MSc Mathematical and Theoretical Physics
The theories of general relativity and quantum mechanics have helped scientists and mathematicians understand nature and the universe at the most fundamental level. They can explain some of the biggest scientific discoveries of the century, from gravitational waves to the Higgs boson, and have brought us ever closer to a 'theory of everything'.
This one-year masters course has been created to equip graduates with advanced mathematical tools that can be used to explain the complex physical principles that describe the world around us. There is a wide range of optional modules for you to choose from, so you can focus on the topics that excite 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 the University of Sheffield's School of Mathematics and Statistics or Department of Physics and Astronomy. This in-depth research experience is great preparation for a PhD. 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 computing and banking.
If you are looking for a masters course that covers more of the practical skills needed to run experiments, visit our particle physics course webpage: MSc(Res) Particle Physics
To apply for this course, complete the University of Sheffield's postgraduate online application form.
You can find more information about the application process on the University's postgraduate webpages.
Deadlines for 2019 entry
Students requiring visas: Friday 2 August
Course Director: Dr Sam Dolan
For general queries contact:
You can also visit us throughout the year:
|About the course||
This one-year course gives you the freedom to choose from a selection of topics in applied mathematics, theoretical physics, statistics and data science. These modules are designed so that you can study the fundamental theories that govern the universe, and develop mathematical skills to explain and predict physical phenomena.
Topics include general relativity, quantum mechanics, particle physics and solar physics, alongside other options that range from geometry and probability to biological physics and modelling natural systems. There are also modules on machine learning, if you are interested in a career in computing, or finance, if you plan to enter the financial sector after graduation.
The biggest part of your degree will be your research project. You'll choose your own topic, and work closely with a member of academic staff from the School of Mathematics and Statistics or the Department of Physics and Astronomy, who is an expert in the area you want to explore. Possible topics include:
You will also take part in a research training programme that teaches you how to interpret and evaluate research papers, and how to communicate your own findings.
|After your degree||
The advanced topics covered and the extensive research training make this degree programme great preparation for a PhD:
Mathematics and physics graduates also develop numerical, problem solving and data analysis skills that are useful in many careers, such as computing or finance. Below are some examples of the kinds of roles and organisations our students end up in.
The University's Careers Service runs workshops on CV and application writing, job hunting and preparing for interviews. They offer events where you can meet employers, and opportunities to get work experience while you study. The Careers Service will even continue to support you for three years after you graduate.
For this course, we usually ask for an upper second class (2:1) degree in mathematics or physics.
We can also accept qualifications from other countries. You can find out which qualifications we accept from your country on the University's webpages for international students.
English Language Requirements
If you have not already studied in a country where English is the majority language, it is likely that you will need to have an English language qualification. We usually ask for:
You can find out whether you need to have an english language qualification, and which other English language qualifications we accept, on the University's webpages for international students.
The English Language Teaching Centre offers English language courses for students who are preparing to study at the University of Sheffield.
|Funding and scholarships||
Funding is available, depending on your fee status, where you live and the course you plan to study. You could also qualify for a repayable postgraduate masters loan to help fund your studies.
Up-to-date fees can be found on the University of Sheffield's webpages for postgraduate students:
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The modules listed below are examples from the current academic year. There may be some changes before you start your course.
Compulsory modules – students take both:
|Dissertation (60 credits)||
Module leader: Dr Paul Mitchener
The dissertation gives the student the opportunity to study an advanced mathematical topic in depth and write a dissertation about what they have learned.
The student selects a topic offered by a member of staff and writes a dissertation on this under the supervision of the member of staff.
|Research Skills (30 credits)||
Module leader: Dr Matthew Mears
This module is designed to allow students to explicitly reflect on various aspects of the research process and its communication. Students will be required to keep a diary of their project and reflect on their progress; write a literature review of the project area reflecting on how and why they chose their sources; reflect on the process of learning a new skill for their project; communicate what their research is about and why it is important to a general audience; consider how to teach what they are researching at undergraduate level.
