Mathematics and Statistics MMath
School of Mathematics and Statistics
You are viewing this course for 202122 entry. 202223 entry is also available.
Key details
 A Levels AAA
Other entry requirements  UCAS code G110
 4 years / Fulltime
 Accredited
 Find out the course fee
Course description
This degree is intended for those who wish to specialise in statistics, whilst developing broader mathematical understanding.
You'll learn statistical data analysis and computing skills, and have the opportunity to apply these in project work.
The course includes statistics, probability, core mathematics, pure mathematics and applied mathematics, with increasing emphasis on statistics and probability as you progress.
In the fourth year, you'll complete your own extended project.
Accredited by the Royal Statistical Society (RSS) for the purpose of eligibility for Graduate Statistician status.
Modules
The modules listed below are examples from the last academic year. There may be some changes before you start your course. For the very latest module information, check with the department directly.
Choose a year to see modules for a level of study:
UCAS code: G110
Years: 2021
Core modules:
 Introduction to Probability and Statistics

The module provides an introduction to the fields of probability and statistics, which form the basis of much of applicable mathematics and operations research. The theory behind probability and statistics will be introduced, along with examples occurring in diverse areas. Some of the computational statistical work may make use of the statistics package R.
20 credits  Mathematical Investigation Skills

This module introduces topics which will be useful throughout students time as undergraduates and beyond. These skills fall into two categories: computer literacy and presentation skills. Various computer packages are introduced in other modules; these share some programming capabilities, and one aim of this module is to develop programming techniques to perform mathematical investigations within the context of these mathematical packages. In addition, spreadsheets have substantial scientific capabilities, and Excel is the program of choice within industry. Finally, students will meet the typesetting package LaTeX, preparing reports and presentations into mathematical topics.
20 credits  Mathematics Core 1

The module explores topics in mathematics which will be used throughout many degree programmes. The module will consider techniques for solving equations, special functions, calculus (differentiation and integration), differential equations, Taylor series, complex numbers and finite and infinite series.
20 credits  Mathematics Core II

The module continues the study of core mathematical topics begun in MS4F1015, which will be used throughout many degree programmes. The module will discuss 2dimensional coordinate geometry, discussing the theory of matrices geometrically and algebraically, and will define and evaluate derivatives and integrals for functions which depend on more than one variable, with an emphasis on functions of two variables.
20 credits  Numbers and Groups

The module provides an introduction to more specialised Pure Mathematics. The first half of the module will consider techniques of proof, and these will be demonstrated within the study of properties of integers and real numbers. The second semester will study symmetries of objects, and develop a theory of symmetries which leads to the more abstract study of groups.
20 credits
Optional modules:
 Vectors and Mechanics

The module begins with the algebra of vectors, essential for the study of many branches of applied mathematics. The theory is illustrated by many examples, with emphasis on geometry including lines and planes. Vectors are then used to define the velocity and acceleration of a moving particle, thus leading to an introduction to Newtonian particle mechanics. Newton's laws are applied to particle models in areas such as sport, rides at theme parks and oscillation theory.
20 credits  Principles of Ecology and Conservation

This course is an introduction to the principles of ecology and conservation. It covers ecological concepts about the abundance and distribution of species and key ideas about conserving populations, communities and habitats.
20 credits  Climate Change and Sustainability

This course introduces the core scientific issues required to understand climate change and sustainability. Students will learn the causes of climate change, its impacts in natural and agricultural ecosystems, the influence of biogeochemical cycles in these ecosystems on climate, and strategies for sustainably managing ecosystems in future. Learning will be achieved via lectures and videos, practicals and independent study.
20 credits  Foundations of Computer Science

The course consists of (around) 10 blocks of 23 weeks work each. Each block develops mathematical concepts and techniques that are of foundational importance to computing. Lectures and problem classes will be used. The intention is to enthuse about these topics, to demonstrate why they are important to us, to lay the foundations of their knowledge and prepare students for future computing courses. It is not expected that the course will cover ALL of the maths that is needed later either in terms of depth or scope.
20 credits  Introduction to Algorithms and Data Structures

Algorithms and algorithmic problem solving are at the heart of computer science. This module introduces students to the design and analysis of efficient algorithms and data structures. Students learn how to quantify the efficiency of an algorithm and what algorithmic solutions are efficient. Techniques for designing efficient algorithms are taught, including efficient data structures for storing and retrieving data. This is done using illustrative and fundamental problems: searching, sorting, graph algorithms, and combinatorial problems such as finding the shortest paths in networks.
10 credits  Elementary Logic

The course will provide students with knowledge of the fundamental parts of formal logic. It will also teach them a range of associated formal techniques with which they can then analyse and assess arguments. In particular, they will learn the languages of propositional and firstorder logic, and they will learn how to use those languages in providing formal representations of everyday claims. They will also learn how to use truthtables and truthtrees.
10 credits  Introduction to Astrophysics

