Mathematics MMath

Core modules:

Introduction to University Mathematics

This core module is designed to consolidate A-level material and explore topics in mathematics that you'll use throughout your degree. You'll also be introduced to core skills, such as mathematical literacy, communication and problem-solving.  

Throughout this module you'll develop a strong foundation in core mathematics. You'll consider techniques for solving equations, special functions, calculus, vectors, complex numbers, and finite and infinite series.

20 credits

Geometry, Matrices and Multivariate Calculus

This core module is designed to further develop your knowledge of the core mathematics you'll use across your degree.

You'll learn about two-dimensional coordinate geometry, discussing the theory of matrices geometrically and algebraically. You'll also define and evaluate derivatives and integrals for functions that depend on more than one variable, with an emphasis on functions of two variables.

Throughout this module you'll continue to develop your employability skills, exploring the career options open to mathematics graduates. You'll also work with your coursemates to undertake a group project on sustainability.

20 credits

Foundations of Pure Mathematics

The module aims to give an overview of basic constructions in pure mathematics; starting from the integers, we develop some theory of the integers, introducing theorems, proofs, and abstraction.  This leads to the idea of axioms and general algebraic structures, with groups treated as a principal example.  The process of constructing the real numbers from the rationals is also considered, as a preparation for “analysis”, the branch of mathematics where the properties of sequences of real numbers and functions of real numbers are considered.

20 credits

Probability and Data Science

Probability theory is branch of mathematics concerned with the study of chance phenomena. Data science involves the handling and analysis of data using a variety of tools: statistical inference, machine learning, and graphical methods. The first part of the module introduces probability theory, providing a foundation for further probability and statistics modules, and for the statistical inference methods taught here. Examples are presented from diverse areas, and case studies involving a variety of real data sets are discussed. Data science tools are implemented using the statistical computing language R.

20 credits

Optional modules:

A student will take 40 credits (two modules) from this group.

Mathematical Investigation Skills

This module introduces topics which will be useful throughout students’ time as undergraduates and in employment. These skills fall into two categories: computer literacy and presentation skills.  One aim of this module is to develop programming skills within Python to perform mathematical investigations.  Students will also meet the typesetting package LaTeX, the web design language HTML, and Excel for spreadsheets.  These will be used for making investigations, and preparing reports and presentations into mathematical topics.

20 credits

Mathematical modelling

Mathematics is the language of science.  By framing a scientific question in mathematical language, it is possible to gain deep insight into the empirical world.  This module aims to give students an appreciation of this astonishing phenomenon.  It will introduce them to the concept of mathematical modelling via examples from throughout science, which may include biology, physics, environmental sciences, and more.  Along the way, a range of mathematical techniques will be learned that tend to appear in empirical applications.  These may include (but not necessarily be limited to) difference and differential equations, calculus, and linear algebra.

20 credits

You can also select 20 credits of Languages for All modules.

Core modules:

Core Mathematics I

Linear algebra and calculus are fundamental to most advanced work in pure and applied mathematics and to much of statistics.

This module will provide you with basic tools and techniques from linear algebra and calculus. You'll also develop an understanding of the theory underpinning these, enabling you to use these methods in a variety of situations beyond the module.

Building on your first year, you'll also continue to develop employability skills throughout this module.

20 credits

Core Mathematics II

This module will develop your understanding of theory covered in first year modules. 

You'll explore analysis, which underpins core concepts across the mathematical sciences.  You'll examine why familiar tools, like differentiation and integration, actually work, allowing us to prove their formal properties. This rigorous foundation in analysis will enable you to tackle more complex problems in the future.

You'll extend your understanding of ideas from calculus to higher dimensions, considering differentiation of functions of many variables as linear transformations.

You'll also have opportunities to reflect on social, ethical, and historical aspects of mathematics, enriching your understanding of the importance of mathematics in the modern world.

