Mathematics MMath
202425 entryPreparation for a career in research, whether you want to work on solutions to abstract mathematics problems or apply your problem solving skills to challenges in industry. In your final year, you’ll complete a major research project.
Key details
 A Levels AAA
Other entry requirements  UCAS code G103
 4 years / Fulltime
 September start
 Find out the course fee
 Optional placement year
 Study abroad
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Course description
This course is great preparation for a career in research, whether you want to work on solutions to abstract mathematics problems or apply your problem solving skills to challenges in industry. There is a wide variety of options to choose from across pure mathematics, applied mathematics, and probability and statistics. In your final year, you’ll complete a major research project.
We have a small but focused number of modules in the first year, that cover all the essentials you’ll need for the rest of your degree. You can develop programming skills using Python and R, which can be applied to lots of jobs that involve data, and learn to use the typesetting software LaTeX, which mathematicians and statisticians use to present their work.
In your second year, you’ll continue to build a powerful toolbox of mathematical techniques, which you can apply to increasingly complex problems. There is less compulsory maths and more options, so as well as calculus and algebra, you can study topics including differential equations, the mechanics of motion in fluids and solids, statistical modelling and computer simulations.
Some module options include more project work. This gives you the chance to put your mathematics skills into practice in different contexts and scenarios that you might encounter when you start work after graduation. A module on careers development gives you the chance to find out about different career paths, learn about potential employers, write an impressive CV and sell yourself at job interviews.
By your third year, you’ll have the skills, knowledge and experience to explore many different areas of mathematics. We’ll give you lots of optional modules to choose from, so you can study the topics that are most useful to the career path you want to take or that you enjoy the most.
You’ll have a similar range of options to choose from in your final year. You’ll also spend a third of your time working on your own research project. You’ll choose a topic in an area of mathematics that interests you, and work closely with one of our staff who is an expert in the field. You’ll write up your findings and give a presentation about what you’ve learned.
Modules
A selection of modules are available each year  some examples are below. There may be changes before you start your course. From May of the year of entry, formal programme regulations will be available in our Programme Regulations Finder.
Choose a year to see modules for a level of study:
UCAS code: G103
Years: 2023
Core modules:
 Mathematics Core

Mathematics Core covers topics which continue school mathematics and which are used throughout the degree programmes: calculus and linear algebra, developing the framework for higherdimensional generalisation. This material is central to many topics in subsequent courses. At the same time, weekly smallgroup tutorials with the Personal Tutor aim to develop core skills, such as mathematical literacy and communication, some employability skills and problemsolving skills.
40 credits  Foundations of Pure Mathematics

The module aims to give an overview of basic constructions in pure mathematics; starting from the integers, we develop some theory of the integers, introducing theorems, proofs, and abstraction. This leads to the idea of axioms and general algebraic structures, with groups treated as a principal example. The process of constructing the real numbers from the rationals is also considered, as a preparation for “analysis”, the branch of mathematics where the properties of sequences of real numbers and functions of real numbers are considered.
20 credits  Mathematical modelling

Mathematics is the language of science. By framing a scientific question in mathematical language, it is possible to gain deep insight into the empirical world. This module aims to give students an appreciation of this astonishing phenomenon. It will introduce them to the concept of mathematical modelling via examples from throughout science, which may include biology, physics, environmental sciences, and more. Along the way, a range of mathematical techniques will be learned that tend to appear in empirical applications. These may include (but not necessarily be limited to) difference and differential equations, calculus, and linear algebra.
20 credits  Probability and Data Science

Probability theory is branch of mathematics concerned with the study of chance phenomena. Data science involves the handling and analysis of data using a variety of tools: statistical inference, machine learning, and graphical methods. The first part of the module introduces probability theory, providing a foundation for further probability and statistics modules, and for the statistical inference methods taught here. Examples are presented from diverse areas, and case studies involving a variety of real data sets are discussed. Data science tools are implemented using the statistical computing language R.
20 credits  Mathematical Investigation Skills

This module introduces topics which will be useful throughout students’ time as undergraduates and in employment. These skills fall into two categories: computer literacy and presentation skills. One aim of this module is to develop programming skills within Python to perform mathematical investigations. Students will also meet the typesetting package LaTeX, the web design language HTML, and Excel for spreadsheets. These will be used for making investigations, and preparing reports and presentations into mathematical topics.
20 credits
In your second year, you’ll continue to build a powerful toolbox of mathematical techniques, which you can apply to increasingly complex problems. There is less compulsory maths and more options, so as well as calculus and algebra, you can study topics including differential equations, the mechanics of motion in fluids and solids, statistical modelling and computer simulations.
Core modules:
 Mathematics Core II