One of four halfmodules forming the Level1 Astronomy course, PHY104 aims to equip students with a basic understanding of the important physical concepts and techniques involved in astronomy with an emphasis on how fundamental results can be derived from fairly simple observations. The module consists of three sections:
10 credits
(i) Basic Concepts, Fluxes, Temperatures and Magnitudes;
(ii) Astronomical Spectroscopy;
(iii) Gravitational Astrophysics.
Parts (i), (ii) and (iii) each comprise some six lectures. The lectures are supported by problems classes, in which you will learn to apply lecture material to the solution of numerical problems.  The Solar System

One of the four halfmodules forming the Level 1 astronomy course, but may also be taken as a standalone module. PHY106 covers the elements of the Solar System: the Sun, planets, moons and minor bodies. What are their structures and compositions, and what dothey tell us about the formation and history of the Solar System?
10 credits  Our Evolving Universe

The course provides a general overview of astronomy suitable for those with no previous experience of the subject. The principal topics covered are (1) how we deduce useful physical parameters from observed quantities, (2) the structure and evolution of stars, (3) the structure of the Milky Way, and the classification, structure and evolution of galaxies in general, (4) an introduction to cosmology and (5) extrasolar plantets and an introduction to astrobiology. All topics are treated in a descriptive manner with minimal mathematics.
10 credits  Social Psychology I

This module will provide an overview of the fundamentals of social psychology. The module will introduce and explain key theories and research, and their application, for understanding social psychological phenomena. Content is organised around two themes: How people think (Semester 1), and how people feel and behave (Semester 2). The module will include lectures that will provide opportunities to learn how to critically evaluate social psychological research and theories, as well as to describe how social psychology theory can be applied to address real world issues.
20 credits  Cognitive Psychology I

This unit provides an overview of core components of cognition, and principles of their investigation. The module covers perception, attention, performance, cognitive neuroscience, language, learning, memory and reasoning. It introduces and explored key concepts, theoretical perspectives and foundational methods. Examples of key studies in cognitive psychology will be considered critically.
20 credits  Neuroscience and Clinical Psychology I

This unit aims to provide students with an understanding of the key principles within neuroscience and clinical psychology. The module will introduce students to the basic structure and function of the brain, techniques and assessments used within neuroscience and clinical psychology, and an awareness of the ethical issues. The module will cover the aetiology, development, assessment and treatment of specific psychological and neurological disorders. Students will develop their knowledge, skills and understanding by attending lectures, engaging with activities/discussions within the lectures and engaging with the reading for this module.
20 credits
Core modules:
 Advanced Calculus and Linear Algebra

Advanced Calculus and Linear Algebra are basic to most further work in pure and applied mathematics and to much of statistics. This course provides the basic tools and techniques and includes sufficient theory to enable the methods to be used in situations not covered in the course.
20 credits
The material in this course is essential for further study in mathematics and statistics.  Analysis

This course is a foundation for the rigorous study of continuity, differentiation and integration of functions of one real variable. As well as providing the theoretical underpinnings of calculus, we develop applications of the theory and examples that show why the rigour is needed, even if we are focused on applications.
20 credits
The material in this course is vital to further studies in metric spaces, measure theory, parts of probability theory, and functional analysis.  Differential Equations

The module aims at developing a core set of advanced mathematical techniques essential to the study of applied mathematics. Topics include the qualitative analysis of ordinary differential equations, solutions of second order linear ordinary differential equations with variable coefficients, first order and second order partial differential equations, the method of characteristics and the method of separation of variables.
20 credits  Statistical Inference and Modelling

This unit develops methods for analysing data, and provides a foundation for further study of probability and statistics at Level 3. It introduces some standard distributions beyond those met in MAS113, and proceeds with study of continuous multivariate distributions, with particular emphasis on the multivariate normal distribution. Transformations of univariate and multivariate continuous distributions are studied, with the derivation of sampling distributions of important summary statistics as applications. The concepts of likelihood and maximum likelihood estimation are developed. Data analysis is studied within the framework of linear models. There will be substantial use of the software package R.
20 credits  Probability Modelling

The course introduces a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might be the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. The aim is to familiarize students with an important area of probability modelling.
10 credits  Scientific Computing and Simulation

The module further develops the students' skills in computer programming and independent investigations. The students will learn how to solve algebraic and differential equations using the solvers in Python as well as Python codes developed by themselves. The students will learn basic computing methods and methods to visualize and analyze numerical results, and then apply the knowledge to explore the physical behaviors of model equations.
10 credits
Optional modules:
 Algebra

This unit continues the study of abstract algebra begun in MAS114, going further with the study of groups, and introducing the concepts of a ring, which generalises the properties of the integers, and a vector space, which generalises the techniques introduced in linear algebra to many more examples.
20 credits
As well as demonstrating the interest and power of abstraction, this course is vital to further studies in most of pure mathematics, including algebraic geometry and topology, functional analysis and Galois theory.  Career Development Skills