20 credits

Statistical Inference and Modelling

Statistical inference and modelling are at the heart of data science, a field of rapidly-growing importance in the modern word.  This module develops methods for analysing data, and provides a foundation for further study of probability and statistics at higher Levels.  You will learn about a range of standard probability distributions beyond those met at Level 1, including multivariate distributions. You will learn about sampling theory and summary statistics, and their relation to data analysis. You will discover how to parametrise various types of statistical model, learn techniques for determining whether one model is 'better' than another for understanding a dataset, and learn how to ascertain how good a statistical model is at explaining trends in data. The software package R will be used throughout.

20 credits

Differential equations

Differential equations are perhaps the most important tool in applied mathematics. They are foundational for modelling all kinds of physical and natural phenomena, including fluids and plasmas, populations of animals or cells, cosmological objects (via relativity), subatomic particles (via quantum mechanics), epidemics, even political and social opinions have been modelled using differential equations. This module will build on the tools learned at Level 1 for analysing differential equations, extending them in a variety of ways. This may include topics such as bifurcation analysis, partial differential equations (which are particularly valuable for modelling things that vary in both space and time), and the effects of boundaries on the dynamics of differential equations. it will provide the foundation for essentially all applied maths modules taught at Levels 3 and 4.

20 credits

Optional modules:

You'll take 40 credits (two modules) from this group.

Algebra

This module will further develop your understanding of abstract algebra. The theory of groups, met at Level 1, will be developed in more generality, with further theory as well as applications to combinatorial problems.

You'll also meet the abstract concepts of rings and fields.  A ring is similar to a group, but where we have two operations, of addition and multiplication, like the integers, while a field also has a division operation.  You'll see concrete examples of both, realising that a number of particular situations you've previously encountered are examples of this more general concept, and that results you've seen before are examples of abstract results that hold in wider generality.  Finally, you will see that the results about vectors that you have seen in core modules can be generalised to the setting of vector spaces over arbitrary fields.

20 credits

Stochastic Modelling

This module examines stochastic processes. These are models of natural and physical processes that incorporate randomness, to reflect the way that life can change unpredictably over time. 

You'll explore a number of general models for processes where the state of a system is fluctuating randomly. This might include the length of a queue, the size of a reproducing population, or the amount of payouts on insurance policies. 

You'll learn various techniques for the analysis of these models, preparing you for further study of stochastic processes and probability in later years.

20 credits

Dynamics, Fluids and Relativity

This module builds on first-year mathematical modelling. You'll develop sophisticated models of the universe, grow your skills with vectors and calculus for two or more variables, and learn to translate physical phenomena into solvable mathematical equations.

You'll explore the motion of point particles in two dimensions and rigid body rotation. Using Newton's laws, you'll study the power of vectors, and explore the consequences of the conservation of energy and angular momentum in orbital dynamics.

You'll investigate fast-moving bodies and the speed of light, re-imagine classical concepts of absolute time and space, and explore the radical implications of time dilation and length contraction. 

You'll study continuous systems, such as fluids and electromagnetic fields. By applying mass conservation and momentum balance, you'll derive the equations governing the behavior of ideal fluids.

20 credits

Scientific Data Processing and Python

This module further develops practical programming and data analysis using Python. 

You'll learn to select and apply suitable Python functions and libraries, develop efficient programming workflows, and perform statistical analysis on data sets of varying sizes. 

We'll focus on fitting complex models, interpreting results, and evaluating analytical reliability. You'll also gain experience assessing and integrating external Python code, and build the skills to adapt, reuse, and document code responsibly.

20 credits

You can also select 20 credits of Languages for All modules.

Core module:

Mathematical and Physical Sciences Projects and Professional Skills

Through this module you'll hone the skills and knowledge required of a graduate-level professional.

You'll undertake extended project work, which will include relating project work to the literature, setting project aims and objectives, planning and carrying out the work, and reporting it using disciplinary conventions.