Building on Level 1 Mathematics Core, Mathematics Core II will focus on foundational skills and knowledge for both higher mathematics and your future life as a highly skilled, analyticallyastute worker. Mathematical content will focus on topics that are vital for all areas of the mathematical sciences (pure, applied, statistics), such as vector calculus and linear algebra. This will help develop your analytic and problem solving skills. Alongside this, you will continue to develop employability skills, building on Level 1 Core. Finally, there will be opportunities to learn and reflect on social, ethical, and historical aspects of mathematics, which will enrich your understanding of the importance of mathematics in the modern world.
30 credits  Differential equations

Differential equations are perhaps the most important tool in applied mathematics. They are foundational for modelling all kinds of physical and natural phenomena, including fluids and plasmas, populations of animals or cells, cosmological objects (via relativity), subatomic particles (via quantum mechanics), epidemics, even political and social opinions have been modelled using differential equations. This module will build on the tools learned at Level 1 for analysing differential equations, extending them in a variety of ways. This may include topics such as bifurcation analysis, partial differential equations (which are particularly valuable for modelling things that vary in both space and time), and the effects of boundaries on the dynamics of differential equations. It will provide the foundation for essentially all applied maths modules taught at Levels 3 and 4.
20 credits  Analysis and Algebra

This module will build on the theory built in Level 1 'Foundations of Pure Mathematics', focusing on the twin pillars of analysis and algebra. These are not only fundamental for pure mathematics at higher levels, but provide rigorous theory behind core concepts that are used throughout the mathematical sciences. Whilst to some extent you have been doing analysis and algebra since you were at school, here you will be going much deeper. You will examine why familiar tools, like differentiation and integration, actually work. Familiar objects, such as vectors, differential operators, and matrices, will be unpacked; powerful, formal properties of these objects proved. Ultimately, this rigorous foundation will enable you to extend these tools and concepts to tackle a far greater set of problems than before.
20 credits  Statistical Inference and Modelling

Statistical inference and modelling are at the heart of data science, a field of rapidlygrowing importance in the modern word. This module develops methods for analysing data, and provides a foundation for further study of probability and statistics at higher Levels. You will learn about a range of standard probability distributions beyond those met at Level 1, including multivariate distributions. You will learn about sampling theory and summary statistics, and their relation to data analysis. You will discover how to parametrise various types of statistical model, learn techniques for determining whether one model is 'better' than another for understanding a dataset, and learn how to ascertain how good a statistical model is at explaining trends in data. The software package R will be used throughout.
20 credits
Optional modules:
 Stochastic Modelling

Many things about life are unpredictable. Consequently, it often makes sense to incorporate some randomness in mathematical models of natural and physical processes. Such models are called 'stochastic models' and are the study object of this module. We will learn about a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might include the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. We will cover various techniques for analysis of such models, setting the student up for further study of stochastic processes and probability at levels 3 and 4.
10 credits  Vector Calculus and Dynamics

Vector calculus is a fundamental tool for modelling the dynamics of all kinds of objects, both solid and fluid. In this module, you will build on the tools of vector calculus from Mathematics Core II, combining them with tools of differential equations from the L1 Mathematical Modelling module, and applying them to understand the dynamics of physical systems. Possible examples might include liquid, gases, plasmas, and/or planetary motion. The tools developed here will build valuable knowledge for the study of fluid dynamics and other applied mathematics modules at higher levels.
10 credits  Group Theory

A group is one of the most foundational objects in mathematics. It just consists of a set, together with a way of combining two objects in that set to create another object in an internallyconsistent fashion. Familiar examples abound: integers with addition, real numbers with multiplication, symmetries of the square, and so on. In this module, you will learn about formal properties of groups in general, including famous results like the orbitstabiliser theorem. You will also learn about important foundational examples, such as number, matrices, and symmetries. You will learn how the general framework of groups allows you to prove theorems that pertain to all these examples in one go. This provides a great example of the power and beauty of abstraction, a feature of pure mathematics that underlies the entire module.
10 credits  Mathematics and Statistics in Action

In this project module, you will investigate one or more case studies of using mathematics and statistics for solving empirical (i.e. 'real world') problems. These case studies will illustrate the process of mathematical and statistical modelling, whereby realworld questions are translated to mathematical and/or statistical questions. Students will see how techniques learned earlier in their degree can be used to explore these problems. There will be a mix of individual and group projects to choose from, and some projects may involve the use of R or Python, but 'MAS116 Mathematical Investigations Skills' is not a prerequisite. There are no lectures or tutorials in this module, so students will be expected to work independently (either individually or in a small group). However, the topic and scope of each piece of project work will be clearly defined by the lecturer in charge of the topic.
10 credits  Scientific Computing

