Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning.

You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem.

Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving techniques across multiple disciplines, preparing you for advanced studies while developing your analytical reasoning abilities.

Statistics allows us to estimate trends and patterns in data and gives a principled way to quantify uncertainty in these estimates. The findings can lead to new insights and support decision-making in fields as diverse as cyber security, human behaviour, finance and economics, medicine, epidemiology, environmental sustainability and many more.

Dive into the behaviour of multivariate random variables and asymptotic probability theory, both of which are central to statistical inference. You will then be equipped to explore one of the most fundamental statistical models, the linear regression model, and learn how to apply general statistical inference techniques to multi-parameter statistical models. Statistical computing is embedded in the module, allowing you to investigate multivariate probability distributions, simulate random data, and implement statistical methods.

Researching, collaborating, writing and presenting are key skills for all students. Collaborating with fellow students, you will investigate a chosen mathematical or statistical subject and produce a report and presentation to share your findings. As part of this, you will learn how to format and structure scientific reports and papers, use specialised documentation software like LaTeX, conduct research, cite and reference sources.

Continuing with your study into real numbers, you will explore their completeness (the idea that there are no ‘gaps’, unlike in the rationals). This completeness will be used to understand the limits of sequences, convergence of series, and power series.

This framework will allow for precision when exploring continuity, differentiability, and integrability of functions of a real variable, providing an improved foundation for calculus. That will enable you to understand when it is appropriate to use calculus; for instance, in proving theorems in other areas of mathematics, such as mathematical physics, probability and number theory.

The cornerstone of mathematical analysis is the construction of proofs involving arbitrarily small numbers, so-called epsilons and deltas. You will have opportunities to practise and improve your management of these quantities, in the process developing your skills in logic, communication and problem-solving.

Ever wondered about the hidden structures that govern mathematics? Algebra is more than just equations, it's the language of symmetry and structure, underpinning subjects ranging from geometry and quantum mechanics to number theory and cryptography. The main frameworks for modern algebra are group theory and ring theory.

Group theory topics include classifying symmetries, the symmetric group, Lagrange's theorem and the first isomorphism theorem. Similarly, ring theory explores the notions of subrings, ideals, and homomorphisms in an example-driven methodology, using abstract number systems, polynomial structures, and matrices.

This module introduces the essential theory and techniques for algebra, laying a solid foundation for further study in mathematics, physics, and other related fields.

Never has the collection of data been more widespread than it is now. The extraction of information from massive, often complex and messy, datasets brings many challenges to fields such as statistics, mathematics and computing.

Develop the skills and understanding to apply modern statistical and data-science tools to gain insight from contemporary data sets. By addressing challenges from a variety of applications, such as social science, public health, industry and environmental science, you will learn how to perform and present an exploratory data analysis, deploy statistical approaches to analyse data and draw conclusions, as well as developing judgement to critically evaluate the appropriateness of chosen methods for real-world challenges.

The success of Newton/Leibniz’s calculus raises the question: what happens if we replace the real numbers with the complex numbers? Afterall, their arithmetic structure is similar, and we can measure distances between points in both. You will learn how to define the derivative of a complex function as usual and explore the behaviour of functions that are complex differentiable. Everything resembles the real case, ultimately leading to the astonishing result that if a complex function can be differentiated once, it can be differentiated infinitely often and is expressed by its Taylor series. Integral calculus for complex functions opens a route towards evaluating definite integrals that cannot be reached by real variables.

Applications of these results include a proof of the fundamental theorem of algebra, which states that every non-constant complex polynomial has a root.

This module lays foundations for further studies of mathematical analysis, pure and applied.

Machine learning is at the heart of modern AI systems, and it is a fundamentally mathematical subject. You will learn this mathematics by discovering how techniques are deployed in several AI systems, including the neural networks that have revolutionised the field.

You’ll start by building connections with previously encountered approaches through the unifying concept of a loss function of a parameter vector. For example, with a neural network model the vector input is the set of weights, and the loss function might be the prediction error on a dataset.

