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    2023 start September 


    School of Mathematics and Statistics, Faculty of Science

    Choose from a variety of advanced lecture modules across pure and applied mathematics, and build the foundations for a career in mathematics research.
    A maths lecturer at a blackboard

    Course description

    This one-year course is designed to help you build the foundations for a successful career in mathematics research. You'll have the freedom to choose from a variety of advanced lecture modules across pure and applied mathematics. Possible topics range from algebra, geometry and topology, to the ways that mathematics can be used in finance or studies of nature.

    You'll be able to get valuable mathematics research experience by working with an experienced mathematician on a dissertation topic of your choice. Throughout the course, you'll have lots of opportunities to improve your problem solving and presentation skills, and learn how to create persuasive and logical arguments.

    Specialist lectures have small class sizes so that they are more informal, with closer interactions between staff and students. We also have an optional directed reading module, individually tailored, to help you develop your understanding in the areas you're most interested in. You'll also be supported through regular meetings with your academic supervisor.


    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.

    Core modules:


    The dissertation is piece of extensive work (10-20,000 words) which provides students' with the opportunity to synthesise theoretical knowledge on a subject that is of interest to them. Students will gain experience of the phases of a relatively large piece of work: planning to a deadline; researching background information; problem specification; the carrying through of relevant analyses; and reporting, both at length through the dissertation and in summary through an oral presentation.

    60 credits

    Optional modules - six or seven from:

    Topics in Advanced Fluid Mechanics

    This module is lectured with the undergraduate module MAS411 of the same name. 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, we describe the three-dimensional flows in terms of vortex dynamics. Minimally required mathematical tools are explained during the course in a self-contained manner. Candidates are directed to read key original papers on some topics to deepen their understanding.

    20 credits
    Special Topics (Autumn)

    There may be cases where students have already covered some of the material in the standard slate of courses, or where students need further background to cover prerequisite material. In addition, MAGIC courses, given to PGT students, may come under this heading.

    10 credits
    Mathematical methods and modelling of natural systems

    This module will develop mathematical methods, including integral transforms, asymptotics and perturbation, applied to problems including differential equations and evaluation of integrals.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 advanced mathematical and computational methods. Study systems will be drawn from throughout the environmental and life sciences.

    20 credits
    Special Topics (Spring)

    There may be cases where students have already covered some of the material in the standard slate of courses, or where students need further background to cover prerequisite material. In addition, MAGIC courses, given to PGT students, may come under this heading.

    10 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
    Algebra I

    This module is lectured with the undergraduate modules MAS438 and MAS442 and considers Fields and Galois Theory. The first half of the module covers basic field theory (field extensions, constructability, etc.) and the second gives the applications to the theory of equations: Galois groups of extensions, and of polynomials, culminating in one of the crowning glories of Galois's work classifying polynomials which are soluble by radicals.

    20 credits
    Algebra II

    This module is lectured with the undergraduate modules on commutative algebra (MAS437 and MAS434), and considers rings and commutative algebra. The first half of the module covers basic ring and module theory (e.g., modules and ideals) and the second develops the theory of commutative algebra, an essential prerequisite for modern algebraic geometry, as well as being important in its own right.

    20 credits
    Analysis I

    This module is lectured with the undergraduate module MAS435 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 course 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
    Analysis II

    This module is lectured with the undergraduate modules MAS331 Metric Spaces and MAS350 Probability and Measure. 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. It also will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory.

    20 credits
    Algebraic Topology I

    This will be a first course on algebraic topology, following on from the introduction to topology given in the Level 3 course 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 then the challenge is 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 us to use familiar algebraic techniques from group theory to study spaces and their deformations.