You'll investigate how your academic studies relate to either research, society, or industry. You'll develop an understanding of where your degree could lead you and reflect on your career ambitions.You'll also undertake activities to develop the professional skills needed to complete applications for employment or further study.

40 credits

Optional modules:

You'll take 80 credits (four modules) from this group.

Metric Spaces, Topology and Measure

This module will explore how the notion of the metric allows measuring distances on sets of functions, matrices, or even graphs and networks. 

You'll learn to generalise the concepts of convergence and continuity. You'll see how this theory has far-reaching implications, from existence of solutions to differential equations, to equilibria in financial markets.

You'll study open, closed, and compact sets, first relying on metrics, before exploring how objects can be defined in the more general setting of a topological space. You'll also develop an understanding of a consistent theory of 'size' (or volume) and probability.

Through this module, you'll gain the knowledge needed to tackle advanced topics in analysis, while content on topological spaces theory will also prepare you for topics in algebraic topology and differential geometry.

20 credits

Fields and Galois Theory

The main goal of this module is to prove the wonderful classification of Galois of those polynomials whose roots can be expressed by radicals, that is, in terms of square, cube and higher roots.

You'll first enhance your understanding of fields, met in your second year, with some further theoretical results, such as studying ruler-and-compass constructions, and the classical problems of antiquity: doubling the cube and trisecting an angle. Next, you'll consider extensions of fields, where one field contains another. The Galois group of a field extension is a sort of symmetry group: if the roots of a polynomial can be expressed by radicals, then the Galois group of the field extension containing the roots of the polynomial has to have a particular structure, and you'll see that this is not generally true for quintic polynomials.

20 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

Medical and Actuarial Statistics

This module covers statistical ideas that are important in medicine and the actuarial sector. 

You'll learn about the design and statistical analysis of clinical trials used to license new drugs. These trials have their own distinct methodology, due to ethical and regulatory constraints involved in experimentation on human subjects.

You'll develop an understanding of survival analysis, which involves analysing 'time-to-event' data. An example of this could include how long an individual lives and the factors that may increase or decrease life expectancy.

You'll also explore statistical methods with actuarial applications, such as extreme value theory (EVT), which considers the likelihood and magnitude of rare events.

Throughout the module you'll implement statistical methodologies using packages in the programming language R.

20 credits

Bayesian and Computational Inference

This module will develop your understanding of the Bayesian approach to statistical inference, which is fundamentally different to the approach taken in earlier statistics modules.

The Bayesian method is more general and more powerful. While widely used, it relies on modern computers for much of its implementation.  It's based on the idea that if we take a (random) statistical model, and condition this model on the event that it generated the data that we actually observed, then we will obtain a better model.

Through this module, you'll explore the foundations of Bayesian statistics and the incorporation of prior beliefs. You'll study the computational tools important in modern applied statistics, including those important for Bayesian inference such as Markov Chain Monte Carlo. You'll also learn to implement computational methods using R and Python.

20 credits

Time Series and Generalised Linear Models

This module introduces two important areas of modern statistics.

Time series are observations made in time, for which the time aspect is potentially important for understanding and use. You'll be introduced to modern methods of time series analysis and forecasting, as applied in economics and finance, environmental sciences, medical and social sciences. You'll gain practical techniques for data analysis and a firm basis for practical modelling.

Generalised linear models extend linear models, such as regression-type models, in order to accommodate non-normal (non-Gaussian) observations. Through this module, you'll be introduced to generalised linear models and explore inference, including model building and goodness of fit. 

You'll also learn to implement computational methods using a programming language such as R.

20 credits

Mathematical Physics and Analytical Dynamics

This module will provide a deep dive into the core mathematical tools that we use in physics.

You'll learn how to set up a mathematical model of a physics problem using the variational principle and Lagrangian dynamics, and solve some of the resulting differential equations. 

You'll discover how linear finite and infinite dimensional spaces of functions and operators are used in physics, particularly in quantum mechanics but also in other areas of mathematical physics.