The goal is to find a vector input that produces a small loss; in the above example, this is known as training the neural net. You will learn and deploy some of the key mathematical ideas and numerical techniques, such as back propagation and stochastic gradient descent, that enable the automated iterative learning of a good vector input.

Many real-world problems give rise to integrals and ordinary differential equation (ODEs) that cannot be solved analytically. To start, you will be introduced to techniques for tackling such problems, beginning with fundamental numerical methods, such as the trapezium rule and Euler’s method, before progressing to more advanced techniques and quantifying the accuracy, stability and limitations of these methods. Alongside numerical approaches, you will also develop heuristic methods to characterise a system's limiting behaviour.

Many familiar phenomena, from the pulses of light down a fibre optic cable to the shudder of turbulence on a plane, involve multiple variables, such as time and position. Their mathematical description requires differentiation with respect to each of these independent variables, leading to partial differential equations (PDEs). You will learn how to formulate PDEs for complex, real-world problems and practice core techniques for solving them.

Differential equations are fundamental to mathematical modelling, with multiple applications in engineering, biology, and the environment.

Explore both ordinary and partial differential equations (ODEs and PDEs), with an introduction to advanced solution techniques and real-world applications. You will understand the deeper theory of ODEs and how solutions lead to special functions through series expansion, including Bessel functions. You will be introduced to Fourier series as a foundational tool in modern science.

Shifting your focus to PDEs, you will classify second-order equations and solving key equations in diverse geometries, noting the importance of boundary conditions. Applications will include acoustics, pollution dispersal, groundwater flow and forms of medical imaging. By the end of the module, you will have mastered analytical methods for solving differential equations and gained the intuition to interpret their solutions in scientific and engineering contexts.

Commutative rings generalise both integers and polynomials and they play a very important role in a wide area of mathematics. As well as being important in algebra, they sit at the heart of algebraic approaches including geometry and number theory, in part because rings of functions occur so naturally there, as they do in analysis. At this stage, you will already know how to factor and divide integers and polynomials. Therefore, a crucial question is to understand the factorisability and divisibility properties in more general commutative rings. For example, what is the analogue of the set of prime integers, or which are the invertible elements?

You will seek to answer these questions, beginning by looking at rings with certain properties and finding the key examples of these, continuing to describe several constructions that allow us to produce rings with properties we would like. You will conclude by discussing the applications to the areas mentioned above.

An introduction to a variety of methods that are useful for analysing environmental data, such as air temperatures, rainfall or wildfire locations. Spatial dependence is a key feature of many environmental datasets, and the Gaussian process will be introduced as a model for continuous spatial processes. You will learn about the properties of the Gaussian process and implement this model for spatial data analysis, before investigating methods for point-reference data, such as earthquake or wildfire locations.

You will also dip into natural hazard risk management, which seeks to mitigate the effects of events, such as flooding or storms, in a manner that is proportionate to the risk. You will learn basic concepts from extreme value theory, including the appropriate distributions for extremes, and how to use these as statistical models for estimating the probability of events more extreme than those in the dataset.

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance.

You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proofs of correctness and efficiency of algorithms.

You will gain experience in following and constructing mathematical proofs, correctly and coherently using mathematical notation, and choosing and carrying out appropriate algorithms to solve problems. The module will enable you to develop an appreciation for a range of discrete mathematical techniques.

An inner product space is a real or complex vector space, equipped with certain extra structure that formalises the geometrical notion of orthogonality. It turns out that each inner product space has an intrinsic notion of distance, allowing us to discuss convergence and completeness. Complete inner product spaces are known as Hilbert spaces.

The theory of Hilbert spaces blends linear algebra and (real) analysis. It is a natural and powerful tool for studying problems of quantitative approximation. Furthermore, it provides an abstract framework that can be applied to diverse areas of maths, from differential equations and spectral theory to quantum mechanics and stochastic processes.

This module will introduce you to the theory of Hilbert spaces and prepare for advanced study in functional analysis, approximation theory, signal processing, and statistical learning.

