Mathematics and Statistics MMath
School of Mathematics and Statistics
Explore this course:
You are viewing this course for 2023-24 entry. 2024-25 entry is also available.
Key details
- A Levels AAA
Other entry requirements - UCAS code G110
- 4 years / Full-time
- September start
- Accredited
- Find out the course fee
- Optional placement year
- Study abroad
Course description
This course is designed to prepare you for a career as a statistics researcher, where you can help solve problems in areas ranging from banking and finance to healthcare and medicine. You’ll be trained to use the tools and software professional statisticians use and complete a major research project in your final year.
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 will start to develop programming skills using Python and R, which can be applied to lots of jobs that involve data. You’ll 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. You’ll also learn how to apply your statistics knowledge to increasingly complex problems through statistical modelling and computer simulations. There are optional modules on topics including algebra, mechanics and fluids.
Some modules 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 the third year, you’ll have the skills, knowledge and experience to go in lots of different directions. There is training in how to design experiments and collect data, how statistics are used in clinical trials of new drugs and how data can predict the likely outcome of an event. 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.
Accredited by the Royal Statistical Society (RSS) for the purpose of eligibility for Graduate Statistician status
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: G110
Years: 2022, 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 higher-dimensional generalisation. This material is central to many topics in subsequent courses. At the same time, weekly small-group tutorials with the Personal Tutor aim to develop core skills, such as mathematical literacy and communication, some employability skills and problem-solving 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. You’ll also learn how to apply your statistics knowledge to increasingly complex problems through statistical modelling and computer simulations. There are optional modules on topics including algebra, mechanics and fluids.
A full list of modules for this year will be available soon.
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 infinite-dimensional 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 three-dimensional 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 self-contained 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 self-originated 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 post-optimality 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 L-series. 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 'error-correcting code' (more accurately, an error-detecting 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 real-life 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 far-reaching 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 complex-valued 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 Black-Scholes 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 Black-Scholes 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 (Navier-Stokes) 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 group-theoretic 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 counter-intuitive. 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, anti-dynamo 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 post-optimality 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, super-conductors, 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 Shannon-Whittaker 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
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 infinite-dimensional 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 three-dimensional 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 self-contained 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 post-optimality 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 chromo-dynamics
- 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 Clebsch-Gordan coefficients is presented. Next, we discuss the relativistic extension of quantum mechanics. The Klein-Gordon 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 L-series. 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 'error-correcting code' (more accurately, an error-detecting 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 real-life 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 far-reaching 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 complex-valued 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 Black-Scholes 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 Black-Scholes 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 (Navier-Stokes) 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 group-theoretic 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 counter-intuitive. 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, anti-dynamo 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 post-optimality 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, super-conductors, 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 Shannon-Whittaker 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 up-to-date and relevant. Individual modules are occasionally updated or withdrawn. This is in response to discoveries through our world-leading 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.
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 a relevant subject 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
Other requirements-
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 a relevant subject 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
Other requirements-
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.
School of Mathematics and Statistics
Staff in the school 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
QS World University Rankings 2023
92 per cent of our research is rated as world-leading or internationally excellent
Research Excellence Framework 2021
Top 50 in the most international universities rankings
Times Higher Education World University Rankings 2022
No 1 Students' Union in the UK
Whatuni Student Choice Awards 2022, 2020, 2019, 2018, 2017
A top 10 university targeted by employers
The Graduate Market in 2022, High Fliers report
School of Mathematics and Statistics
Research Excellence Framework 2021
Student profiles
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.
Placements and study abroad
Placement
Study abroad
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
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 post-16 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 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
Not ready to apply yet? You can also register your interest in this course.
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.
Any supervisors and research areas listed are indicative and may change before the start of the course.