You'll also explore how to work with complex functions, which are indispensable for classical and quantum field theories.

20 credits

Quantum Theory and Electrodynamics

This module develops your understanding of quantum theory from the five postulates to operator algebras, uncertainty relations and entanglement theory.

You'll explore the structure of electrodynamics as a relativistic theory, uncovering a special symmetry called gauge invariance and exploring the behaviour of electromagnetic fields at boundaries and in different materials.

20 credits

Mathematical Biology

This module focuses on the mathematical modelling of biological phenomena.

You'll learn how to construct, analyse and interpret mathematical models, using a combination of differential equations, scientific computing and mathematical reasoning.

You'll focus 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.

You'll investigate the dynamic consequences of feedback interactions within biological systems. In cases where explicit solutions are not readily obtainable, you'll learn to use techniques that give a qualitative picture of the model dynamics (including numerical simulation).

20 credits

Waves and Fluid Dynamics

This module will introduce you to the mathematical ideas used to describe waves and fluids, which play a central role in many physical systems. 

You'll explore wave motion in nature, to illustrate wave propagation, dispersion, and energy transport in fluids. You'll consider shock wave solutions to simple nonlinear partial differential wave equations, learning how basic differential equations can be used to model real-life phenomena. 

Starting from basic conservation principles, you'll also see how simple mathematical models arise for fluid flows, introducing  Euler's equations and Navier-Stokes equations. You'll learn how to identify the dominant physical effects in different situations and choose suitable models.Through this module, you'll develop analytical techniques and modelling skills that are used in applied mathematics and beyond, in areas such as engineering, climate and finance.

20 credits

Game Theory and Optimisation

In this module, you'll study two topics in operations research, which involve developing strategies to make decisions about the optimal ways to respond to certain situations. 

You'll explore game theory, learning about optimal responses to competitive situations.

You'll also study optimisation, by finding and analysing optimal solutions to certain kinds of mathematical problems.

Through this module, you'll develop an understanding of strategies with far-reaching applications, from engineering and computing to economics and business management.

20 credits

Combinatorics and Graph Theory

In this module, you'll investigate the mathematics of combinatorics and graph theory, and use them in a wide range of applications.

Combinatorics is the mathematics of selections and combinations. You'll explore counting and pairing problems from across mathematics, developing a toolbox of techniques to solve problems, such as the inclusion-exclusion principle and binomial coefficients.

Graph theory involves the study of graphs, which are a simple mathematical structure consisting of a collection of points, some pairs of which are joined by lines. You'll develop an understanding of their basic nature and explore how they can be used to illustrate a wide range of situations.

20 credits

Number Theory and Cryptography

This module will develop your understanding of number theory, which involves the study of integers, primes and equations.

You'll explore topics such as linear and quadratic congruences, Fermat's Little Theorem and Euler's Theorem, the RSA cryptosystem, quadratic reciprocity, perfect numbers, and continued fractions.

You'll also study codes and cryptography. Codes are used to store information reliably, while cryptography is used to transmit information securely.

In addition to the RSA cryptosystem, you'll also see other examples where number theory is used in the construction of cryptographic protocols.

20 credits

Geometry, Knots and Surfaces

This module explores various ways that mathematicians model shapes like curves and surfaces in two and three dimensions.

You'll study differential geometry, considering curves in the plane and surfaces in three-dimensional space as rigid things. You'll look at how we can model how 'curved' they are and learn mathematical notions of curvature. You'll also look at the interplay between geometric intuition of curvature and the formal mathematical definition.

You'll consider knotted curves and surfaces in three-dimensional space as 'floppy' rather than 'rigid' and consider the notion of algebraic invariants (analogous to curvature), which can be used to distinguish shapes. You'll also meet the Jones polynomial of knots and the Euler characteristic of surfaces.

20 credits

Machine Learning

This module provides a practical toolkit for modelling and understanding complex datasets by integrating modern AI tools. 

