Mathematics with a Year Abroad MMath
School of Mathematics and Statistics
You are viewing this course for 2021-22 entry. 2022-23 entry is also available.
Your study will include core mathematics, pure mathematics, applied mathematics and probability and statistics. You can specialise later in your course, and can often switch between our degrees.
Your third year is spent at a university in Australia or North America learning mathematics. You'll continue your studies at Sheffield in the fourth year, where you'll complete an extended individual project.
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: G106
- 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
- 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 2-dimensional co-ordinate 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
- 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
- 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
- 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. The material in this course is essential for further study in mathematics and statistics.20 credits
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. 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.20 credits
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. The material in this course is vital to further studies in metric spaces, measure theory, parts of probability theory, and functional analysis.20 credits
- 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
- 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 real-world problems. The module will illustrate the process of mathematical and statistical modelling, whereby real-world 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 work-energy 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
- 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
You will spend your third year studying physics at one of our partner universities in Australia, Canada, Hong Kong, New Zealand, Singapore or the USA.
- Study Abroad
Contact department for more information.120 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
- 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
- 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
- Algebraic Topology
This unit will cover algebraic topology, following on from MAS331: Metric Spaces. 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.20 credits
- Analytical Dynamics and Classical Field Theory
Newton formulated his famous laws of mechanics in the late 17th century. Later, mathematicians like Lagrange, Hamilton and Jacobi discovered that underlying Newton's work are wonderful mathematical structures. In the first semester we discuss this work, its influence on the subsequent formulation of field theory, and Noethers theorem relating symmetries and conservation laws. 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.20 credits
- 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
- 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
- Medical Statistics
This module introduces an important application of statistics: medical research, specifically, the design and analysis of clinical trials. For any new drug to be approved by a regulator (such as the Medicines and Healthcare products Regulatory Agency in the UK) for use on patients, the effectiveness of the drug has to be demonstrated in a clinical trial. This module explains how clinical trials are designed and how statistical methods are used to analyse the results, with a particular focus on ‘survival’ or ‘time-to-event’ analysis.15 credits
- Sampling Theory and Design of Experiments
Whereas most statistics modules are concerned with the analysis of data, this module is focussed on the collection of data. In particular, this module considers how to collect data efficiently: how to ensure the quantities of interest can be estimated sufficiently accurately, using the smallest possible sample size. Three settings are considered: sample surveys (for example when conducting an opinion poll), physical experiments, as may be used in industry, and experiments involving predictions from computer models, where there is uncertainty in the computer model prediction.15 credits
- 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: • 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 physics10 credits
- 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
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
- 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
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 dynamo10 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
- Optics and Symplectic Geometry
This course is an introduction to some of the areas of pure mathematics which have evolved from the mathematical study of optics. Optics provides a unifying thread, but no prior knowledge of the properties of light is required. Mathematical topics covered include symplectic structures on vector spaces, symplectic maps and matrices, Lagrangian subspaces and characteristic functions and, if time permits, an introduction to the Maslov class and/or Symplectic manifolds. In terms of optics we cover Gaussian, linear and geometrical optics and (if time permits) an introduction to aberration.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
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
You'll learn through lectures, problems classes in small groups and research projects. Some modules also include programming classes.
This tells you the aims and learning outcomes of this course and how these will be achieved and assessed.
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:
The A Level entry requirements for this course are:
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
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.
Why 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
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
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.
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.
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.
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 run regularly throughout the year, at 1pm every Monday, Wednesday and Friday.
Apply for this course
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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.