This unit will equip students with the necessary skills to support them in gaining employment upon graduation. Students will learn how to construct covering letters, CV writing and complete applications to enhance their success when applying for jobs. Skills such as how to communicate mathematics to nonmathematicians and the need for attention to detail will also be introduced.
10 credits  Mathematics and Statistics in Action

This module will demonstrate, in a series of case studies, the use of applied mathematics, probability and statistics in solving a variety of realworld problems. The module 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, as well as simple computer programming, can be used to explore these problems. There will be a mix of individual and group projects, and some projects will involve the use of R or Python, but MAS115 is not a prerequisite.
10 credits  Mechanics and Fluids

This module extends the Newtonian mechanics studied in MAS112. The main topics treated are (i) extensions of the workenergy principle and conservation of energy, (ii) a full treatment of planetary and satellite motion, (iii) the elements of rigid body motion, and (iv) inviscid (frictionless) fluid motions. The course is a prerequisite for students wishing to pursue higher level modules in fluid mechanics.
10 credits  Ecosystems in a Changing Global Environment

Human impacts on the world¿s ecosystems are profound and without precedent in Earth's history. The urgent need to understand the impacts of overexploitation, landuse change and anthropogenic climate change has meant that ecosystem science has become one of the most important biological disciplines. This module will introduce students to the fundamental principles of ecosystem science by exploring human impacts on key marine and terrestrial ecosystems and their feedbacks on global climate. In doing so, it will cover the interacting roles of (1) climatic tolerance, trophic interactions, carbon sequestration and fire on land, and (2) biodiversity, energy, nutrients and extinction in the sea.
10 credits  Population and Community Ecology 2

This unit will examine major themes in population and community ecology, across plants, animals and their
10 credits
interactions with each other and their environment. It focuses on crosscutting themes in ecology and
evolution including life history, predation, competition, disease and biodiversity. It builds deep, conceptual and
theoretical understanding of life cycles, population growth, and species interactions. It provides insight into
common patterns and unique properties among plants and animals of the factors that determine the
abundance, diversity and distribution of species. It provides insight into the role of species interactions and
the environment in controlling biodiversity and ecosystem function.  Data Driven Computing

This module is intended to serve as an introduction to machine learning and pattern processing, but with a clear emphasis on applications. The module is themed around the notion of data as a resource; how it is acquired, prepared for analysis and finally how we can learn from it. The module will employ a practical Pythonbased approach to try and help students develop an intuitive grasp of the sophisticated mathematical ideas that underpin this challenging but fascinating subject.
20 credits  Logic in Computer Science

This module introduces the foundations of logic in computer science. The first part introduces the syntax and semantics of propositional and predicate logics, natural deduction, and notions such as soundness, completeness and (un)decidability. The second part covers applications in computer science and beyond, such as automated reasoning and decision procedures, modal and temporal logics for the verification of computing systems, and type systems for programming languages.
10 credits  Critical Issues in Teaching

This module introduces you to key issues and roles involved in being a teacher. It is suitable for those who definitely want to teach and those who have not yet considered teaching as a career. The focus of the module is teaching in England. It covers teaching across the age range, with sessions devoted to early years, primary, secondary and further and higher education. The module also deals with issues such as assessing students' learning, managing challenging behaviour, working with parents and other professionals. By the end of the module you should have a clear idea of what's involved in `being a teacher'.
20 credits  Topics in Political Philosophy

This module will investigate a broad range of topics and issues in political philosophy and through doing so provide students with a broad understanding of those. It will include both historical and foundational matters and recent state of the art research.
20 credits  Religion and the Good Life

What, if anything, does religion have to do with a welllived life? For example, does living well require obeying God's commands? Does it require atheism? Are the possibilities for a good life enhanced or only diminished if there is a God, or if Karma is true? Does living well take distinctive virtues like faith, mindfulness, or humility as these have been understood within religious traditions? In this module, we will examine recent philosophical work on questions like these while engaging with a variety of religions, such as Buddhism, Christianity, Confucianism, Daoism, Islam, and Judaism.
20 credits  Social Psychology II

The module continues from the linked first year social psychology module, Social Psychology 1 to cover specific social psychological topics in greater detail and depth. Lectures cover key theories and empirical research in social psychology. Lecture activities provide opportunities for applying social psychological theory and empirical research to explain or solve real world issues.
20 credits  Cognitive Psychology II

This unit builds on the overview of core components of cognition established at level 1 (Cognitive Psychology I). The module covers the same broad topics as Cognitive Psychology II  perception, attention, performance, cognitive neuroscience, language, learning, memory and reasoning. It expands on key concepts and introduced additional details of experimental methods and theoretical nuance. Applications of fundamental science are discussed.
20 credits  Neuroscience and Clinical Psychology II