The ability to programme is a central skill for any highlynumerate person in the 21st century. This module builds on skills learned at Level 1 in 'Mathematical Investigation Skills' by developing skills in computer programming and independent investigation. You will learn how to solve various mathematical problems in programming languages commonlyused by mathematicians, for example Python. You will learn basic computational and visualisation methods for exploring numerical solutions to equations (including differential equations), and then apply this knowledge to explore the behaviour of example physical systems that these equations might model.
10 credits
Optional modules:
 Undergraduate Ambassadors Scheme in Mathematics

This module provides an opportunity for Level Three students to gain first hand experience of mathematics education through a mentoring scheme with mathematics teachers in local schools. Typically, each student will work with one class for half a day every week for 11 weeks. The classes will vary from key stage 2 to sixth form. Students will be given a range of responsibilities from classroom assistant to the organisation and teaching of selforiginated special projects. Only a limited number of places are available and students will be selected on the basis of their commitment and suitability for working in schools.
20 credits  Bayesian Statistics

This module develops the Bayesian approach to statistical inference. The Bayesian method is fundamentally different in philosophy from conventional frequentist/classical inference and is becoming the approach of choice in many fields of applied statistics. This course will cover both the foundations of Bayesian statistics, including subjective probability, inference, and modern computational tools for practical inference problems, specifically Markov Chain Monte Carlo methods and Gibbs sampling. Applied Bayesian methods will be demonstrated in a series of case studies using the software package R.
10 credits  Codes and Cryptography

The word 'code' is used in two different ways. The ISBN code of a book is designed in such a way that simple errors in recording it will not produce the ISBN of a different book. This is an example of an 'errorcorrecting code' (more accurately, an errordetecting code). On the other hand, we speak of codes which encrypt information  a topic of vital importance to the transmission of sensitive financial information across the internet. These two ideas, here labelled 'Codes' and 'Cryptography', each depend on elegant pure mathematical ideas: codes on linear algebra and cryptography on number theory. This course explores these topics, including the reallife applications and the mathematics behind them.
10 credits  Combinatorics

Combinatorics is the mathematics of selections and combinations. For example, given a collection of sets, when is it possible to choose a different element from each of them? That simple question leads to Hall's Theorem, a farreaching result with applications to counting and pairing problems throughout mathematics.
10 credits  Complex Analysis

It is natural to use complex numbers in algebra, since these are the numbers we need to provide the roots of all polynomials. In fact, it is equally natural to use complex numbers in analysis, and this course introduces the study of complexvalued functions of a complex variable. Complex analysis is a central area of mathematics. It is both widely applicable and very beautiful, with a strong geometrical flavour. This course will consider some of the key theorems in the subject, weaving together complex derivatives and complex line integrals. There will be a strong emphasis on applications. Anyone taking the course will be expected to know the statements of the theorems and be able to use them correctly to solve problems.
10 credits  Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the BlackScholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the BlackScholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance.
10 credits  Game Theory

The module will give students the opportunity to apply previously acquired mathematical skills to the study of Game Theory and to some of the applications in Economics.
10 credits  Generalised Linear models

This module introduces the theory and application of generalised linear models. These models can be used to investigate the relationship between some quantity of interest, the "dependent variable", and one more "explanatory" variables; how the dependent variable changes as the explanatory variables change. The term "generalised" refers to the fact that these models can be used for a wide range of different types of dependent variable: continuous, discrete, categorical, ordinal etc. The application of these models is demonstrated using the programming language R.
10 credits  Graph Theory

A graph is a simple mathematical structure consisting of a collection of points, some pairs of which are joined by lines. Their basic nature means that they can be used to illustrate a wide range of situations. The aim of this course is to investigate the mathematics of these structures and to use them in a wide range of applications.
10 credits  Introduction to Relativity

Einstein's theory of relativity is one of the corner stones of our understanding of the universe. This course will introduce some of the ideas of relativity, and the physical consequences of the theory, many of which are highly counterintuitive. For example, a rapidly moving body will appear to be contracted as seen by an observer at rest. The course will also introduce one of the most famous equations in the whole of mathematics: E=mc2.
10 credits  Knots and Surfaces

The course studies knots, links and surfaces in an elementary way. The key mathematical idea is that of an algebraic invariant: the Jones polynomial for knots, and the Euler characteristic for surfaces. These invariants will be used to classify surfaces, and to give a practical way to place a surface in the classification. Various connections with other sciences will be described.
10 credits  Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of 'Big Data', statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.
10 credits  Topics in Mathematical Biology

