 academic year.

Tuition fees cover the cost of your teaching and assessment, operating facilities such as libraries, IT equipment, and access to academic and personal support.

If you're a UK national, or have settled status in the UK, you may be eligible to apply for a Postgraduate Loan worth up to £12,167 to help with course fees and living costs. Learn more about paying for your studies..

Additional costs

We understand that budgeting for your time at university is important, and we want to make sure you understand any course-related costs that are not covered by your tuition fee. This could include buying a laptop, books, or stationery.

Find out more about the additional study costs that may apply to this course.

Additional study costs

We understand that budgeting for your time at university is important, and we want to make sure you understand any course-related costs that are not covered by your tuition fee. This could include buying a laptop, books, or stationery.

Find out more about additional study costs.

Scholarships and bursaries

We offer a range of scholarships and bursaries that could help pay your tuition and living expenses.

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Postgraduate Global Advancement Scholarship

  • International students

If you’re a new international student starting this course with us from September 2024, you could be eligible to receive a discount of £5,000 off your tuition fees.

Graduate Loyalty Advancement Scholarship

  • Home and international students
  • University of Liverpool current students and alumni only

Completed your undergraduate degree, or studied as an undergraduate exchange student, at the University of Liverpool?

You could get a loyalty discount of up to £2,500 off the tuition fees for this course from September 2024 entry.

  • £1,500 tuition fee discount for eligible UK University of Liverpool graduates
  • £2,500 tuition fee discount for eligible international University of Liverpool graduates.

ANID Chile Scholarship

  • International students
  • Chile

Postgraduate taught and research students from Chile are eligible for this scholarship.

Chevening Scholarships

  • International students

The University, in partnership with Chevening, is delighted to offer this generous scholarship to students who are studying a master’s programme and who have future leadership potential. Please note that there is a fee cap applied to MBA programmes that requires applicants to cover any additional tuition costs over £18,000. You will still receive all additional allowances.


  • International students
  • Mexico

The University of Liverpool has an agreement with CONACYT to support postgraduate taught and research students from Mexico.


  • International students
  • Mexico

20% reduction in tuition fees for postgraduate taught and research programmes. Must be Mexico national.

Fulbright Scholarship

  • International students
  • University of Liverpool alumni only
  • United States

One scholarship is available for a master’s student from the US and another is available for a postgraduate research student to undertake a three to six month research stay from the US

FUNED Awards

  • International students
  • Mexico

Up to ten awards are available for Masters or Research students from Mexico in receipt of FUNED loans. The award gives students a 20% reduction in fees for all applications received.

Graduate Association Hong Kong & Tung Postgraduate Scholarships

  • International students
  • China
  • Hong Kong

The University is able to offer competitive scholarships for both postgraduate taught master’s and research programmes.

HRM Princess Sirindhorn University of Liverpool Scholarship (Thailand)

  • International students
  • Thailand

The University is able to offer one award to a new postgraduate taught master’s student from Thailand.

The scholarship is open to all subjects offered as a one-year taught master’s programme.  However, priority will be given to those students who wish to study in a subject area associated with HRH Princess Sirindhorn such as science, IT, medicine, the arts, geography, history and languages.

JuventudEsGto Scholarship

  • International students
  • Mexico

Residents of State of Guanajuato, Mexico, wishing to study at postgraduate taught and research levels are eligible for this scholarship.

Marshall Scholarship

  • International students
  • University of Liverpool alumni only
  • United States

One scholarship is available for a master’s student from the US to cover the cost of tuition fees. Another, to the value of £20,000, is available for Doctoral study visit:

Turkish Ministry of Education Scholarship

  • International students
  • Turkey

Postgraduate taught and research students from Turkey are eligible for this scholarship, see the Turkish Ministry of Education website for more information.

University of Liverpool Humanitarian Scholarships for Master’s Programmes

  • International students

The three awards available cover full tuition fees, visas and support for accommodation and living expenses.

This scholarship is open to support people who have recognised status as either refugees or are under humanitarian protection under the 1951 Refugee Convention. This status must be held outside of the UK.

The scholarship is open for all postgraduate-taught programmes, excluding medicine, dentistry, veterinary and nursing.

University of Liverpool International College Excellence Scholarship

  • International students

The University of Liverpool will award five University of Liverpool International College students, who achieve the highest academic excellence (minimum 75%) in their UoLIC Pre-Master’s programme, the prestigious UoLIC Excellence scholarship.

University of Liverpool International College Impact Progression Scholarships

  • International students

University of Liverpool International College recipients of the Kaplan awards will receive the £3,000 Progression Impact Scholarship, deducted from first-year tuition fees, on successful progression to their UoL degree programme.