This unit will build on the content of Neuroscience and Clinical Psychology I, to provide students with a more in depth understanding of principles within neuroscience and clinical psychology. The module will cover the aetiology, development, assessment and treatment of more complex psychological and neurological disorders. Students will develop and build on their knowledge, skills and understanding by attending lectures, engaging with activities/discussions within the lectures and engaging with the reading for this module.
20 credits
Core modules:
 Practical and Applied Statistics

The overall aim of the course is to give students practice in the various stages of dealing with a real problem: objective definition, preliminary examination of data, modelling, analysis, computation, interpretation and communication of results. It could be said that while other courses teach how to do statistics, this teaches how to be a statistician. There will be a series of projects and other exercises directed towards this aim. Projects will be assessed, but other exercises will not.
20 credits  Applied Probability

The unit will link probability modelling to Statistics. It will explore a range of models that can be constructed for random phenomena that vary in time or space  the evolution of an animal population, for example, or the number of cancer cases in different regions of the country. It will illustrate how models are built and how they might be applied: how likelihood functions for a model may be derived and used to fit the model to data, and how the result may be used to assess model adequacy. Models examined will build on those studied in MAS275
10 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  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  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
Optional modules:
 Algebraic Topology

This unit will cover algebraic topology, following on from MAS331: Metric Spaces.
20 credits
Topology studies the shape of space, with examples such as spheres, the Mobius Band, the Klein bottle, the torus and other surfaces. The first task is to formalise this notion of space, and to work out when a given space can be smoothly deformed into another, where stretching and bending is allowed, but cutting and glueing is not. Algebraic topology builds a powerful bridge between space and algebra, enabling the use familiar algebraic techniques from group theory to study spaces and their deformations.  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 Noethers theorem relating symmetries and conservation laws.
20 credits
In the second semester, Einsteins 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.  Commutative Algebra and Algebraic Geometry

This module develops the theory of algebraic geometry, especially over complex numbers, from both a geometrical and algebraic point of view. The main ingredient is the theory of commutative algebra, which is developed in the first part of the module.
20 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 mainly on the study of Hilbert spaces complete vector spaces equipped with an inner product and linear maps between Hilbert spaces. Applications of the theory considered include Fourier series, differential equations, index theory, and the basics of wavelet analysis.
20 credits  Stochastic Processes and Finance

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.
20 credits  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  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  Advanced Operations Research

Mathematical Programming is concerned with the algorithms that deal with constrained optimisation problems. We consider only constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable and so they do not fall into the category of problems considered in organisation; 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)? Additional topics will include the transportation and assignment problems.
10 credits  Analytic Number Theory

The module will discuss the distribution of prime numbers (Bertrand's Postulate, prime counting function, the statement of the Prime Number Theorem and some of its consequences), basic properties of the Riemann zeta function, and Euler products of Lseries. A big chunk of the module will be dedicated to Dirichlet's Theorem on primes in arithmetic progressions and it's proof.
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  Continuum Mechanics

Continuum mechanics is concerned with the mechanical behaviour of solids and fluids which change their shape when forces are applied. For example, rubber extends when pulled but behaves elastically returning to its original shape when the forces are removed. Water starts to move when the external pressure is applied. This module aims to introduce the basic kinematic and mechanical ideas needed to model deformable materials and fluids mathematically. They are needed to develop theories which describe elastic solids and fluids like water. In this course, a theory for solids which behave elastically under small deformations is developed. This theory is also used in seismology to discuss wave propagation in the Earth. An introduction in theory of ideal and viscous, incompressible and compressible fluids is given. The theory is used to solve simple problems. In particular, the propagation of sound waves in the air is studied.
10 credits  Differential Geometry

What is differential geometry? In short, it is the study of geometric objects using calculus. In this introductory course, the geometric objects of our concern are curves and surfaces. Besides calculating such familiar quantities as lengths, angles and areas, much of our focus is on how to measure the 'curvature' of a geometric object. The story is relatively simple for curves, but naturally becomes more involved for surfaces  and more interesting too.
10 credits  Fields

A field is a set where the familiar operations of arithmetic are possible. It often happens, particularly in the theory of equations, that one needs to extend a field by forming a bigger one. The aim of this course is to study the idea of field extension and various problems where it arises. In particular, it is used to answer some classical problems of Greek geometry, asking whether certain geometrical constructions, such as angle trisection or squaring the circle, are possible.
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  Fluid Mechanics I

The way in which fluids move is of immense practical importance; the most obvious examples of this are air and water, but there are many others such as lubricants in engines. Moreover, the scientific principles and mathematical techniques needed to explain fluid motion are of intrinsic interest. This module builds on Level 2 work (MAS222 Differential Equations; MAS280 Mechanics and Fluids). The first step is to derive the equation (NavierStokes) governing the motions of most common fluids. This serves as a basis for the remainder of MAS320, with the main addition to MAS280 being that it covers viscous (frictional) fluids.
10 credits  Galois Theory