This module focuses on the mathematical modelling of biological phenomena. The emphasis will be on deterministic models based on systems of differential equations. Examples will be drawn from a range of biological topics, which may include the spread of epidemics, predatorprey dynamics, cell biology, medicine, or any other biological phenomenon that requires a mathematical approach to understand. Central to the module will be the dynamic consequences of feedback interactions within biological systems. In cases where explicit solutions are not readily obtainable, techniques that give a qualitative picture of the model dynamics (including numerical simulation) will be used. If you did not take Scientific Computing at Level 2, you may still be able to enrol on this module, but you will need to obtain permission from the module leader first.
10 credits  Mathematical Methods

This course introduces methods which are useful in many areas of mathematics. The emphasis will mainly be on obtaining approximate solutions to problems which involve a small parameter and cannot easily be solved exactly. These problems will include the evaluation of integrals. Examples of possible applications are: oscillating motions with small nonlinear damping, the effect of other planets on the Earth's orbit around the Sun, boundary layers in fluid flows, electrical capacitance of long thin bodies, central limit theorem correction terms for finite sample size.
10 credits  Mathematical modelling of natural systems

Mathematical modelling enables insight in to a wide range of scientific problems. This module will provide a practical introduction to techniques for modelling natural systems. Students will learn how to construct, analyse and interpret mathematical models, using a combination of differential equations, scientific computing and mathematical reasoning. Students will learn the art of mathematical modelling: translating a scientific problem into a mathematical model, identifying and using appropriate mathematical tools to analyse the model, and finally relating the significance of the mathematical results back to the original problem. Study systems will be drawn from throughout the environmental and life sciences.
10 credits  Probability with Measure

Probability is a relatively new part of mathematics, first studied rigorously in the early part of 20th century. This module introduces the modern basis for probability theory, coming from the idea of 'measuring' an object by attaching a nonnegative number to it. This might refer to its length or volume, but also to the probability of an event happening. We therefore find a close connection between integration and probability theory, drawing upon real analysis. This rigorous theory allows us to study random objects with complex or surprising properties, which can expand our innate intuition for how probability behaves. The precise material covered in this module may vary according to the lecturer's interests.
10 credits  Medical Statistics

This course comprises sections on Clinical Trials and Survival Data Analysis. The special ethical and regulatory constraints involved in experimentation on human subjects mean that Clinical Trials have developed their own distinct methodology. Students will, however, recognise many fundamentals from mainstream statistical theory. The course aims to discuss the ethical issues involved and to introduce the specialist methods required. Prediction of survival times or comparisons of survival patterns between different treatments are examples of paramount importance in medical statistics. The aim of this course is to provide a flavour of the statistical methodology developed specifically for such problems, especially with regard to the handling of censored data (eg patients still alive at the close of the study). Most of the statistical analyses can be implemented in standard statistical packages.
10 credits  Metric Spaces

This unit explores ideas of convergence of iterative processes in the more general framework of metric spaces. A metric space is a set with a distance function which is governed by just three simple rules, from which the entire analysis follows. The course follows on from MAS207 'Continuity and Integration', and adapts some of the ideas from that course to the more general setting. The course ends with the Contraction Mapping Theorem, which guarantees the convergence of quite general processes; there are applications to many other areas of mathematics, such as to the solubility of differential equations.
10 credits  Operations Research

Mathematical Programming is the title given to a collection of optimisation algorithms that deal with constrained optimisation problems. Here the problems considered will all involve constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable; special algorithms have to be developed. The module considers not only the solution of such problems but also the important area of postoptimality analysis; i.e. given the solution can one answer questions about the effect of small changes in the parameters of the problem (such as values of the cost coefficients)?
10 credits  Quantum Theory

The development of quantum theory revolutionized both physics and mathematics during the 20th century. The theory has applications in many technological advances, including: lasers, superconductors, modern medical imaging techniques, transistors and quantum computers. This course introduces the basics of the theory and brings together many aspects of mathematics: for example, probability, matrices and complex numbers. Only first year mechanics is assumed, and other mathematical concepts will be introduced as they are needed.
10 credits  Sampling Theory and Design of Experiments

The results of sample surveys through opinion polls are commonplace in newspapers and on television. The objective of the Sampling Theory section of the module is to introduce several different methods for obtaining samples from finite populations. Experiments which aim to discover improved conditions are commonplace in industry, agriculture, etc. The purpose of experimental design is to maximise the information on what is of interest with the minimum use of resources. The aim of the Design section is to introduce some of the more important design concepts.
10 credits  Time Series