Bridging computer science, statistics, and physics, you'll learn to implement industry-standard algorithms, from linear models to deep neural networks, using Python and modern libraries like Keras and scikit-learn.

You'll not only learn to code, but also effectively co-pilot with AI agents for debugging and code generation, preparing you for the future of technical work.

We'll focus on intuition and implementation. You'll learn to understand when to use a method, how to implement it, and why it works, enabling you to solve real-world problems in science, finance, and business.

20 credits

Undergraduate Ambassadors Scheme in Mathematics

This module provides an opportunity for Level Three students to gain first hand experience of mathematics education through a mentoring scheme with mathematics teachers in local schools. Typically, each student will work with one class for half a day every week for 11 weeks. The classes will vary from key stage 2 to sixth form. Students will be given a range of responsibilities from classroom assistant to the organisation and teaching of self-originated special projects. Only a limited number of places are available and students will be selected on the basis of their commitment and suitability for working in schools.

20 credits

Core module:

Research Project (60 credits)

Optional modules:

A student will take 60 credits (four modules) from this group.

General Relativity

Einstein's theory of General Relativity is one of the most accurate and successful theories in physics, and stands as one of the foundational pillars of modern physics. In this module you will learn how General Relativity is built up, starting with the Equivalence Principle and how it leads to the fundamental laws of General Relativity, namely the Einstein equations. You will study the solutions to these equations, including Schwartzschild black holes, the Robertson-Walker expanding universe and gravitational waves. You will study aspects of differential geometry, which is the mathematical framework of General Relativity, and encounter objects such as the metric tensor and the Riemann curvature tensor. Finally, you will learn about the two predictions of General Relativity that convinced the world that the theory is, in essence, correct: the bending of light around stars and the anomalous precession of Mercury's orbit.

15 credits

Fundamental Physics from Symmetries

In this module you will learn how symmetries under rotations, translations, and Lorentz boosts lead to the mathematical structure of the fundamental physical theories of Nature. We will develop the formalism of Lagrangian densities and prove Noether's theorem that links symmetries to physical conservation laws. We introduce Lie theory, which provides an elegant framework for capturing the consequences of symmetries for our physical theories. You will apply this framework to examples that include scalar fields such as the Higgs field, vector fields such as Yang-Mills fields with additional gauge symmetries, and classical Dirac spinor fields. Finally, you will explore the role of symmetry breaking.

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

Quantum Field Theory

In this module you will learn how the fundamental theories of physics are turned into quantum field theories via a process called 'canonical quantisation'. The extended version of Noether's theorem is proved, which determines the conserved quantities in quantum field theory. You will study the free scalar quantum field and the interacting scalar field, which leads to the Feynman rules and diagrams used in particle scattering calculations. You will encounter the Poincaré group, which will lead to the formulation of the Lagrangian of quantum electrodynamics (QED) and its Feynman rules. You will discover how the infinities that crop up in the theory can be avoided using renormalisation theory.

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 and Random Graphs

Random graphs were studied by mathematicians as early as the 1950s. The field has become particularly important in recent decades as modern technology gives rise to a vast range of examples, such as social and communication networks, or the genealogical relationships between organisms. This course studies a range of models of random trees, graphs and networks, alongside probabilistic ideas that are needed to analyse their different properties. The precise material covered in this module may vary according to the lecturer's interests.

15 credits

- Algebraic Topology (15 credits)
- Further Topics in Algebra (15 credits)
- Computational Finance (15 credits)
- Advanced Bayesian Statistics (15 credits)
- Fluids and Magnetohydrodynamics (15 credits)
- Scientific Computing (15 credits)
- Ecological and Environmental Statistics (15 credits)
- Mathematics of Data Science (15 credits)
- Dynamics and Chaos (15 credits)
- Methods in Numerical Modelling (15 credits)
- Asymptotic Methods and Perturbation Theory (15 credits)