Given a field K (as studied in MAS333/MAS438) one can consider the group G of isomorphisms from K to itself. In the cases of interest, this is a finite group, and there is a tight link (called the Galois correspondence) between the structure of G and the subfields of K. If K is generated over the rationals by the roots of a polynomial f(x), then G can be identified as a group of permutations of the set of roots. One can then use the Galois correspondence to help find formulae for the roots, generalising the standard formula for the roots of a quadratic. It turns out that this works whenever the degree of f(x) is less than five. However, the fifth symmetric group lacks certain grouptheoretic properties that lie behind these formulae, so there is no analogous method for solving arbitrary quintic equations. The aim of this course is to explain this theory, which is strikingly rich and elegant.
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  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  History of Mathematics

The course aims to introduce the student to the history of mathematics. The topics discussed are Egyptian and Babylonian mathematics, early Greek mathematics, Renaissance mathematics, and the early history of the calculus.
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  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  Mathematical Biology

This module provides an introduction to the mathematical modelling of the dynamics of biological populations. The emphasis will be on deterministic models based on systems of differential equations that encode population birth and death rates. Examples will be drawn from a range of different dynamic biological populations, from the species level down to the dynamics of molecular populations within cells. Central to the module will be the dynamic consequences of feedback interactions within the populations. 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.
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  Mathematical modelling of natural systems (Advanced)

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  Measure and Probability

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a useful precursor or companion course to the Level 4 courses MAS436 (Functional Analysis) and MAS452 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas
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  Signal Processing

The transmission reception and extraction of information from signals is an activity of fundamental importance. This course describes the basic concepts and tools underlying the discipline, and relates them to a variety of applications. An essential concept is that a signal can be decomposed into a set of frequencies by means of the Fourier transform. From this grows a very powerful description of how systems respond to input signals. Perhaps the most remarkable result in the course is the celebrated ShannonWhittaker sampling theorem, which tells us that, under certain conditions, a signal can be perfectly reconstructed from samples at discrete points. This is the basis of all modern digital technology.
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  Waves

Studying wave phenomena has had a great impact on Applied Mathematics. This module looks at some important wave motions with a view to understanding them by developing from first principles the key mathematical tools. We begin with waves on strings (e.g. a piano or violin), developing the concept of standing and progressive waves, and normal modes. Fourier series are used to solve problems relating to waves on strings and membranes. Sound waves and water waves are considered. The concepts of dispersion and group velocity are introduced. The course concludes with consideration of 'traffic waves' as the simplest example of nonlinear waves.
10 credits  Human Planet

This course examines the historical, social, cultural and political dimensions of sustainability, focusing on food production and natural resource management on the land and in the oceans. Students will learn how key historical developments led to sustainability issues, how geopolitics perpetuates these in the modern world, and how an understanding of these issues can help us to develop more sustainable ways to live in future. Learning will be achieved through lectures and videos, independent study and classroom discussion sessions.
10 credits  Evolution of Terrestrial Ecosystems

This module examines the evolution of terrestrial ecosystems, from the invasion of the land by plants and animals in the Ordovician (475 million years ago) up to the present day. All of the major events will be covered: the origin of land plants; the invasion of the land by invertebrate animals (worms, insects, etc); the first forests; the origin of amphibians, reptiles, mammals and birds; beginnings of phtogeographical differentiation; origin of the flowering plants etc. Throughout the course the evolution of terrestrial ecosystems will be considered in light of: (i) the interrelationships between global change and evolving terrestrial ecosystems; (ii) plantfungalanimal interactions and coevolution.
10 credits  Topics in Evolutionary Genetics

This course aims to provide the opportunity for students to develop (i) their knowledge of current leadingedge research areas in evolutionary genetics and (ii) their skills in accessing, interpreting and synthesising the primary scientific literature in this field. This will be achieved by examining three areas of current research activity in evolutionary genetics though detailed analysis of the questions, methods and interpretations in groups of recent publications.
10 credits  Sustainable AgroEcosystems

This module highlights the threats to global sustainability, with a particular focus on food production and ecosystem functioning, being caused by human impacts on the environment. The module considers how we have got into the present unsustainable mess ¿ of poor land and natural resource management, under valuing of farmers, lifethreatening soil degradation causing flooding, pollution of fresh water and soil insecurity, as well as large numbers of people overconsuming and wasting food whilst others don¿t have enough. It shows that how we sustainably manage agroecosystems now, and in the immediate future, will determine the fate of humanity. Soils are the foundations of terrestrial ecosystems, food and biofuel production, but are amongst the most badly abused and damaged components of the ecosphere. Climate change, agricultural intensification, biofuels and unsustainable use of fertilizers and fossil fuels pose critical threats to global food production and sustainable agroecosystems  and their impacts on soil ecosystems are central to these threats. The module considers soil ecosystems function in nature and the lessons that we can then apply to develop more sustainable agriculture and ecosystem management.
10 credits  Text Processing