To be eligible for our Impact Progression Scholarships, students must apply for one of the Kaplan Impact Scholarships demonstrating their commitment to making an impact across issues of importance to the University and Kaplan. Themes include:

Women in STEM
Career Focus (Employability)

Vice-Chancellor’s International Attainment Scholarship for China

  • International students
  • China

Details are:
1 (one) Full scholarship
2 (two) £10,000 scholarships
5 (five) £5,000 scholarships
10 (ten) £1,000 scholarships
All scholarships will be awarded after the formal registration at the University and will take the form of a fee waiver.

Please note: This scholarship cannot be combined with any other scholarships or bursaries provided by the University.

Entry requirements

The qualifications and exam results you'll need to apply for this course.

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Your qualification Requirements

About our typical entry requirements

Postgraduate entry requirements

You will normally need a 2:1 honours degree or above, or equivalent. This degree should be in mathematics.

We also encourage applications from those with degrees in subjects where mathematics is a major component, for example physics and engineering. In these circumstances, we may look for higher marks to offset the lower number of credits earned for mathematics modules. Each application will be assessed on its own merits.

International qualifications

If you hold a bachelor’s degree or equivalent, but don’t meet our entry requirements, a Pre-Master’s can help you gain a place. This specialist preparation course for postgraduate study is offered on campus at the University of Liverpool International College, in partnership with Kaplan International Pathways. Although there’s no direct Pre-Master’s route to this MSc, completing a Pre-Master’s pathway can guarantee you a place on many other postgraduate courses at The University of Liverpool.

English language requirements

You'll need to demonstrate competence in the use of English language, unless you’re from a majority English speaking country.

We accept a variety of international language tests and country-specific qualifications.

International applicants who do not meet the minimum required standard of English language can complete one of our Pre-Sessional English courses to achieve the required level.

English language qualification Requirements
IELTS 6.5 overall, with no component below 6.0
TOEFL iBT 88 overall, with minimum scores of listening 19, writing 19, reading 19 and speaking 20
Duolingo English Test 120 overall, with no component below 105
Pearson PTE Academic 61 overall, with no component below 59
LanguageCert Academic 70 overall, with no skill below 65
PSI Skills for English B2 Pass with Merit in all bands
INDIA Standard XII National Curriculum (CBSE/ISC) - 75% and above in English. Accepted State Boards - 80% and above in English.
WAEC C6 or above


Do you need to complete a Pre-Sessional English course to meet the English language requirements for this course?

The length of Pre-Sessional English course you’ll need to take depends on your current level of English language ability.

Find out the length of Pre-Sessional English course you may require for this degree.

Pre-sessional English

If you don’t meet our English language requirements, we can use your most recent IELTS score, or the equivalent score in selected other English language tests, to determine the length of Pre-Sessional English course you require.

Use the table below to check the course length you're likely to require for your current English language ability and see whether the course is available on campus or online.

Your most recent IELTS score Pre-Sessional English course length On campus or online
6.0 overall, with no component below 6.0 6 weeks On campus
6.0 overall, with no component below 5.5 10 weeks On campus and online options available
6.0 overall, with no more than one component below 5.5, and no component below 5.0 12 weeks On campus and online options available
5.5 overall, with no more than one component below 5.5, and no component below 5.0 20 weeks On campus
5.0 overall, with no more than one component below 5.0, and no component below 4.5 30 weeks On campus
4.5 overall, with no more than one component below 4.5, and no component below 4.0 40 weeks On campus

If you’ve completed an alternative English language test to IELTS, we may be able to use this to assess your English language ability and determine the Pre-Sessional English course length you require.

Please see our guide to Pre-Sessional English entry requirements for IELTS 6.5, with no component below 6.0, for further details.

About our entry requirements

Our entry requirements may change from time to time both according to national application trends and the availability of places at Liverpool for particular courses. We review our requirements before the start of the new application cycle each year and publish any changes on our website so that applicants are aware of our typical entry requirements before they submit their application.

We believe in treating applicants as individuals, and in making offers that are appropriate to their personal circumstances and background. Therefore the offer any individual applicant receives may differ slightly from the typical offer quoted on the website.

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Discover more about the city and University.

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Why Liverpool?

Liverpool bursts with diversity and creativity which makes it ideal for you to undertake your postgraduate studies and access various opportunities for you and your family.

Accommodation Postgraduate students walking through the campus.


To fully immerse yourself in the university experience living in halls will keep you close to campus where you can always meet new people. Find your home away from home.

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Fees and Finance

Discover what expenses are covered by the cost of your tuition fees and other finance-related information you may need regarding your studies at Liverpool.

Changes to Mathematical Sciences MSc

See what updates we've made to this course since it was published. We document changes to information such as course content, entry requirements and how you'll be taught.

23 March 2023: New postgraduate taught course pages

New course pages launched.