Time series are observations made in time, for which the time aspect is potentially important for understanding and use. The course aims to give an introduction to modern methods of time series analysis and forecasting as applied in economics, engineering and the natural, medical and social sciences. The emphasis will be on practical techniques for data analysis, though appropriate stochastic models for time series will be introduced as necessary to give a firm basis for practical modelling. Appropriate computer packages will be used to implement the methods.
10 credits  Topics in Number Theory

In this module we study intergers, primes and equations. Topics covered include linear and quadratic congruences, Fermat Little Theorem and Euler's Theorem, the RSA cryptosystem, Quadratic Reciprocity, perfect numbers, continued fractions and others.
10 credits
Core modules:
 Mathematics and Statistics Project

This module forms the final part of the SoMaS project provision at Level 4 and involves the completion, under the guidance of a research active supervisor, of a substantial project on an advanced topic in Mathematics or Statistics. Training is provided in the use of appropriate computer packages for the presentation of mathematics and statistics and guidance on the coherent and accurate presentation of technical information.
45 credits
A student will take 75 credits from this group:
 Topics in Advanced Fluid Mechanics

This module aims to describe advanced mathematical handling of fluid equations in an easily accessible fashion. A number of topics are treated in connection with the mathematical modelling of formation of the (near)singular structures with concentrated vorticity in inviscid flows. After discussing prototype problems in one and two dimensions, the threedimensional flows in terms of vortex dynamics are described. Key mathematical tools, for example, singular integrals and calculus inequalities, are explained during the unit in a selfcontained manner. Candidates are directed to read key original papers on some topics to deepen their understanding.
20 credits  Topics in Mathematical Physics

This unit will introduce students to advanced concepts and techniques in modern mathematical physics, in preparation for researchlevel activities.
15 credits
It is assumed that the student comes equipped with a working knowledge of analytical dynamics, and of nonrelativistic quantum theory.
We will examine how key physical ideas are precisely formulated in the language of mathematics. For example, the idea that fundamental particles arise as excitations of relativistic quantum fields finds its mathematical realisation in Quantum Field Theory. In QFT, particles can be created from the vacuum, and destroyed, but certain other quantities such as charge, energy, and momentum are conserved (after averaging over quantum fluctuations).
We will examine links between conservation laws and invariants, and the underlying (discrete or continuous) symmetry groups of theories. We will also develop powerful calculation tools. For example, to find the rate of creation of new particles in a potential, one must evaluate the terms in a perturbative (Feynmandiagram) expansion.
For details of the current syllabus, please consult the module leader.  Magnetohydrodynamics

Magnetohydrodynamics has been successfully applied to a number of astrophysical problems (eg to problems in Solar Magnetospheric Physics), as well as to problems related to laboratory physics, especially to fusion devices. This module gives an introduction to classical magnetohydrodynamics. Students will get familar with the system of magnetohydrodynamic equations and main theorems that follow from this system (e.g. conservation laws, antidynamo theorem). They will study the simplest magnetic equilibrium configurations, propagation of linear waves, and magnetohydrodynamic stability. The final part of the module provides an introduction to the theory of magnetic dynamo
10 credits  Generalised Linear Models

This module introduces the theory and application of generalised linear models. These models can be used to investigate the relationship between some quantity of interest, the "dependent variable", and one or more "explanatory" variables; how the dependent variable changes as the explanatory variables change. The term "generalised" refers to the fact that these models can be used for a wide range of different types of dependent variable ,continuous, discrete, categorical, ordinal etc. The application of these models is demonstrated using the programming language R.
15 credits  Medical Statistics

This module introduces an important application of statistics: medical research, specifically, the design and analysis of clinical trials. For any new drug to be approved by a regulator (such as the Medicines and Healthcare products Regulatory Agency in the UK) for use on patients, the effectiveness of the drug has to be demonstrated in a clinical trial. This module explains how clinical trials are designed and how statistical methods are used to analyse the results, with a particular focus on 'survival' or 'timetoevent' analysis.
15 credits  Sampling Theory and Design of Experiments

Whereas most statistics modules are concerned with the analysis of data, this module is focussed on the collection of data. In particular, this module considers how to collect data efficiently: how to ensure the quantities of interest can be estimated sufficiently accurately, using the smallest possible sample size. Three settings are considered: sample surveys (for example when conducting an opinion poll), physical experiments, as may be used in industry, and experiments involving predictions from computer models, where there is uncertainty in the computer model prediction.
15 credits  Bayesian Statistics and Computational Methods

