- Entry requirements: Related 2:1 degree (or equivalent)
- Full-time: 12 months
- Part-time: 24 months
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.
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.
Discover what you'll learn, what you'll study, and how you'll be taught and assessed.
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.
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.
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.
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.
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.
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.
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.
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.
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. Some knowledge of Vector Calculus (e.g. MATH225 Vector calculus with applications in fluid mechanics) is useful.
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.
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. Together with MATH325 (Quantum mechanics) it covers the basics of modern theoretical physics. Possible follow up modules in theoretical physics are MATH423 (Introduction to string theory), MATH425 (Quantum field theory) and MATH431 (Introduction to modern particle theory). MATH326 is essential for students who consider doing a project on black holes or cosmology. Students following a pure mathematics or applied mathematics pathway might be interested in MATH326 because of its applications of differential geometry, and take it together with MATH349 (Differential geometry).
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.
To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for dynamic events occurring over time. To familiarise students with the usual techniques in the area of probability modelling.
This module follows on directly from MATH263 (Statistical Theory and Methods I), extending the work there on linear regression and analysis of variance, and then going beyond these to generalised linear models. The module emphasises applications of statistical methods, while the companion module MATH361 (Theory of Statistical Inference) focuses on more theoretical aspects. Statistical software is used throughout as familiarity with its use is a valuable skill for those interested in a career in a statistical field. It is helpful, though not essential, to have taken MATH264 (Statistical Theory and Methods II).
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.
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).
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. The module is complementary to MATH362 (Applied probability), in which discrete-time processes are studied. Those who plan to go on to MSc study in financial mathematics will find this module a very useful preparation for modules such as MATH481 (Interest rate theory), MATH482 (Stochastic modelling in finance), MATH483 (Stochastic analysis and its applications) and MATH484 (Advanced numerical analysis for financial mathematics).
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.
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.
REPRESENTATION THEORY OF FINITE GROUPS
This module provides an introduction to topics in Analytic Number Theory, including the worst and average case behaviour of arithmetic functions, properties of the Riemann zeta function, and the distribution of prime numbers.
Introduction to String Theory.
This module introduces some of the generic ideas that underpin the analysis of waves in physical systems. Both linear and nonlinear models are discussed. Quasi-linear hyperbolic first-order systems of equations are introduced leading to the study of Riemann invariants, simple waves and shock solutions. Some knowledge of Vector Calculus (e.g. MATH225 Vector calculus with applications in fluid mechanics) is useful.
Modern particle theory is combining special relativity, quantum mechanics and field theory to describe all the fundamental subatomic particles and their interactions. The module develops the relevant concepts that enter into the Standard Model of particle physics. The key concept in modern physics is that of invariance under local symmetries and the conservation laws that they give rise to. The module covers the basic elements that describe modern particle theory, including: Lorentz and Poincare symmetries, which underlie special relativity; Hamilton and Lagrange formalism of classical mechanics and fields, which underlie the modern formalism; basic elements of relativistic quantum mechanics including the Dirac and Klein-Gordon equations; field quantisation; global and local symmetries; global and local symmetry breaking and the Higgs mechanism; unitary groups and the classification of elementary particles; basic elements of grand unified theories and phenomenological aspects. The students will be introduced to many of the modern ideas in Particle Physics at an accessible level.
This module provides an introduction into perturbation theory for partial differential equations. This theory has a wide, and growing, range of applications in the study of electro-magnetism, elasticity, heat conduction, the propagation of waves, and the study of cracks in materials.
This module provides an introduction to the problems and methods in the theory of elliptic curves.
This module covers a wide variety of methods of geometric continued fraction theory.
This module provides the foundations of stochastic analysis. Many of the basic results are considered in detail, in particular the ones that play a crucial role in applications such as mathematical finance. Students taking this module will study conditional expectations, martingales, Brownian motion, Brownian bridge, the reflection principle and scaling, stopping times, Ito’s integral and stochastic calculus, stochastic differential equations (linear and nonlinear), martingale representation, Girsanov theorem, and Feynman-Kac formula. Applications include stochastic control, optimal investment, and mathematical finance. All the theoretical results are illustrated with numerical examples from various fields of applications.
Algebraic geometry is a classical and nowadays vast area of mathematics. It deals with geometric figures given as roots of polynomial equations. Such figures live in projective spaces and are called algebraic varieties. Because of the algebraic nature of the defining equations, varieties can be considered over different ground fields.