Optional modules – students take 90 credits:
|Advanced Electrodynamics (10 credits)||
Module leader: Dr Pieter Kok
In this course, our starting point is Maxwell's equations, after a brief recap of vector calculus. We describe the electric and magnetic fields in terms of potentials, and present two general classes of solving Maxwell's equations. We treat fields in macroscopic media, waveguides and cavities, and we end with the relativistic formulation of electrodynamics.
|Advanced Particle Physics (10 credits)||
Module leader: Professor Dan Tovey
The module provides students with a comprehensive understanding of modern particle physics. Focussing 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.
|Advanced Quantum Mechanics (10 credits)||
Module leader: Dr Pieter Kok
In this course, we will build quantum mechanics from the ground up, starting with linear vector spaces and Hilbert space. We introduce the postulates of quantum mechanics and explore important topics in modern quantum mechanics such as mixed states, decoherence, entanglement, and quantum teleportation. We also give a thorough treatment of spin and orbital angular momentum. In the second part, we look at many body problems in quantum mechanics. We explore the physics of identical particles, and apply this theory to many-electron atoms, spin waves in solids, and atoms in an optical cavity. Finally, we will briefly introduce the basic principles behind quantum field theory.
|Biological Physics (10 credits)||
Module leader: Dr Rhoda Hawkins
This module will introduce students to biological physics, that is, the application of principles and tools from physics to biological systems. Biological materials are often soft condensed matter with properties between those of simple liquids and solids. In addition biological matter is usually out of equilibrium due to internal biochemical sources of energy. Students will begin to explore the world of biological cells and biopolymer macromolecules, such as DNA. They will see how physics can help understand biological systems through mathematical models and experimental imaging techniques and how this can lead to new physics and applications in biology.
|Field Theory and General Relativity (20 credits)||
Module leader: Dr Sam Dolan
Newton formulated his famous laws of mechanics in the late 17th century. Only later it became obvious through the work of mathematicians like Lagrange, Hamilton and Jacobi that underlying Newton's work are wonderful mathematical structures. In the first semester, the work of Lagrange, Hamilton and Jacobi will be discussed and how it has later affected the formulation of field theory. We will also discuss Noether's theorem, which relates symmetries of a system to the conservation law of certain quantities (such as energy and momentum). In the second semester, Einstein's theory of gravity, General Relativity, will be introduced. The physical principles of General Relativity and mathematical concepts from differential geometry presented. Some consequences of this theory, such as black holes and the expanding universe, will be discussed.
|Geometry I (20 credits)||
Module leader: Dr Kirill Mackenzie
The aim of this module is to introduce the students to the theory of differential geometry, of crucial importance in modern mathematical physics, and to give some applications involving optics and symplectic geometry.
|Machine Learning (10 credits)||
Module leader: Dr Frazer Jarvis
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. Although other aspects of machine learning will be mentioned, the module focuses on the problem of classification; other topics in machine learning are covered by modules in Computer Science.
|Mathematical Methods of Modelling Natural Systems (20 credits)||
Module leader: Dr Alex Best
Part 1: This part of the 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 and the solution of differential equations. 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.
Part 2: Mathematical modelling enables insight into a wide range of scientific problems. This part 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.
|Measure and Probability (10 credits)||
Module leader: Professor David Applebaum
Measure theory is that branch of mathematics which evolves from the idea of "weighing" a set by attaching a non-negative number to it which signifies its worth. This generalises the usual physical ideas of length, area and mass as well as probability. It turns out (as we will see in the course) that these ideas are vital for developing the modern theory of integration.
The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory.
|Statistical Physics (10 credits)||
Module leader: Dr Buddhapriya Chakrabarti
Statistical physics is the derivation of the thermal properties of matter using the underlying microscopic Hamiltonians. The aims of this course are to introduce the techniques of statistical mechanics, and to use them to describe a wide variety of phenomena from physics, chemistry and astronomy.
|Stochastic Processes and Finance (20 credits)||
Module leader: Dr Nic Freeman
A stochastic process is a mathematical model for a randomly evolving system. In this course we study several examples of stochastic process and analyse their behaviour. We apply our knowledge of stochastic processes to mathematical finance, in particular to asset pricing and the Black-Scholes model.
|Topics in Advanced Fluid Mechanics (20 credits)||
Module leader: Professor Koji Ohkitani
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.
|Waves and Magnetohydrodynamics (20 credits)||
Module leader: Dr Rekha Jain
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.
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 timetable teaching across the whole of our campus, the details of which can be found on our campus map. Teaching may take place in a student’s home department, but may also be timetabled to take place within other departments or central teaching space.
Scientist explains the physics of time travel
Dr Pieter Kok, who leads the physics teaching on this course, takes inspiration from his favourite sci-fi films to explain time travel.