    20 credits
    Analytical Dynamics and Classical Field Theory

    This module is lectured alongside the undergraduate module MAS412 of the same name. Mathematicians like Lagrange, Hamilton and Jacobi discovered that underlying Newton's mechanics are wonderful mathematical structures. In the first semester we discuss this work, its influence on the subsequent formulation of field theory, and Noether's theorem relating symmetries and conservation laws.In the second semester, Einstein's theory of gravity, General Relativity, will be introduced, preceded by mathematical tools such as covariant derivatives and curvature tensors. Einstein's field equations, and two famous solutions, will be derived. Two classic experimental tests of General Relativity will be discussed.

    20 credits
    Waves and Magnetohydrodynamics

    This module is lectured with the undergraduate modules MAS315 Waves and MAS422 Magnetohydrodynamics. 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. In the second part it gives an introduction to classical magnetohydrodynamics. Magnetohydrodynamics has been successfully applied to a number of astrophysical problems (e.g., to problems in Solar and Magnetospheric Physics), as well as to problems related to laboratory physics, especially to fusion devices.

    20 credits
    Directed Reading in Mathematics

    This module will be used as a reading module for the MSc and may cover material deemed useful by a dissertation supervisor, or background material for a student. Precise content and subject matter will depend on the interests of the student, as well as the level and background of the individual student.

    20 credits
    Special Topics I

    There may be cases where students have already covered some of the material in the standard slate of courses, or where students need further background to cover prerequisite material. In addition, MAGIC courses, given to PGT students, may come under this heading.

    20 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
    Machine Learning

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

    10 credits
    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
    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
    Applied Probability

    The unit will link the idea of 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 probability 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 and suggest improvements.

    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.

    Open days

    An open day gives you the best opportunity to hear first-hand from our current students and staff about our courses. You'll find out what makes us special.

    Online postgraduate open day - Wednesday 26 April 2023


    1 year full-time


    There are lectures and seminars.


    You’re assessed by exams, coursework and a dissertation.

    Your career

    The advanced topics you'll cover and the extensive research training make this course great preparation for a PhD. Sheffield maths graduates have secured postgraduate research positions at many of the world's top 100 universities.

    Mathematics graduates also develop numerical, problem solving and data analysis skills that are useful in many careers. This can help you stand out in job markets where maths graduates thrive, such as computing, banking and data science.

    Employers that have hired Sheffield maths graduates include Amazon, Barclays, Dell, Goldman Sachs, IBM, PwC, Sky, the NHS and the Civil Service.


    A lecturer stood at the front of a seminar by a blackboard

    The School of Mathematics and Statistics is one of the biggest departments at the University of Sheffield. It’s home to more than 50 academic staff with expertise across many areas of pure mathematics, applied mathematics, probability and statistics. We aspire to be an inclusive and welcoming environment for all who enjoy mathematics.

    Our mathematics and statistics researchers work on a wide variety of topics, from the most abstract questions in algebraic geometry and number theory, to the calculations behind infectious disease, black holes and climate change. 

    In the Research Excellence Framework 2021, 96 per cent of our research was rated in the highest two categories as world-leading or internationally excellent.

    Staff in the School of Mathematics and Statistics have received honours from the Royal Society, the Society for Mathematical Biology and the Royal Statistical Society, who also provide professional accreditation for our statistics courses. 

    With the support of the London Mathematical Society, we are an organiser of the Transpennine Topology Triangle – a key focal point for topology research in the UK. 

    We also have strong links with the Society for Industrial and Applied Mathematics, the Institute of Mathematics and its Applications, the European Physical Society and the International Society on General Relativity and Gravitation.

    Entry requirements

    Minimum 2:1 undergraduate honours degree with a substantial maths component.

    We also consider a wide range of international qualifications:

    Entry requirements for international students

    Overall IELTS score of 6.5 with a minimum of 6.0 in each component, or equivalent.

    If you have any questions about entry requirements, please contact the department.


    You can apply for postgraduate study using our Postgraduate Online Application Form. It's a quick and easy process.

    Apply now


    +44 114 222 3789

    Any supervisors and research areas listed are indicative and may change before the start of the course.

    Our student protection plan

    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.

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