This module introduces the Bayesian approach to statistical inference. The Bayesian method is fundamentally different in philosophy from conventional frequentist/classical inference, and has been the subject of some controversy in the past, but is now widely used. The module also presents various computational methods for implementing both Bayesian and frequentist inference, in situations where obtaining results 'analytically' would be impossible. The methods will be implemented using the programming languages R and Stan, and some programming is taught alongside the theory lectures.
30 credits  Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of “Big Data”, statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.
15 credits  Advanced Topics in Algebra A

Algebra is a very broad topic, relating many disparate areas of mathematics, from mathematical physics to abstract computer science. It was noticed that the same underlying structures arose in a number of different areas, and this led to the study of the abstract structures. This module studies some of the algebraic structures involved: fields, groups and Galois Theory; rings and commutative algebra, and gives applications to, for example, number theory, roots of polynomials and algebraic geometry.
30 credits  Advanced Topics in Algebra B

Algebra is a very broad topic, relating many disparate areas of mathematics, from mathematical physics to abstract computer science. It was noticed that the same underlying structures arose in a number of different areas, and this led to the study of the abstract structures. This module studies some of the algebraic structures involved: fields, groups and Galois Theory; rings and commutative algebra, and gives applications to, for example, number theory, roots of polynomials and algebraic geometry.
30 credits  Algebraic Topology

This unit will cover algebraic topology, following on from metric spaces. Topology studies the shape of spaces, with examples such as spheres, the Möbius Band, the Klein bottle, the torus and other surfaces. The first task is to formalise the notion of space, and to work out when a given space can be continuously deformed into another, where stretching and bending is allowed, but cutting and glueing is not. Algebraic topology builds a powerful bridge between shapes and algebra, enabling the use of familiar algebraic techniques from linear algebra and group theory to study spaces and their deformations.
30 credits  Analytical Dynamics and Classical Field Theory

Newton formulated his famous laws of mechanics in the late 17th century. Later, mathematicians like Lagrange, Hamilton and Jacobi discovered that underlying Newton's work are wonderful mathematical structures. In the first semester we discuss this work, its influence on the subsequent formulation of field theory, and Noether's theorem relating symmetries and conservation laws. In the second semester, Einstein's theory of gravity, General Relativity, will be introduced, preceded by mathematical tools such as covariant derivatives and curvature tensors. Einstein's field equations, and two famous solutions, will be derived. Two classic experimental tests of General Relativity will be discussed.
30 credits  Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the BlackScholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the BlackScholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance, and includes a computational project where students further explore some of the ideas of option pricing.
15 credits  Functional Analysis

Functional analysis is the study of infinitedimensional vector spaces equipped with extra structure. Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right, functional analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, study of the solutions of certain differential equations, stochastic processes, and in quantum physics. In this unit we focus on Banach spaces and especially Hilbert spaces  complete vector spaces equipped with an inner product  and linear maps between Hilbert spaces. Applications of the theory we examine include Fourier series and the Fourier transform, and differential equations.
15 credits  Mathematical Modelling of Natural Systems

Mathematical modelling enables insight into a wide range of scientific problems. This module will provide a practical introduction to techniques for modelling natural systems. Students will learn how to construct, analyse and interpret mathematical models, using a combination of differential equations, scientific computing and mathematical reasoning. Students will learn the art of mathematical modelling: translating a scientific problem into a mathematical model, identifying and using appropriate mathematical tools to analyse the model, and finally relating the significance of the mathematical results back to the original problem. Study systems will be drawn from throughout the environmental and life sciences.
15 credits  Further Topics in Number Theory

Elementary number theory has been seen in a number of earlier modules. To go further, however, additional input is needed from other areas of pure mathematics  analysis and algebra. For example, the distribution of prime numbers is intricately related to the complex analytic properties of the Riemann zeta function And one can ask similar questions to those we ask about prime numbers for the rational numbers over, for example, quadratic fields. This module will treat examples of further topics in number theory, accessible with the aid of advanced mathematical background.
15 credits  Probability and Random Graphs

Random graphs were studied by mathematicians as early as the 1950s. The field has become particularly important in recent decades as modern technology gives rise to a vast range of examples, such as social and communication networks, or the genealogical relationships between organisms. This course studies a range of models of random trees, graphs and networks, alongside probabilistic ideas that are needed to analyse their different properties. The precise material covered in this module may vary according to the lecturer's interests.
15 credits  Probability with Measure Theory

Probability is a relatively new part of mathematics, first studied rigorously in the early part of 20th century. This module introduces the modern basis for probability theory, coming from the idea of 'measuring' an object by attaching a nonnegative number to it. This might refer to its length or volume, but also to the probability of an event happening. We therefore find a close connection between integration and probability theory, drawing upon real analysis. This rigorous theory allows us to study random objects with complex or surprising properties, which can expand our innate intuition for how probability behaves. The precise material covered in this module may vary according to the lecturer's interests.
15 credits  Stochastic Processes and Finance