Knots play a fundamental role in many areas of mathematics, from pure topology and algebra through to quantum field theory and protein-folding.

Develop tools to measure knottedness, including geometrical ideas like curvature, knot invariants like the Jones polynomial, and the crucial concept of the fundamental group, which has applications in topology far beyond detecting knots.

Linear systems of differential and integral equations provide a mathematical model for a wide range of real-world devices, including communication systems, 5G networks, electrical circuits, heating systems and economic processes. Mathematical analysis of these models gives insight into the behaviour of these devices, with applications in automatic control, signal processing, wireless communications and numerous other areas.

Linear systems are considered in continuous time that reduce to a standard (A,B,C,D) state space representation. Via the Laplace transform, these are reduced further to rational transfer functions. Linear algebra enables us to classify and solve (A,B,C,D) models, while we describe their properties via diagrams in standard computer software. You will consider feedback control for linear systems, describing the rational controllers that stabilise an (A,B,C,D) system. Alongside the development of analytic methods to study linear systems, you will also gain experience in modelling real-world devices by such systems.

The module commences by looking at classical methods of encryption, discussing their advantages, disadvantages and efficiency. You will also investigate statistical attacks on these methods of encryption and the need for better methods.

After this, you will explore modern methods of encryption that are used in the real-world and rely on the robustness of modular arithmetic. While most encryption methods are still considered secure, you will review potential attacks on these systems (e.g. factorisation algorithms) and situations where bad key generation or implementation has occurred.

Production of a big enough quantum computer renders the above schemes useless. Therefore, you will dive into a short introduction to post-quantum cryptography, including the production of next-gen cryptographic schemes considered to be impenetrable to both classical and quantum computers. You will also explore the theory of lattices and see how these can be used to produce new schemes that may be quantum secure (e.g. NTRU).

Consider the key issues in the teaching and learning of mathematics. Develop an excellent foundation for a PGCE by engaging with educational literature and gain experience in writing academically.

Having studied mathematics for many years, you will be well-placed to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. Throughout this module, you will prepare to become a mathematics graduate who can contribute knowledgeable to future debates about the ways in which maths is treated within the education system.

Put theory into practice as you participate in a semester-long, part-time placement in a local primary or secondary school. During your placement you will have the opportunity to take part in classroom observation and assistance, develop classroom resources, host one-on-one or small group support sessions and possibly even teaching parts of a lesson to the class.

This module is recommended if you have an interest in, or are curious about, a career in teaching or educational research. During regular meetings, an academic will help you to develop your skills in critical reflection as you relate your placement experiences to theoretical frameworks. Alongside your personal development, it is also imperative that you provide genuine assistance to local teachers by bringing your mathematical knowledge and enthusiasm into their classroom to help encourage future mathematicians.

Lay the mathematical foundations necessary to model certain transactions in the world of finance. You will study some stochastic models for financial markets and investigate the pricing of European and American options and other financial products.

You will consider two discrete models, the binomial model and finite market model, and one continuous model. In particular, you’ll deduce the Black Scholes formula following an introduction to some probabilistic terminology, such as sigma algebras and martingales, and some financial terminology such as arbitrage opportunities and self-financing trading strategies. You will also gain a brief overview of Brownian motion.

AI is suddenly everywhere and the methods for training and using AI tools are fundamentally mathematical, and fascinating in their own right. By understanding what goes on ‘under the hood’ you will open up a plethora of exciting opportunities for both further study and employment.

We will introduce you to the deep learning architectures used in modern AI. You will investigate how different architectures work with different data types and tasks, and what the computationally specified architectures actually mean in a modelling sense.

However, deep neural nets need more than just an appropriate architecture; they need to be both trained and deployed. You’ll study the interesting maths at each of these stages: the most recent approaches to loss function minimisation, and the techniques to sequentially learn and adapt to new data and observations, a critical component of modern AI methods.

Statistical methods play a crucial role in health research. This module introduces you to the key study designs used in health investigations, such as randomised controlled trials and various types of observational study.