Non-singular projective varieties over complex numbers are complex manifolds. As such, they are also smooth manifolds, so that the methods of complex-analytic and differential geometry are applicable to them. If varieties are given by polynomials with coefficients in the field of rational numbers, they encode deep arithmetical phenomena. This is why algebraic geometry marvelously merges different kinds of geometry and number theory into one big field. Algebraic geometry itself splits into several major subareas, such as birational geometry, intersection theory, deformation theory, moduli problems, arithmetic geometry, non-commutative geometry and much more. In the last decades the role of algebraic geometry in theoretical physics is steadily increasing. Within this advanced and demanding one-semester module the students will learn basics of algebraic geometry, being concentrated on the detailed elaboration of some instructive examples illustrating fundamental concepts and phenomena. Our purpose would be to train our algebraic-geometrical intuition working both synthetically, i.e. without coordinates, and in coordinates in terms of polynomials with coefficients in an algebraically closed field.
We will discuss some basic constructions of algebraic geometry, such as projections, blowing up, Segre embedding and products, families of varieties, linear series, tangent and vector bundles, differential forms and the canonical class. In the end we will also touch upon cohomology groups and how they can help to understand the geometry of algebraic varieties in the context of their topology and differential forms.
Combinatorics is a part of mathematics in which mathematicians deal with discrete and countable structures by means of various combinations, such as permutations, ordered and unordered selections, etc. The seemingly simple methods of combinatorics can raise highly non-trivial mathematical questions and lead to deep mathematical results, which are, in turn, closely related to some fundamental phenomena in number theory
This module is important for students who are interested in the abstract theory of integrating and in the deep theoretical background of the probability theory. It will be extremely useful for those who plan to do MSc and perhaps PhD in Probability, including financial applications. If you plan to take level 4 module(s) on Financial Mathematics next year, MATH365 can be very helpful.
MATH367 aims to develop an appreciation of optimisation methods for real-world problems using fundamental tools from network theory; to study a range of ‘standard problems’ and techniques for solving them. Thus, network flow, shortest path problem, transport problem, assignment problem, and routing problem are some of the problems that are considered in the syllabus. MATH367 is a decision making module, which fits well to those who are interested in receiving knowledge in graph theory, in operational research, in economics, in logistics and in finance.
Statistical Physics is a core subject in Physics and a cornerstone for modern technologies. To name just one example, quantum statistics is informing leading edge developments around ultra-cold gases and liquids giving rise to new materials. The module will introduce foundations of Statistical Physics and will develop an understanding of the stochastic roots of thermodynamics and the properties of matter. After successfully completing this module students will understand statistical ensembles and related concepts such as entropy and temperature, will understand the properties of classical and quantum gases, will be know the laws of thermodynamics and will be aware of advanced phenomena such as phase transition. The module will also develop numerical computer programming skills for the description of macroscopic effects such as diffusion by an underlying stochastic process.
In this module you will explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur and see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc.
A “dynamical system” is a system that changes over time according to a fixed rule. In complex dynamics, we consider the case where the state of the system is described by a single (complex) variable, and the rule of evolution is given by a holomorphic function. It turns out that this seemingly simple setting leads to very rich, subtle and intricate problems, some of which are still the subject of ongoing mathematical research, both at the University of Liverpool and internationally. This module will provide an introduction to this fascinating subject, and introduce students to some of these problems. In the course of this study, we will encounter many results about complex functions that may seem “magic” when compared with what might be expected from real analysis. A highlight of this kind is the theorem that every polynomial is “chaotic” on its Julia set. We will also see how this “magic” can help us understand phenomena that at first seem to have no connection with complex numbers at all.
Topology is the mathematical study of space. It is distinguished from geometry by the fact that there is no consideration of notions of distance, angle or other similar quantities. For this reason topology is sometimes popularly referred to as ‘rubber sheet’ geometry. It was introduced by Poincaré, under the name of analysis situs, in 1895 and became one of the most successful areas of 20th century mathematics. It continues to be an active research area to this day, and its insights and methods underlie many areas of modern mathematics. More recently, new applications of topological ideas outside mathematics have been developed, in particular to provide qualitative analysis of large data sets. This module introduces the basic notions of topological space and continuous map, illustrating them with many examples from different areas of mathematics. It also introduces homotopy theory, the study of paths in a space, which has become one of the most fundamental areas of modern mathematics.
This module introduces fundamental topics in mathematical statistics, including the theory of point estimation and hypothesis testing. Several key concepts of statistics are discussed, such as sufficiency, completeness, etc., introduced from the 1920s by major contributors to modern statistics such as Fisher, Neyman, Lehmann and so on. This module is absolutely necessary preparation for postgraduate studies in statistics and closely related subjects.
In recent years a culture of evidence-based practice has become the norm in the medical profession. Central to this is the medical statistician, who is required to not only analyse data, but to design research studies and interpret the results. The aim of MATH364 is to provide the student with the knowledge to become part of a “team” to enhance and improve medical practice. This is done by demonstrating the design of studies, methods of analysis and interpretation of results through a number of real-world examples, covering epidemiology, survival analysis, clinical trials and meta-analysis.
