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    2023 start September 

    Mathematical and Theoretical Physics

    School of Mathematics and Statistics, Faculty of Science

    Expand your understanding of general relativity and quantum mechanics to tackle some of the biggest ideas in science. You can work with researchers on topics that range from dark matter to quantum computing.
    Students in a maths lecture

    Course description

    This course has been created to equip graduates with advanced mathematical tools that can be applied in major areas of scientific intrigue, from black holes and dark matter to quantum computing.

    There is a wide range of optional modules for you to choose from, so you can focus on the topics that interest you most: general relativity, field theory, quantum mechanics, geometry, electrodynamics, solar physics, particle physics and more.

    You’ll spend around one-third of your time working on your own research project, under the supervision of an expert from our School of Mathematics and Statistics or Department of Physics and Astronomy. This in-depth research experience is great preparation for a PhD. Possible topics include:

    • Cosmology, dark matter and dark energy 
    • Quantum information processing and communication
    • Magnetohydrodynamics in the stellar atmosphere
    • Gravitational wave astronomy, black holes and neutron stars
    • Fluid mechanics and the Kolmogorov theory of turbulence
    • Chaos and fractal boundaries in dynamical systems
    • The search for dark matter at underground laboratories

    There are also modules on machine learning, finance and statistics, so you can develop skills to help you stand out in job markets where maths and physics graduates thrive, such as data science and banking.


    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:

    Analytical Dynamics and Classical Field Theory

    Newton formulated his famous laws of mechanics and gravitation in the 17th century, but only later did it become clear, through the work of mathematicians Lagrange, Hamilton and Jacobi, that underlying Newton's work are deep mathematical structures. In the first semester, we will reformulate and unify mechanics and classical field theory (e.g. Maxwell's equations), starting with the stationary-action principle. We will derive Noether's theorem, which relates the symmetries of a system to conserved quantities in physics (such as energy, momentum and charge). In the second semester, we will explore Einstein's general theory of relativity, in which gravity arises through the curvature of spacetime. We will apply the mathematical tools of Riemannian geometry to explore some radical consequences of the theory, such as black holes, the big bang, and the expanding universe.

    30 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

    Maths options - between 25 and 70 credits from:

    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
    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
    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
    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
    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
    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

    Physics options – between 20 and 65 credits from:

    Advanced Electrodynamics

    This module gives a detailed mathematical foundation for modern electrodynamics, starting from Maxwell's equations, charge conservation and the wave equation, to gauge invariance, waveguides, cavities and antennas, and an introduction to quantum electrodynamics. After a brief recap of vector calculus, we explore the role of the scalar and vector potential, the multi-pole expansion of the field, the Poisson and Laplace equations, energy and momentum conservation of the fields, and waveguides and cavities. After a relativistic treatment of the fields we consider the quantisation of the electromagnetic field modes, the Hamiltonian for the dipole coupling between a field and a radiation emitter, and finally we explore the Aharonov-Bohm effect.

    15 credits
    Advanced Quantum Mechanics

    Quantum mechanics at an intermediate to advanced level, including the mathematical vector space formalism, approximate methods, angular momentum, and some contemporary topics such as entanglement, density matrices and open quantum systems. We will study topics in quantum mechanics at an intermediate to advanced level, bridging the gap between the physics core and graduate level material.  

    The syllabus includes a formal mathematical description in the language of vector spaces; the description of the quantum state in Schrodinger and Heisenberg pictures, and using density operators to represent mixed states; approximate methods: perturbation theory,  variational method and time-dependent perturbation theory;  the theory of angular momentum and spin; the treatment of identical particles; entanglement; open quantum systems and decoherence. The problem solving will provide a lot of practice at using vector and matrix methods and operator algebra techniques.

    The teaching will take the form of traditional lectures plus 

    weekly problem classes where you will be provided with support and feedback on your attempts.

    15 credits
    Advanced Soft Matter and Biological Physics

    Fascinating behaviour of soft matter and biological systems often occurs at thermal energy scales and can be described by statistical mechanical models. In addition, living biological matter is driven out of equilibrium due to internal biochemical sources of energy. Mathematical models and modern advanced experimental techniques are revealing the physical principles underpinning the biological world and the technological possibilities of complex soft materials.Much recent progress in soft matter and biology has been made thanks to the advent of advanced experimental  techniques which we will show are based on elegant physical principles. We will also study the physical principles underpinning the behaviour of complex soft matter and biological materials. We will describe phase transitions in multiple soft matter systems by calculating free energies. We will use random walk models to describe the shape of polymer molecules and the Brownian motion of colloids. We will also study the dynamics of polymers and the kinetics of polymerisation. We will then consider how polymerisation of protein filaments and action of molecular motors can generate forces in biological cells. This will involve us introducing concepts of systems that are in equilibrium versus out of equilibrium. Using a mathematical  framework we can describe behaviour at different length scales for example from the cytoskeleton to tissues, bacteria colonies and flocking. We will also investigate how the energy required for life is captured in photosynthesis.