Issues of study design will be covered from both a practical and theoretical perspective, aiming to identify the most efficient design which adheres to ethical principles and can be carried out in a feasible amount of time, or using a feasible number of patients. Various approaches to controlling for confounding will be discussed, including both design and analysis-based methods. You will also explore different types of response data, including introducing time-to-event data and the resulting challenges presented by censoring.

Real-world studies and published articles will be used to illustrate the concepts, and reference will be made to the ICH guidelines for pharmaceutical research and STROBE guidelines for epidemiological studies.

A metric space consists of a set, whose elements are called points, and a notion of distance between points governed by three simple rules, abstracted from basic properties of Pythagorean distance in the Euclidean plane. In examples, ‘points’ may be functions where uniformity of convergence can be captured, or binary sequences with applications in computer science, or even subsets of a Euclidean space delivering fractal sets as limits.

Topology goes further, abstracting the notions of continuity and convergence, rendering a teacup and doughnut indistinguishable. A topological space equips each of its ‘points’ with its so-called ‘neighbourhoods’. The few simple principles governing these unlock a robust theory that now pervades the mathematical sciences and theoretical physics.

You will learn the fundamental concepts of completeness, total boundedness for metric spaces, compactness, and the Hausdorff property and metrisability for topological spaces.

Study the structure of intricate mathematical objects, such as groups and rings, by looking at linear approximations of them. Linear approximation is such a fundamental idea that it extends throughout mathematical sciences, cropping up in quantum physics and topological data analysis.

Explore representations of finite groups before passing to algebras and modules, which are ‘vector spaces’ over rings. You will look at the atomic theory of representations: the simple and indecomposable representations that are their building blocks. Can we describe all the building blocks? Attempting to answer this leads us to complete reducibility for representations of finite groups (Maschke's theorem) and to representations of directed graphs. We see how Jordan Normal Form in linear algebra is a first theorem of representation theory and methods for complete classification of the building blocks when complete reducibility fails, culminating in connections to Lie theory.

Building on the statistical techniques explored so far, you will gain an understanding of both the theoretical underpinnings and practical application of frequentist statistical inference. You will then be introduced to an alternative paradigm: Bayesian statistics.

The frequentist perspective views all probabilities in terms of the proportions of outcomes over repeated experimentation and has been the foundation of hypothesis testing and experimental design in years of data-driven science and research. Meanwhile, the increasingly popular Bayesian approach arises directly from Bayes theorem, avoiding hypothetical repeated sampling. As a result, Bayesian statistics is often more intuitive and easier to communicate and naturally takes all forms of uncertainty into account.

With this in mind, you will compare and contrast these two perspectives and their associated tools. You will learn to select and justify an appropriate methodology for inference and model selection, and to reason about the uncertainty in your findings within each paradigm.

Stochastic processes are fundamental to probability theory and statistics and appear in many places in both theory and practice. For example, they are used in finance to model stock prices and interest rates, in biology to model population dynamics and the spread of disease, and in physics to describe the motion of particles.

During this module, you will focus on the most basic stochastic processes and how they can be analysed, starting with the simple random walk. Based on a model of how a gambler's fortune changes over time, it questioned whether there are betting strategies that gamblers can use to guarantee a win. We will focus on Markov processes, which are natural generalisations of the simple random walk, and the most important class of stochastic processes. You will discover how to analyse Markov processes and how they are used to model queues and populations.

Statistics and machine learning share the goal of extracting patterns or trends from very large and complex datasets. These patterns are used to forecast or predict future behaviour or interpolate missing information. Learn about the similarities and differences between statistical inference and machine learning algorithms for supervised learning.

You will explore the class of generalised linear models, which is one of the most frequently used classes of supervised learning model. You will learn how to implement these models, how to interpret their output and how to check whether the model is an accurate representation of your dataset. Lastly, you will have the opportunity to see how these models can be extended to the case of the ‘large p, small n’ question. This phrase refers to the situation in which there are many more variables than there are samples, something which is now commonplace.