Stochastic processes are models that reflect the wide variety of unpredictable ways in which reality behaves. In this course we study several examples of stochastic processes, and analyse the behaviour they exhibit. We apply this knowledge to mathematical finance, in particular to arbitrage free pricing and the BlackScholes model.
30 credits  Further Topics in Mathematical Biology

This module focuses on the mathematical modelling of biological phenomena. The emphasis will be on deterministic models based on systems of differential equations. Examples will be drawn from a range of biological topics, which may include the spread of epidemics, predatorprey dynamics, cell biology, medicine, or any other biological phenomenon that requires a mathematical approach to understand. Central to the module will be the dynamic consequences of feedback interactions within biological systems. In cases where explicit solutions are not readily obtainable, techniques that give a qualitative picture of the model dynamics (including numerical simulation) will be used. If you did not take Scientific Computing at Level 2, you may still be able to enrol on this module, but you will need to obtain permission from the module leader first.
15 credits  Advanced Particle Physics

The module provides students with a comprehensive understanding of modern particle physics. Focusing on the standard model, it provides a theoretical underpinning of this model and discusses its predictions. Recent developments including the discovery of the Higgs Boson and neutrino oscillation studies are covered. A description of the experiments used to probe the standard model is provided. Finally the module looks at possible physics beyond the standard model.
15 credits  Advanced Quantum Mechanics

Quantum mechanics at an intermediate to advanced level, including the mathematical vector space formalism, approximate methods, angular momentum, and some contemporary topics such as entanglement, density matrices and open quantum systems. We will study topics in quantum mechanics at an intermediate to advanced level, bridging the gap between the physics core and graduate level material. The syllabus includes a formal mathematical description in the language of vector spaces; the description of the quantum state in Schrodinger and Heisenberg pictures, and using density operators to represent mixed states; approximate methods: perturbation theory, variational method and timedependent perturbation theory; the theory of angular momentum and spin; the treatment of identical particles; entanglement; open quantum systems and decoherence. The problem solving will provide a lot of practice at using vector and matrix methods and operator algebra techniques. The teaching will take the form of traditional lectures plus weekly problem classes where you will be provided with support and feedback on your attempts.
15 credits
The content of our courses is reviewed annually to make sure it's uptodate and relevant. Individual modules are occasionally updated or withdrawn. This is in response to discoveries through our worldleading research; funding changes; professional accreditation requirements; student or employer feedback; outcomes of reviews; and variations in staff or student numbers. In the event of any change we'll consult and inform students in good time and take reasonable steps to minimise disruption.
Learning and assessment
Learning
You'll learn through lectures, problems classes in small groups and research projects. Some modules also include programming classes.
Assessment
You will be assessed in a variety of ways, depending on the modules you take. This can include quizzes, examinations, presentations, participation in tutorials, projects, coursework and other written work.
Programme specification
This tells you the aims and learning outcomes of this course and how these will be achieved and assessed.
Entry requirements
With Access Sheffield, you could qualify for additional consideration or an alternative offer  find out if you're eligible.
The A Level entry requirements for this course are:
AAA
including Maths
 A Levels + additional qualifications
 AAB, including A in Maths + A in a relevant EPQ; AAB, including A in Maths + B in A Level Further Maths
 International Baccalaureate
 36, with 6 in Higher Level Maths (Analysis and Approaches)
 BTEC Extended Diploma
 D*DD in Engineering with Distinctions in all Maths units
 BTEC Diploma
 DD + A in A Level Maths
 Scottish Highers + 1 Advanced Higher
 AAAAB + A in Maths
 Welsh Baccalaureate + 2 A Levels
 A + AA, including Maths
 Access to HE Diploma
 Award of Access to HE Diploma in a relevant subject, with 45 credits at Level 3, including 39 at Distinction (to include Maths units), and 6 at Merit

We will give your application additional consideration if you have passed the Sixth Term Examination Paper (STEP) at grade 3 or above or the Test of Mathematics for University Admissions (TMUA) at grade 5 or above
The A Level entry requirements for this course are:
AAB
including A in Maths
 A Levels + additional qualifications
 AAB, including A in Maths + A in a relevant EPQ; AAB, including A in Maths + B in A Level Further Maths
 International Baccalaureate
 34, with 6 in Higher Level Maths (Analysis and Approaches)
 BTEC Extended Diploma
 DDD in Engineering with Distinctions in all Maths units
 BTEC Diploma
 DD + A in A Level Maths
 Scottish Highers + 1 Advanced Higher
 AAABB + A in Maths
 Welsh Baccalaureate + 2 A Levels
 B + AA, including Maths
 Access to HE Diploma
 Award of Access to HE Diploma in a relevant subject, with 45 credits at Level 3, including 36 at Distinction (to include Maths units), and 9 at Merit