This module raises the awareness of students on how mathematical methods from stochastics can help to deal with problems arising in a variety of areas, ranging from quantifying uncertainty, to problems in physics, to optimisation and decision making, among others. The module summarises probability theory, explain the basics of simulation and sampling and then focuses on learning theory and methods. Specific topics and examples will be presented along with the theory and computer experiments.
Many real-world systems in mathematics, physics and engineering can be described by differential equations. In rare cases these can be solved exactly by purely analytical methods, but much more often we can only solve the equations numerically, by reducing the problem to an iterative scheme that requires hundreds of steps. We will learn efficient methods for solving ODEs and PDEs on a computer.
To provide an understanding of the mathematical risk theory used in practise in non-life actuarial depts of insurance firms, to provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities), to develop skills of calculating the ruin probability and the total claim amount distribution in some non ‐ life actuarial risk models with applications to insurance industry, to prepare the students adequately and to develop their skills in order to be exempted for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).
Networks are familiar to us from many real-world systems such as the internet, power grids, transportation and biological networks. The underpinning mathematical concept is called a graph an it is no surprise that the same issues arise in each area, whether this is to identify the most important or influential individuals in the network, or to prevent dynamics on the network (e.g. epidemics) or to make the network robust to the dynamics it supports (e.g. power grids and transportation). In this module, we learn about different classes of networks and how to quantify and describe them including their structures and their nodes. Much of our detailed understanding of networks and their features will come from analysis of idealised random networks which nevertheless are often good representations of those seen in the real world. We will consider real-world biological applications of network theory, in particular focusing on epidemics.
This module introduces the theory of polynomial equations of one variable: Galois Theory. This theory provides criteria when a polynomial equation can be solved in radicals, when a geometric construction can be performed by a ruler and a compass.
This module provides students with an introduction to mathematical research, through guided reading, writing a report and preparing and delivering a presentation.
This is your opportunity to study in depth a mathematical topic which interests you, and produce a substantial written report upon it. Your supervisor will help you select a suitable topic, and guide you in your studies and writing. For some students this is a prelude to embarking on a research degree, and for others it is an end in its own right, but for all it should be both stimulating and exciting.
The programme is delivered through a combination of lectures and tutorials, typically totalling four hours per module per week. Some modules will use a traditional face-to-face format while others may adopt a hybrid approach with some online activity and interactive face-to-face classes.
You will also be expected to attend research seminars in the department. These are currently operating in a hybrid format, with some seminars delivered in person and some online. These seminars may be difficult to understand at first but you will find that they become much clearer as the year progresses. Attending will give you an opportunity to see what is happening at research level in the department and to have regular contact with people in the relevant area.
We are proud of our record on teaching quality, with many of our MSc modules delivered by world-leading experts, some of whom have received the prestigious Sir Alastair Pilkington Award for Teaching. The Mathematics Centre for Enhancement in Education helps staff to develop innovative techniques for teaching mathematics.
Mathematics modules on this course have a continuous assessment component and a final assessment component. The final assessment component is taken in January and May exam periods. Assessments may take the form of class tests, coursework or project-based tasks. The programme also includes a summer dissertation.
We have a distinctive approach to education, the Liverpool Curriculum Framework, which focuses on research-connected teaching, active learning, and authentic assessment to ensure our students graduate as digitally fluent and confident global citizens.
Studying with us means you can tailor your degree to suit you. Here's what is available on this course.
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A mathematically-based degree opens up a wide range of career opportunities, including some of the most lucrative professions.
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.
Your tuition fees, funding your studies, and other costs to consider.
|UK fees (applies to Channel Islands, Isle of Man and Republic of Ireland)|
|Full-time place, per year||£11,250|
|Part-time place, per year||£5,625|
|Full-time place, per year||£21,000|
|Part-time place, per year||£10,500|
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 tuition fees, funding and Postgraduate Loans.
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.
We offer a range of scholarships and bursaries to help cover tuition fees and help with living expenses while at university.
The qualifications and exam results you'll need to apply for this course.
My qualifications are from: United Kingdom.
|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.
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.
You'll need to demonstrate competence in the use of English language. 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|
View our IELTS academic requirements key.
Standard Level 5
|TOEFL iBT||88 or above with minimum scores in components as follows: Listening and Writing 19, Reading 19, Speaking 20.|
|INDIA Standard XII||70% or above from Central and Metro State Boards|
|Hong Kong use of English AS level||C|
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Last updated 11 May 2023 / / Programme terms and conditions /