    15 credits
    Advanced Particle Physics

    The module provides students with a comprehensive understanding of modern particle physics. Focusing on the standard model, it provides a theoretical underpinning of this model and discusses its predictions. Recent developments including the discovery of the Higgs Boson and neutrino oscillation studies are covered. A description of the experiments used to probe the standard model is provided. Finally the module looks at possible physics beyond the standard model.

    15 credits
    Introduction to Cosmology

    The aim of this course is to provide students with an understanding of the Universe as its own entity. Students will learn how the contents of the Universe affect its dynamic evolution, and how we can use observations of Type 1a Supernovae and the Cosmic Microwave Background to constrain the properties of the Universe. Students will also learn about key epochs during the history of the Universe, from inflation through to nucleosynthesis, recombination, and reionisation, before learning how the first stars and galaxies started to form.Throughout a series of lectures, students will first learn that spacetime forms the fabric of the Universe, and how the contents of the Universe in the form of dark energy, dark and baryonic matter, and radiation dictate the dynamic evolution of the Universe. Students will next learn about modern precision cosmology, whereby cosmologists use observations of Type 1a Supernovae and the Cosmic Microwave Background to measure various cosmological parameters. This aspect of the course will form the basis of a computer programming-based assessment. Toward the end of the lecture course, students will learn about the epochs of inflation, nucleosynthesis, recombination and reionisation, before learning how today’s stars and galaxies began to form. Finally, students will learn about current cosmological research via a literature review.

    15 credits
    Quantum Optics and Quantum Computing

    Quantum computing is introduced through the fundamental concepts of quantum gates and circuits before moving to cover more advanced topics such as quantum programming, quantum algorithms and quantum error correction. These concepts are then applied by studying how programming quantum circuits can be done using cloud computers (e.g. using openQASM format) and the implementation of quantum algorithms (including examples) and quantum error correction using stabiliser formalism and graph states and quantum error correction codes.

    The second part of the module covers quantum optics and quantum optical applications at the forefront of current research in the field. This includes topics such as weak and strong coupling of dipole sources in a cavity, single photon sources, protocols of quantum optical communications and linear optics computation. The module then progresses to quantum optical applications. Cavity electrodynamics is studied in the regimes of strong and weak coupling of matter excitations to the electromagnetic field in optical microstructures. This will lead to the physics of highly efficient single photon devices necessary for linear optics quantum computation. The effects of entanglement and quantum teleportation will be also considered.

    15 credits
    Directed reading in Physics and Astrophysics

    This short module gives Masters students the opportunity to explore in detail a topic of interest to them, making use of research literature and/or graduate texts as appropriate.  Students will select the topic (subject to approval by the module leader) and be assigned a supervisor with appropriate expertise.  (Note: suggested topics may have to be rejected if no suitably qualified supervisor is available.)  Students will develop their understanding of the chosen topic by reading appropriate literature, as identified by the student with guidance from the supervisor.  If appropriate to the topic, students may also undertake other activities, particularly coding. 

       The module aims to encourage students to learn independently using research-level sources.  Consequently, there is no formal teaching.  Students are guided by weekly supervision sessions in which supervisors will discuss that week’s reading with the student, provide feedback and suggestions for further reading, help with any points that the student has found difficult to understand, and correct any misconceptions.  Students will keep a study diary in which they keep notes of their sources, explain topics in their own words, and perhaps work through problems.  In some cases, other activities, especially coding, may be undertaken to explore aspects of the topic. 

    5 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.

    Upcoming open days and campus tours


    1 year full-time


    There is series of lectures, tutorials, practical tasks, problem-solving classes and a research project.


    You'll be assessed by examinations, assignments 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 and physics graduates also develop numerical, problem solving and data analysis skills that are useful in many other careers. In particular, our optional modules on machine learning, finance and statistics, mean you can develop skills to help you stand out in job markets where maths and physics graduates thrive, such as computing, banking and data science.

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

    Entry requirements

    Minimum 2:1 undergraduate honours degree in mathematics or physics.

    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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