This module introduces fundamental concepts and ideas in natural language text processing, covers techniques for handling text corpora, and examines representative systems that require the automated processing of large volumes of text. The course focuses on modern quantitative techniques for text analysis and explores important models for representing and acquiring information from texts. Students should be aware that there are limited places available on this course
10 credits  Adaptive Intelligence

This course will examine the theme of bioinspired Machine Learning and in particular of Unsupervised and Reinforcement Learning in Neural Networks. The first half of the course covers Unsupervised algorithms (Clustering, Principal Component Analysis) that could potentially have biological counterparts in the human or animal brain. The second half of the course introduces the theory of Reinforcement Learning in a simple and intuitive way, and more specifically Temporal Difference learning and the SARSA algorithm. It also discusses stateoftheart methods (Deep Reinforcement Learning).
10 credits  Speech Processing

This module aims to demonstrate why computer speech processing is an important and difficult problem, to investigate the representation of speech in the articulatory, acoustic and auditory domains, and to illustrate computational approaches to speech parameter extraction. It examines both the production and perception of speech, taking a multidisciplinary approach (drawing on linguistics, phonetics, psychoacoustics, etc.). It introduces sufficient digital signal processing (linear systems theory, Fourier transforms) to motivate speech parameter extraction techniques (e.g. pitch and formant tracking). Students should be aware that there are limited places available on this course.
10 credits  Globalising Education

This module considers the extent to which education might be viewed as a global context with a shared meaning. Moving outwards from the dominant concepts, principles and practices which frame 'our own' national, or regional responses to education, the module explores other possible ways of understanding difference. By examining 'other ways of seeing difference', in unfamiliar contexts, students are able to examine the implications of globalisation for education and explore the opportunities and obstacles for the social justice agendas within a range of cultural settings.
20 credits  Pain, Pleasure, and Emotions

Affective states like pain, pleasure, and emotions have a profound bearing on the meaning and quality of our lives. Chronic pain can be completely disabling, while insensitivity to pain can be fatal. Analogously, a life without pleasure looks like a life of boredom, but excessive pleasure seeking can disrupt decisionmaking. In this module, we will explore recent advances in the study of the affective mind, by considering theoretical work in the philosophy of mind as well as empirical research in affective cognitive science. These are some of the problems that we will explore: Why does pain feel bad? What is the relation between pleasure and happiness? Are emotions cognitive states? Are moral judgments based on emotions? Can we know what other people are feeling?
20 credits  History of Astronomy

The module aims to provide an introduction to the historical development of modern astronomy. After a brief chronological overview and a discussion of the scientific status of astronomy and the philosophy of science in general, the course is divided into a series of thematic topics addressed in roughly chronological order. We will focus on the nature of discovery in astronomy, in particular the interplay between theory and observation, the role of technological advances, and the relationship between astronomy and physics.
10 credits  Current Controversies in the Psychology of Addiction

This module will introduce students to controversial issues related to addiction from a psychological perspective. Topics will cover the nature of addiction, its determinants, underlying neurological basis, treatment, and prognosis. During each lecture, the controversial topic will be introduced before relevant evidence is described and critically evaluated. Clinicians and / or service users with direct experience of addiction may contribute to some of these sessions. Topics covered may vary from year to year as new controversies emerge, but are likely to include critical appraisals of the brain disease model of addiction, and the roles of compulsion and habit.
10 credits
Core modules:
 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  Mathematics and Statistics Project II

This unit 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 and Statistics.
30 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  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  Project Presentation in Mathematics and Statistics

The unit provides training and experience 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.
10 credits
Optional modules:
 Algebraic Topology

This unit will cover algebraic topology, following on from MAS331: Metric Spaces.
20 credits
Topology studies the shape of space, with examples such as spheres, the Mobius Band, the Klein bottle, the torus and other surfaces. The first task is to formalise this notion of space, and to work out when a given space can be smoothly deformed into another, where stretching and bending is allowed, but cutting and glueing is not. Algebraic topology builds a powerful bridge between space and algebra, enabling the use familiar algebraic techniques from group theory to study spaces and their deformations.  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 Noethers theorem relating symmetries and conservation laws.
20 credits
In the second semester, Einsteins 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.  Commutative Algebra and Algebraic Geometry

This module develops the theory of algebraic geometry, especially over complex numbers, from both a geometrical and algebraic point of view. The main ingredient is the theory of commutative algebra, which is developed in the first part of the module.
20 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 mainly on the study of Hilbert spaces complete vector spaces equipped with an inner product and linear maps between Hilbert spaces. Applications of the theory considered include Fourier series, differential equations, index theory, and the basics of wavelet analysis.
20 credits  Stochastic Processes and Finance

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.
20 credits  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  Advanced Operations Research

Mathematical Programming is concerned with the algorithms that deal with constrained optimisation problems. We consider only constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable and so they do not fall into the category of problems considered in organisation; 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)? Additional topics will include the transportation and assignment problems.
10 credits  Advanced Particle Physics

