MSci (Hons) Mathematics with Computer Science Integrated Master's degree at Lancaster University

Computing and data control many critical elements of modern society. It’s vital that there is a strong theoretical foundation to computer science. We begin by examining the hard questions at the centre of computer science. You will cover the fundamentals in logic, sets, and mathematics of vectors, matrices and linear algebra and their practical applications in software, such as computer graphics. Algorithms, abstract data types, and analysis of algorithms is introduced to allow you to make reasonable decisions about the design of your programs. Finally, you will get the chance to investigate the principles of data science to select, process and analyse data, and examine the way programs and

At university, emphasis is placed on understanding general mathematical theorems. They apply in many different cases, and understanding why a result is true enables us to creatively use the underlying ideas to tackle new problems. Study the language and structure of mathematical proofs, illustrated by results from number theory. You will see the concept of congruence of integers, which is a simplified form of arithmetic where seemingly impossible problems become solvable. In relation, you’ll encounter the abstract idea of an equivalence relation. Sets and functions form the basic language of mathematics. You will study functions of a real variable and abstract functions between arbitrary set

Interested in how mathematicians build theories from basic concepts to complex ideas, like eigenvalues and integration? Journey from polynomial operations to matrices and calculus through this module. Starting with polynomials and mathematical induction, you will learn fundamental proof techniques. You will explore matrices, arrays of numbers encoding simultaneous linear equations, and their geometric transformations, which are essential in linear algebra. Eigenvalues and eigenvectors, which characterise these transformations, will be introduced, highlighting their role in applications including population growth and Google's page rankings. Next, we will reintroduce you to calculus, from its

An introduction to the mathematical and computational toolsets for modelling the randomness of the world. You will learn about probability, the language used to describe random fluctuations, statistics and the mathematical techniques used to extract meaning from data. You will explore how computing tools can be used to solve challenges in scientific research, artificial intelligence, machine learning and data science. You will develop the axiomatic theory of probability, discover the theory and uses of random variables and investigate how theory matches intuitions about the real-world. You will then dive into statistical inference, learning to select appropriate probability models to describ

Software forms a central aspect of our lives. From the applications we run on our phones to satellites in space, all modern technology is enabled by software. In this module, you will focus on Software Development, the processes and skills associated with designing and constructing computer programs. Designed with your needs in mind, whether you have previous experience in computing or not, we adapt to ensure you gain the contemporary knowledge, skills and techniques needed to develop high-quality computer software. This includes a thorough treatment of the principles of computer programming and how these principles can be applied using a range of contemporary and established languages such

Symmetry is central to our understanding of a range of subjects, from the structure of molecules to the roots of polynomials. In this module, you will see how group theory naturally appears whenever we look at symmetry. Using familiar examples, including symmetries of regular polygons, rotations and reflection matrices, roots of 1 in the complex plane, and permutations, you will define what makes a group and how this can provide a unifying language, highlighting connections between seemingly different subjects. You will then transition into mathematical analysis, developing an approach to sequences, limits, and continuity that provides the foundation for calculus. Examining a range of exampl

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

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.

We introduce the foundational principles of systems security, focusing on Confidentiality, Integrity and Availability; and Authentication, Authorisation and Accountability (AAA). You will explore access control models, security policies and the mechanisms that underpin secure system design. You will learn about the main categories of cryptosystems (e.g. symmetric, asymmetric) highlighting their practical applications and limitations in real-world contexts. We also investigate common system vulnerabilities and the tools and techniques used by attackers. Through structured, hands-on lab sessions, you will develop practical skills in identifying, analysing and mitigating threats.

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 al

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 and deploy statistical approaches to analyse data and draw conclusions. You will also develop judgement to critically evaluate the appro

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

The success of Newton/Leibniz’s calculus raises the question: what happens if we replace the real numbers with the complex numbers? After all, 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 cann

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 widespread, 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

The internet and the world wide web have now pervaded 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 now commonplace. 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 infrastructur

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 lea

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 m

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

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, begin

Models of dynamical systems are fundamental to our understanding of the physical and natural world. Explore a new class of model, the Markov jump process, for the time evolution of dynamical systems such as the evolution of species populations in the wild and the spread of infectious diseases. You will learn how to simulate from these processes and will study methods for 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

Statistical techniques are often applied to environmental data, such as air temperatures, rainfall or wildfire locations. You will learn about some of the common features of such datasets and how these features are used to design statistical models. You will first be introduced to the Gaussian process 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 i

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, proo

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 mo

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

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 current machine learning 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

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

Mathematical models are central to financial decision making. You will discover the mathematical foundations necessary to model certain transactions in the world of finance. You will then study stochastic models for financial markets and investigate the pricing of European and American options and other financial products. You will explore two discrete models, the binomial model and the finite market model, and one continuous model. 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 deduce the Black Scholes formula. You will also gai

From denoising diffusion to flow matching, modern generative models are governed by elegant mathematics: stochastic differential equations, PDEs for probability evolution and transport on spaces of measures. This module develops that mathematical toolkit and shows how it underpins today’s state-of-the-art image, audio and scientific generative models.? We start from how probability distributions evolve over time (continuity and Fokker–Planck equations) and show how this leads to a reverse-time stochastic differential equation and an equivalent probability-flow ODE. We then look at discrete-time diffusion models and explain why their training objective is a practical stand-in for maximum like

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 time-to-event data and the res

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 unlo

Building on the statistical techniques explored so far, you deepen your 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 over 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

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 questions 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

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 and how the two approaches can be used for classification and prediction. 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

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 int

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 com