We will give your application additional consideration if you have passed the Sixth Term Examination Paper (STEP) at grade 3 or above or the Test of Mathematics for University Admissions (TMUA) at grade 5 or above
You must demonstrate that your English is good enough for you to successfully complete your course. For this course we require: GCSE English Language at grade 4/C; IELTS grade of 6.5 with a minimum of 6.0 in each component; or an alternative acceptable English language qualification
Equivalent English language qualifications
Visa and immigration requirements
Other qualifications  UK and EU/international
If you have any questions about entry requirements, please contact the department.
Graduate careers
School of Mathematics and Statistics
Strong mathematics skills open all kinds of doors for our graduates: from banking, insurance and pensions, to software development at tech companies and encryption services at security agencies. They also work for businesses with vast amounts of data to process and inform new products and services.
Organisations that have hired Sheffield maths graduates include AstraZeneca, BAE Systems, Barclays, Bet365, Dell, Deloitte, Goldman Sachs, GSK, HSBC, IBM, Lloyds, PwC, Unilever, the Civil Service and the NHS. Lots of our students also go on to do PhDs at world top 100 universities.
Your career in mathematics and statistics
School of Mathematics and Statistics
When new students join the School of Mathematics and Statistics, we want them to feel part of a community. At the heart of this is the Sheffield University Mathematics Society, or SUMS, who organise activities throughout the academic year – from charity fundraisers to nights out. Our students also take part in pizza lectures, rocket engineering projects, international maths challenges, and an LGBT+ support group for maths students.
Staff in the School of Mathematics and Statistics work on a wide range of topics, from the most abstract research in areas like algebraic geometry and number theory, to the calculations behind animal movements and black holes. They’ll guide you through the key concepts and techniques that every mathematician needs to understand and give you a huge range of optional modules to choose from.
The department is based in the Hicks Building, which has classrooms, lecture theatres, computer rooms and social spaces for our students. It’s right next door to the Students' Union, and just down the road from the 24/7 library facilities at the Information Commons and the Diamond.
School of Mathematics and StatisticsWhy choose Sheffield?
The University of Sheffield
Number one in the Russell Group
National Student Survey 2023 (based on aggregate responses)
92 per cent of our research is rated as worldleading or internationally excellent
Research Excellence Framework 2021
Top 50 in the most international universities rankings
Times Higher Education World University Rankings 2022
Number one Students' Union in the UK
Whatuni Student Choice Awards 2022, 2020, 2019, 2018, 2017
Number one for teaching quality, Students' Union and clubs/societies
StudentCrowd 2023 University Awards
A top 20 university targeted by employers
The Graduate Market in 2023, High Fliers report
School of Mathematics and Statistics
Research Excellence Framework 2021
Fees and funding
Fees
Additional costs
The annual fee for your course includes a number of items in addition to your tuition. If an item or activity is classed as a compulsory element for your course, it will normally be included in your tuition fee. There are also other costs which you may need to consider.
Funding your study
Depending on your circumstances, you may qualify for a bursary, scholarship or loan to help fund your study and enhance your learning experience.
Use our Student Funding Calculator to work out what you’re eligible for.
Placements and study abroad
Placement
Study abroad
Visit
University open days
We host five open days each year, usually in June, July, September, October and November. You can talk to staff and students, tour the campus and see inside the accommodation.
Subject tasters
If you’re considering your post16 options, our interactive subject tasters are for you. There are a wide range of subjects to choose from and you can attend sessions online or on campus.
Offer holder days
If you've received an offer to study with us, we'll invite you to one of our offer holder days, which take place between February and April. These open days have a strong department focus and give you the chance to really explore student life here, even if you've visited us before.
Campus tours
Our weekly guided tours show you what Sheffield has to offer  both on campus and beyond. You can extend your visit with tours of our city, accommodation or sport facilities.
Apply
Contact us
 Telephone
 +44 114 222 3999
 maths.admiss@sheffield.ac.uk
The awarding body for this course is the University of Sheffield.
Recognition of professional qualifications: from 1 January 2021, in order to have any UK professional qualifications recognised for work in an EU country across a number of regulated and other professions you need to apply to the host country for recognition. Read information from the UK government and the EU Regulated Professions Database.
Any supervisors and research areas listed are indicative and may change before the start of the course.