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    School of Mathematical and Physical Sciences , Faculty of Science

    Choose from a variety of advanced lecture modules across pure and applied mathematics, and build the foundations for a career in pure or applied mathematics.
    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 a directed reading module, individually tailored, to help you develop your understanding in the areas you're most interested in.

    You'll be supported through regular meetings with your academic supervisor.


    A selection of modules is 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:

    Research Skills in Mathematics

    The unit provides training in research-level skills used in mathematics and related technical disciplines, with a particular focus on mathematical/scientific writing and presentation skills. Students will gain experience in the use of appropriate computer packages for the presentation of mathematical and statistical material, and guidance on how to prepare a coherent, structured and accurate report. Topics covered include conducting literature searches, summarising information, accessing papers through preprint servers, and managing reference lists. Students will draw on these skills to write their own literature review in an area of interest to them, and prepare and deliver a talk on this topic.

    15 credits

    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:

    A student will take 105 credits from this group.

    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
    Special Topics

    This module is available to students who have already covered the material in some of the courses offered on the MSc, and who would otherwise have difficulty in finding sufficient credits to complete the programme.  A course of study would be proposed by a member of staff, which may consist, for example, of material from a book, or from a graduate lecture course taken by PhD students.  The student would be expected to write a short report, as part of a portfolio of notes and exercises.

    15 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 project.

    15 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 on Banach spaces and especially Hilbert spaces - complete vector spaces equipped with an inner product - and linear maps between Hilbert spaces. Applications of the theory we examine include Fourier series and the Fourier transform, and differential equations.

    15 credits
    Further Topics in Mathematical Biology

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

    15 credits
    Topics in Mathematical Physics

    This unit will introduce students to advanced concepts and techniques in modern mathematical physics, in preparation for research-level activities.

    It is assumed that the student comes equipped with a working knowledge of analytical dynamics, and of non-relativistic quantum theory.

    We will examine how key physical ideas are precisely formulated in the language of mathematics. For example, the idea that fundamental particles arise as excitations of relativistic quantum fields finds its mathematical realisation in Quantum Field Theory. In QFT, particles can be created from the vacuum, and destroyed, but certain other quantities such as charge, energy, and momentum are conserved (after averaging over quantum fluctuations).

    We will examine links between conservation laws and invariants, and the underlying (discrete or continuous) symmetry groups of theories. We will also develop powerful calculation tools. For example, to find the rate of creation of new particles in a potential, one must evaluate the terms in a perturbative (Feynman-diagram) expansion.

    For details of the current syllabus, please consult the module leader.

    15 credits
    Special Topics 2

    This module is available to students who have already covered the material in some of the courses offered on the MSc, and who would otherwise have difficulty in finding sufficient credits to complete the programme.  A course of study would be proposed by a member of staff, which may consist, for example, of material from a book, or from a graduate lecture course taken by PhD students.  The student would be expected to write a short report, as part of a portfolio of notes and exercises.

    15 credits
    Mathematical Modelling of Natural Systems

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

    15 credits
    Further Topics in Number Theory

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

    15 credits
    Probability with Measure Theory

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

    15 credits
    Directed Reading in Mathematics

    This module serves as a preparation for the MSc dissertation, and will cover material deemed useful by the dissertation supervisor, consisting of specialist 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.  The module will be assessed with a portfolio consisting of a report and exercises set by the supervisor.

    15 credits
    Advanced Topics in Algebra A

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

    30 credits
    Advanced Topics in Waves and Fluid Dynamics A

    Waves and Fluid Dynamics are cornerstones of Applied Mathematics.  Both relate to the flow of fluids, i.e., propagation of information, which include not only gas (e.g., air) and liquid (e.g., water), but also in more complex media (e.g., lubricants and blood), and other materials or even the fourth state of matter: plasma.  The scientific principles and mathematical techniques involved in studying these are of inherent interest.  Wave motions give rise to well-known class of partial differential equations, and relate to concepts such as standing, progressive, and shock (i.e., nonlinear) waves; we can study these using Fourier series, Laplace transform and the powerful method of characteristics.  Viscous fluid flow gives rise to the Navier-Stokes equations.  The first semester will cover some of these ideas, while the second will move onto more advanced topics, such as three-dimensional flows, boundary layers, vortex dynamics, or magnetohydrodynamics.

    30 credits
    Algebraic Topology

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

    30 credits
    Analytical Dynamics and Classical Field Theory

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

    30 credits
    Stochastic Processes and Finance

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

    30 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.

    Open days

    An open day gives you the best opportunity to hear first-hand from our current students and staff about our courses.

    You may also be able to pre-book a department visit as part of a campus tour.Open days and campus tours


    1 year full-time


    You’ll be taught via a variety of lectures and small group seminars.


    Our assessment methods are designed to support the achievement of learning outcomes and develop your professional skills. This includes coursework, exams and a dissertation.

    Regular feedback is also provided, so you can understand your own development throughout the course.

    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.


    School of Mathematical and Physical Sciences

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

    The School of Mathematical and Physical Sciences is leading the way with groundbreaking research and innovative teaching. We provide our students with the skills and knowledge to support them in a wide range of careers.

    Mathematicians at the University of Sheffield have expertise across many areas of pure mathematics, applied mathematics, probability and statistics.
    We want mathematics and statistics students to feel part of a community, and 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 mathematical sciences research was rated in the highest two categories as world-leading or internationally excellent.

    Mathematics and statistics staff 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 now using our Postgraduate Online Application Form. It's a quick and easy process.

    Apply now

    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.