MSc Mathematics
Our masters degree in mathematics builds on the core knowledge of pure and applied maths from your undergraduate degree. You'll get to choose modules that add depth to your understanding of the theory, and complete your own research project. The skills you'll gain will help you build a bridge to a great career or further study at PhD level.
ApplyTo apply for this course, complete the University of Sheffield's postgraduate online application form. 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 |
ContactIf you have any questions about this course, contact our Postgraduate Support Officer, Fiona Maisey. E: postgradmaths-enquiry@sheffield.ac.uk You can also visit us throughout the year: Course Director: Dr Paul Mitchener |
About the course |
This 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. |
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Entry requirements |
For this course, we usually ask for an upper second class (2:1) degree in mathematics, from a three- or four-year course, or equivalent. We can also accept equivalent qualifications from other countries. You can find out which qualifications we accept from your country on the University's webpages for international students. Prospective international students: Your country 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. |
Fees and funding |
Up-to-date fees and funding opportunities can be found on the University of Sheffield's webpages for postgraduate students. These may include scholarships for home and international students and a 10% discount for University of Sheffield graduates. |
Current modules
The modules listed below are examples from the current academic year. There may be some changes before you start your course.
Compulsory modules:
Dissertation |
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. |
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Optional modules – students take six:
Algebra I |
Module leader: Professor Vladimir Bavula This module 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. |
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Algebra II |
Module leader: Dr Paul Johnson This module will develop both the algebraic and geometric theories of commutative rings and modules. The most basic form of interaction between these two subjects can be seen as the relationship between polynomials (seen as elements in a ring) and their graphs. This relationship can then be extended to the relationship between certain kinds of ideals in a ring and the geometric object ("graph") such an ideal describes. At a basic level, this module can be seen as the study of turning algebra into pictures and describing pictures using algebra. To do so, we will study many important properties of commutative rings and their modules, and then explore the geometric objects that arise from various algebraic properties. Interpreted in the context of complex numbers, this analogy between algebra and geometry reflects many of the basic intuitions one has about graphs of polynomial equations, but we will also consider the geometry that comes about in more exotic situations, such as over finite fields. |
Algebraic Topology I |
Module leader: Professor John Greenlees In this course, we will study spaces from a topological point of view. This means we will be interested in some notion of the "shape" of a space rather than distances between points, so the emphasis will no longer be on metrics. We will show how to generalise the notion of metric space to achieve this, giving the notion of topological space. Our examples will include balls, spheres, the n-holed torus, the Möbius strip, the Klein bottle, other surfaces, knots, projective spaces. We will define what it means for two spaces to be homeomorphic, and introduce the more subtle and expressive notion of homotopy equivalence, with some interesting examples. We will study two methods for using algebra to analyse the properties of a space: the fundamental group, and homology. For any space we will define a group, called the fundamental group, which is a beautiful and powerful way of using algebra to detect topological features of spaces; for example we can sometimes use the fundamental group to check whether two spaces are homotopy equivalent. We will calculate the fundamental groups of a number of spaces and give some applications, including a proof of the Fundamental Theorem of Algebra, and the classification of surfaces. In the second part of the course we will study homology groups, which give a more tractable method than homotopy groups for studying higher-dimensional properties of spaces. |
Analysis I |
Module leader: Dr Paul Mitchener 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. |
Analysis II |
Module leader: Professor David Applebaum 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 MAS6340 (Analysis I) and MAS6052 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas. In the first semester, ideas of convergence of iterative processes are explored 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. Semester 2 studies "measure theory", a 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. |
Analytical Dynamics and Classical Field Theory |
Module leader: Professor Carsten van de Bruck 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. |
Directed Reading in Mathematics |
Module leader: Dr Paul Mitchener 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. |
Geometry I |
Module leader: Dr Simon Willerton 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. |
Mathematical Methods of Modelling Natural Systems |
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. |
Special Topics (Autumn / Spring / I / 4) |
These units will allow well-qualified students, who are already familiar with material in existing modules, or for whom it is educationally appropriate, to undertake study in a specialist area. Possibilities for areas include further study of algebra, analysis, topology and magnetohydrodynamics. |
Stochastic Processes and Finance |
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 behavior. 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 |
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 |
Module leader: Professor Robert von Fay-Siebenburgen 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 (eg, 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.