The main aim of the unit is to give a formal overview of modern particle physics. The mathematical foundations of Quantum Field Theory and of the Standard Model will be introduced. The theoretical formulation will be complemented by examples of experimental results from the Large Hadron Collider and Neutrino experiments. The unit aims to introduce students to the following topics:
10 credits
• A brief introduction to particle physics and a review of special relativity and quantum mechanics
• The Dirac Equation
• Quantum electrodynamics and quantum chromodynamics
• The Standard Model
• The Higgs boson
• Neutrino oscillations
• Beyond the Standard Model physics  Advanced Quantum Mechanics

This module presents modern quantum mechanics with applications in quantum information and particle physics. After introducing the basic postulates, the theory of mixed states is developed, and we discuss composite systems and entanglement. Quantum teleportation is used as an example to illustrate these concepts. Next, we develop the theory of angular momentum, examples of which include spin and isospin, and the method for calculating ClebschGordan coefficients is presented. Next, we discuss the relativistic extension of quantum mechanics. The KleinGordon and Dirac equations are derived and solved, and we give the equation of motion of a relativistic electron in a classical electromagnetic field. Finally, we explore some topics in quantum field theory, such as the Lagrangian formalism, scattering and Feynman diagrams, and modern gauge field theory.
10 credits  Analytic Number Theory

The module will discuss the distribution of prime numbers (Bertrand's Postulate, prime counting function, the statement of the Prime Number Theorem and some of its consequences), basic properties of the Riemann zeta function, and Euler products of Lseries. A big chunk of the module will be dedicated to Dirichlet's Theorem on primes in arithmetic progressions and it's proof.
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  Continuum Mechanics

Continuum mechanics is concerned with the mechanical behaviour of solids and fluids which change their shape when forces are applied. For example, rubber extends when pulled but behaves elastically returning to its original shape when the forces are removed. Water starts to move when the external pressure is applied. This module aims to introduce the basic kinematic and mechanical ideas needed to model deformable materials and fluids mathematically. They are needed to develop theories which describe elastic solids and fluids like water. In this course, a theory for solids which behave elastically under small deformations is developed. This theory is also used in seismology to discuss wave propagation in the Earth. An introduction in theory of ideal and viscous, incompressible and compressible fluids is given. The theory is used to solve simple problems. In particular, the propagation of sound waves in the air is studied.
10 credits  Differential Geometry

What is differential geometry? In short, it is the study of geometric objects using calculus. In this introductory course, the geometric objects of our concern are curves and surfaces. Besides calculating such familiar quantities as lengths, angles and areas, much of our focus is on how to measure the 'curvature' of a geometric object. The story is relatively simple for curves, but naturally becomes more involved for surfaces  and more interesting too.
10 credits  Fields

A field is set where the familiar operations of arithmetic are possible. It is common, particularly in the study of equations, that a field may need to be extended. This module will study the idea of field extension and the various problems that may arise as a result. Particular use is made of this to answer some of the classical problems of Greek geometry, to ask whether certain geometrical constructions such as angle trisection or squaring the circle are possible.
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, and includes a computational project where students further explore some of the ideas of option pricing.
10 credits  Fluid Mechanics I

The way in which fluids move is of immense practical importance; the most obvious examples of this are air and water, but there are many others such as lubricants in engines. Moreover, the scientific principles and mathematical techniques needed to explain fluid motion are of intrinsic interest. This module builds on Level 2 work (MAS222 Differential Equations; MAS280 Mechanics and Fluids). The first step is to derive the equation (NavierStokes) governing the motions of most common fluids. This serves as a basis for the remainder of MAS320, with the main addition to MAS280 being that it covers viscous (frictional) fluids.
10 credits  Galois Theory

Given a field K (as studied in MAS333/MAS438) one can consider the group G of isomorphisms from K to itself. In the cases of interest, this is a finite group, and there is a tight link (called the Galois correspondence) between the structure of G and the subfields of K. If K is generated over the rationals by the roots of a polynomial f(x), then G can be identified as a group of permutations of the set of roots. One can then use the Galois correspondence to help find formulae for the roots, generalising the standard formula for the roots of a quadratic. It turns out that this works whenever the degree of f(x) is less than five. However, the fifth symmetric group lacks certain grouptheoretic properties that lie behind these formulae, so there is no analogous method for solving arbitrary quintic equations. The aim of this course is to explain this theory, which is strikingly rich and elegant.
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  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  Groups and Symmetry

Groups arise naturally as collections of symmetries. Examples considered include symmetry groups of Platonic solids. Groups can also act as symmetries of other groups. These actions can be used to prove the Sylow theorems, which give important information about the subgroups of a given finite group, leading to a classification of groups of small order.
10 credits  History of Mathematics

The course aims to introduce the student to the history of mathematics. The topics discussed are Egyptian and Babylonian mathematics, early Greek mathematics, Renaissance mathematics, and the early history of the calculus.
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  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  Mathematical Biology

