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.

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.

An introduction to two essential concepts in modern computing systems, cyber security and data engineering. We explore the building blocks of the Authentication, Authorisation and Accountability (AAA) framework, including access control models, security policies and mechanisms. You will review the main categories of existing cryptosystems to understand their security properties, discuss basic concepts of systems security, study the common approaches and tools that attackers use and gain first-hand experience tackling the weaknesses that can be present in real-world systems through guided work in a highly controlled, small-group practical lab.

You will gain a practical and theoretical background in the design, implementation and use of database management systems. This will incorporate the consideration of information quality and security, Entity-Relationship Models, the relational model and the data normalisation process, and alternative schema definitions, SQL, NoSQL and Object-Oriented data models, big data, as well as transaction processing and concurrency control.

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.

Delve into the key principles of artificial intelligence (AI), touching on the core concepts and philosophy of AI and discussing its presence and ethical challenges in the modern world. Throughout, you will unearth the underlying principles of search spaces, knowledge representation and inference logic that form the core of rule-based systems, before learning the principles of machine learning, clustering, classification, linear regression and neural networks.

From this, you will have the grounding necessary to progress to modules in topics such as machine learning, computer vision, and NLP. You will also gain a deeper understanding of computational problem solving, exploring the very nature of computability, including non-deterministic polynomial (NP) complexity classes such as NP-hard, NP-complete and problems which cannot be solved. Be introduced to classical algorithmic approaches to problem solving including divide and conquer, recursion, and parallel approaches, exploring their relative merits for different classes of problem.

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.

Extended reality (XR) refers to the interactive technologies that blend virtual and physical worlds into a hybrid environment or immersive experience. The technology is based on multi-modal platforms that integrate the use of wearable computing. In this module, you will explore different uses of extended reality within the Reality-Virtuality Continuum and identify the needs and means of augmenting human senses.

You will take an applied approach to the design, implementation, deployment, and evaluation of systems that are used to create an XR environment and deliver an immersive experience. To do this, you will study the latest trends in research, emerging technologies, and novel tools, with an analytical focus that assesses the socio-ethical impacts that may result from widespread usage of XR. A key topic will be the computer graphics technology that enables extended realities to exist visually, exploring the fundamental concepts related to visual content generation through relevant theory and practice using current game engines.

The internet and the world wide web have now invaded every aspect of our lives, from ecommerce and entertainment to logistics and social media. Increasingly, application software is no longer written for specific devices, but for internet web browsers. The internet has replaced operating systems as the de-facto platform for application development, making an already global phenomenon truly impactful.

This module explores the various approaches to the development of internet applications, investigating both the client and server-sides, and discussing the trade-off of performance, scalability, privacy and trust associated with these approaches. You will review the role of ‘cloud infrastructures’ (federated distributed computation) in the provision and management of internet applications. Through interactive lectures and practical sessions, you will study common frameworks for client-side application development and create and deploy an internet application from first principles.

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.

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.

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.

You will spend this year working in a graduate-level placement role. This is an ideal opportunity to gain experience in an industry or sector that you might be considering working in once you graduate.

Although it's up to you to find your placement we'll support you all the way. Our Careers Service will provide guidance on CVs, applications, interview techniques and creating a digital profile.

Dive into alternative programming language paradigms, beyond imperative and object-oriented programming. Emphasis is placed on functional programming languages and their unique constraints and features, such as more expressive type systems, immutability, pure functions and side-effects, lambdas, higher order functions, currying, map/reduce and pattern matching. You will also explore why functional languages bring about increased reliability and scalability and how they are now experiencing a resurgence within the software industry. Through hands-on laboratory sessions, you will learn a functional programming language, such as Haskell, and see how functional programming concepts are being integrated in mainstream programming languages, such as Java, Python and JavaScript, to create versatile multi-paradigm programming environments.

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.

Computer vision is a branch of artificial intelligence which aims to build computer-based systems that can interpret and draw meaning from digital images. This module digs into the fundamentals of image formation, information relating to the human visual system, and image interpretation methodologies including convolution, edge detection and feature extraction, and comparison. You will tackle key problems in current research, including semantic segmentation, object detection and three-dimensional image interpretation. You will cover a range of approaches, from low-level image processing to convolutional neural networks. At the end of the module, you will be equipped to construct software components that implement contemporary image processing and computer vision algorithms and recognise issues within computer vision in order to develop and evaluate solutions.

Digital Health explores the utilisation of digital technologies in healthcare. These technologies have an ever-growing role to play in improving health systems and public health, as well as increasing and improving access to health services. Discover the practical applications, implications, and how to enable technologies of digital health. You will survey sensor technologies that permit remote and automated patient monitoring and study the technologies and processes that enable patient-driven healthcare. You will also investigate the structure of health data in electronic health records and methods for the evaluation of digital health solutions. Alongside these applied topics, you’ll also learn about data governance and the ethical issues surrounding digital health technologies, policy, and regulation.

Models of dynamical systems are fundamental to our understanding of the physical and natural world.

Explore a new class of model for the time evolution of a dynamical system and investigate Markov jump process models for real-world systems, such as the evolution of species populations in the wild and the spread of infectious diseases. Using these processes, you will learn how to simulate and study methods understanding their properties and behaviours. Unlike deterministic differential equation models, Markov jump processes are random, allowing for different behaviour every time they are simulated. You will discover how it is often possible to associate a jump process with a related differential equation approximation and that this can provide important insights into the behaviour of the jump process and the original real-world system.