This module provides an introduction to the mathematical modelling of the dynamics of biological populations. The emphasis will be on deterministic models based on systems of differential equations that encode population birth and death rates. Examples will be drawn from a range of different dynamic biological populations, from the species level down to the dynamics of molecular populations within cells. Central to the module will be the dynamic consequences of feedback interactions within the populations. 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.
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  Mathematical modelling of natural systems (Advanced)

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  Measure and Probability

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a companion course to MAS436 (Functional Analysis) and MAS452 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas.
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  Signal Processing

The transmission reception and extraction of information from signals is an activity of fundamental importance. This course describes the basic concepts and tools underlying the discipline, and relates them to a variety of applications. An essential concept is that a signal can be decomposed into a set of frequencies by means of the Fourier transform. From this grows a very powerful description of how systems respond to input signals. Perhaps the most remarkable result in the course is the celebrated ShannonWhittaker sampling theorem, which tells us that, under certain conditions, a signal can be perfectly reconstructed from samples at discrete points. This is the basis of all modern digital technology.
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  Waves

Studying wave phenomena has had a great impact on Applied Mathematics. This module looks at some important wave motions with a view to understanding them by developing from first principles the key mathematical tools. We begin with waves on strings (e.g. a piano or violin), developing the concept of standing and progressive waves, and normal modes. Fourier series are used to solve problems relating to waves on strings and membranes. Sound waves and water waves are considered. The concepts of dispersion and group velocity are introduced. The course concludes with consideration of 'traffic waves' as the simplest example of nonlinear waves.
10 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. We are no longer offering unrestricted module choice. If your course included unrestricted modules, your department will provide a list of modules from their own and other subject areas that you can choose from.
Learning and assessment
Learning
You'll learn through lectures, problems classes, programming classes and research projects.
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
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 Further Maths AAB, including A in Maths + A in a relevant EPQ; AAB, including A in Maths + B in Further Maths
International Baccalaureate  36, 6 in Higher Level Mathematics (Analysis and Approaches) 34 with 6 in Higher Level Mathematics
BTEC  D*DD in a relevant subject with Distinctions in Maths units DDD in a relevant subject with Distinctions in Maths units
Scottish Highers + 1 Advanced Higher  AAAAB + A in Maths AAABB + A in Maths
Welsh Baccalaureate + 2 A Levels  A + AA, including Maths B + AA, including Maths
Access to HE Diploma  60 credits overall in a relevant subject with Distinctions in 39 Level 3 credits, including Mathematics units, + Merits in 6 Level 3 credits 60 credits overall in a relevant subject with Distinctions in 36 Level 3 credits, including Mathematics units, + Merits in 9 Level 3 credits
Mature students  explore other routes for mature students
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

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
We also accept a range of other UK qualifications and other EU/international qualifications.
If you have any questions about entry requirements, please contact the department.
School of Mathematics and Statistics
Staff in the School of Mathematics and Statistics work on a wide range of topics, from the most abstract research on topics 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
A Top 100 university 2021
QS World University Rankings
Top 10% of all UK universities
Research Excellence Framework 2014
No 1 Students' Union in the UK
Whatuni Student Choice Awards 2019, 2018, 2017
School of Mathematics and Statistics
National Student Survey 2019
Graduate careers
School of Mathematics and Statistics
There will always be a place for maths graduates in banking, insurance, pensions, and financial districts from the City of London to Wall Street. Big engineering companies still need people who can crunch the numbers to keep planes in the sky and trains running on time too. But the 21st century has also created new career paths for our students.
Smartphones, tablets, social networks and streaming services all use software and algorithms that need mathematical brains behind them. In the age of ‘big data’, everyone from rideshare apps to high street shops is gathering information that maths graduates can organise, analyse and interpret. The same technological advances have created new challenges and opportunities in cybersecurity and cryptography.
If the maths itself is what interests you, a PhD can lead to a career in research. Mathematicians working in universities and research institutes are trying to find rigorous proofs for conjectures that have challenged pure mathematicians for decades, or are doing the calculations behind major experiments, like the ones running on the Large Hadron Collider at CERN.
What if I want to work outside mathematics?
A good class of degree from a top university can take you far, whatever you want to do. We have graduates using their mathematical training in everything from teaching and management to advertising and publishing.
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.
Visit us
University open days
There are four open days every year, usually in June, July, September and October. You can talk to staff and students, tour the campus and see inside the accommodation.
Taster days
At various times in the year we run online taster sessions to help Year 12 students experience what it is like to study at the University of Sheffield.
Applicant days
If you've received an offer to study with us, we'll invite you to one of our applicant open days, which take place between November 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
Campus tours run regularly throughout the year, at 1pm every Monday, Wednesday and Friday.
Apply for this course
Make sure you've done everything you need to do before you apply.
How to apply When you're ready to apply, see the UCAS website:
www.ucas.com
Contact us
Telephone: +44 114 222 3999
Email: 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.