Distributed systems are the foundation upon which modern large-scale infrastructures are built, such as Cloud and service-oriented architectures (also known as ‘as a service’). You’ll investigate the cryptographic techniques used to build such systems, and secure distributed systems themselves.

You’ll study the design approaches to constructing a secure distributed system, including the common vulnerabilities and attack surfaces associated with distributed systems, and the widely adopted design patterns used to mitigate them. To ensure the correctness of such systems, you will be introduced to formal verification techniques covering system specification and the verification of their correctness. This is imperative for systems that form the foundation of modern infrastructures or when we require security guarantees in mission-critical scenarios. Formal languages are used to define precise system specifications, and automated verification techniques verify their correctness. The languages enable the modelling of distributed systems and algorithms, and the verification of properties to prove their correctness.

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.

All programming languages are based on theoretical principles of formal language theory. In this module, you dive deep into formal languages representation and grammars, and how they relate to programming language compilers and interpreters. You will study formal language syntax and semantics, phrase structure grammars, and the Chomsky hierarchy. You will learn how to classify languages and explore the concepts of ambiguity in context-free grammar and its implications. In particular, you will learn about the compilation process including lexical analysis and syntactic analysis, recursive descent parsers and semantic analysis. Finally, you get to investigate the synthesis phase, where intermediate representations, target languages and structures lead to code generation.

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.

Delve into machine learning, a fundamental concept in artificial intelligence that enables a computer to learn how to perform a task from data rather than traditional programming. In this module, you will study the key ideas and techniques behind machine learning and develop the practical skills needed to understand the implications and potential of machine learning in business and society. You will begin by looking at real-world problems, challenges, and fundamental techniques in current methodology. Building on this, you will cover a variety of approaches to machine learning, from decision trees to a wide range of deep neural networks, including multilayer perceptrons, convolutional neural networks, long short-term memory, autoencoder and generative adversarial networks.

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).

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.

Gain a broad understanding of Natural Language Processing (NLP), a branch of artificial intelligence where computational methods are used to analyse and understand human languages. Throughout the module, you will be exposed to the core concepts surrounding the NLP pipeline, covering methods and techniques for data collection, cleaning, tokenisation, and annotation using a hierarchy of linguistic levels (e.g. morphology, syntax, and semantics). You will experiment with and comparatively evaluate different methods and techniques, including rule-based, probabilistic, machine learning and deep learning approaches. You will also learn to apply and adapt NLP pipelines and tools to real-world text mining scenarios and problems, including examples such as health and finance. Key issues such as ethical data collection, bias in language models, and employing sustainable computing methods will also be touched upon throughout.

We introduce you to quantum computing's core principles and applications, contrasting its capabilities with classical systems. You will master Dirac notation and essential linear algebra, before examining quantum mechanics' four postulates, including qubits, gates, and circuit models. You will cover fundamental algorithms, including Deutsch's algorithm (implemented via Qiskit), Simon's problem, Bernstein-Vazirani, Grover's search (with BBBV Theorem analysis), and Shor's factorisation algorithm's impact on RSA cryptography.

Quantum cryptography components address post-quantum security and QKD protocols, while quantum information theory explores superdense coding, the no-cloning theorem and teleportation. The module concludes with emerging concepts like quantum money and the Elitzur-Vaidman bomb tester. Combining theoretical foundations with practical programming exercises, you will develop a critical understanding of both quantum computing's potential and current technological limitations, preparing you for advanced study or research in this rapidly evolving field.

Artificial Intelligence (AI) is being rapidly adopted in both research and industry, via technologies such as generative AI and large language models (LLM). They are being used for a range of applications by enhancing cyber security through the detection of anomalies, identifying threats, and monitoring abnormal activities. However, AI itself is susceptible to various attacks, such as prompt injection, data leakages, jailbreaking, bypassing guardrails, model backdoors, and more.

In this module, you will learn the fundamentals of AI for security and security for AI. This encompasses both how AI can be leveraged to augment and improve established cyber security techniques (from firewalls, risk analysis, to attack detection), as well as the emerging attacks against AI itself (data poisoning, extraction, membership inference). You’ll learn how AI is being used to revolutionise the established cyber security field, the emerging threats of adversarial attacks against ML models and data, and how to mitigate those attack.

Understand security threats to cyber physical systems (CPS), such as industrial control systems, Internet of Things and connected vehicles, as well as techniques to mitigate these threats. Compared to traditional computer systems, CPS have limited resources and are typically deployed into a physical environment. This impacts the implementation of security techniques, as due to the environment they are deployed in you must consider both digital and physical attacks.

This module introduces how to identify the appropriate security techniques to use for a CPS. You will come to understand how to write secure applications for CPS and which alternative mitigations are appropriate. You will also learn how the limitations of these systems impact the guarantee of security. In addition to this, you will examine the safety and privacy threats facing CPS and explore the interconnectivity between them and security.

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.

Computing plays a pivotal role in addressing growing energy costs, greenhouse emissions, and the climate crisis. Whilst we can use computing and its associated digital technologies to shape a greener society (as well as create more energy-efficient software and hardware), there exist important trade-offs in respect to economic cost, engineering effort, and environmental impact. Explore key concepts associated with creating sustainable computing, spanning from how a processor uses electricity to how computers shape a greener economy and society. You will study the methods to create more energy-efficient code, energy-aware device mechanisms, as well as the benefits and drawbacks of computing and digital technology with respect to its impacts upon the environment and economy.