BSc (Hons) Mathematics, Artificial Intelligence, and Real-world Systems (MARS) (Placement Year) Bachelor's degree at Lancaster University

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

A mathematical model is a representation of a real-world event, such as a building vibrating during an earthquake or the spread of a disease within a population. In this module, you will investigate mathematical models that lead to ordinary differential equations and will study a variety of core analytical methods for solving them, such as integrating factors and separation of variables. You will learn to develop models by extracting important information from real-world scenarios, which can then be analysed and refined. Many mathematical models, including those used in artificial intelligence, cannot be solved analytically, and to deal with this you will establish and practice fundamental p

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

Modern artificial intelligence relies on multivariate calculus: every time a neural network learns, it does so by computing derivatives in high-dimensional spaces. Many real-world problems seek to understand the function of a vector, where the vector could be a position in space, a direction, or the weights of a neural network. In this module, you will explore the world of multivariate techniques and multivariate calculus, deepening your understanding of vectors, angles, curves, surfaces and volumes, multidimensional space, and alternative co-ordinate systems. You will encounter multidimensional derivatives, integrals and stationary points, and practice multidimensional analogues of techniqu

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

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

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

Many of the most important real-world challenges, from predicting climate change, to modelling the spread of disease, are described by equations 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.? Other familiar phenomena, such as pulses of light down a fibre optic cable to the shudder of turbulence on a p

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

Deepen your understanding of the atmospheric physics and chemistry that govern weather systems, atmospheric composition and air pollution. You will explore the physical and dynamic properties of the atmosphere and how they influence air movement - from small-scale flows to global circulation patterns such as monsoons and El Niño. You will examine the factors affecting important atmospheric trace gases and study the key chemical processes behind urban air pollution, acid rain and stratospheric ozone depletion. Practical sessions and a field trip to our Hazelrigg meteorological station will introduce you to observing and reporting atmospheric conditions. You will also learn how these observati

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

Explore the scientific foundations of conservation biology and ecosystem ecology by examining how species, communities and environments interact. You'll develop an understanding of the significant threats to global biodiversity, such as habitat loss, climate change, invasive species and overexploitation and learn the ecological reasons why conservation is vital. Through a mix of theory, real-world case studies and practical work, you'll learn how to quantify biodiversity, assess population sizes and threats and apply conservation strategies at the genetic, species and ecosystem levels. Key topics include ecosystem resilience, habitat connectivity, conservation prioritisation and designing pr

This module will explore electric and magnetic fields, and their influence on electromagnetic circuits and their relation to voltage and current. The module will cover capacitance and inductance as electromagnetic phenomena and will use this to calculate these for circuit components. This will build to electromagnetic waves, including antennas and transmission lines, which are fundamental building blocks of communication systems. This will be further expanded by looking at communication systems, starting with frequency domain analysis and building to modulation schemes. The students will develop electromagnetic simulation models, leading to the design, build and testing of a microstrip patch

Understanding how fluids move and how mass is transported is essential in engineering as these principles are applied across various industries, from energy production and manufacturing design to chemical and nuclear plants. In this module, you will explore the nature of fluids and their fundamental properties. You will develop a solid understanding of how fluids behave at rest (fluid mechanics) and in motion (fluid dynamics), and you will test your understanding in the lab. You will also gain insight into mass transfer principles, including molecular diffusion, convective mass transfer, and interfacial transport. These concepts are crucial for designing efficient separation processes, optim

Building on your earlier introduction to glacial systems, this module takes a deeper dive into how glaciers shape landscapes and interact with people and ecosystems. You’ll explore glacial processes in greater detail and begin to see how they connect across space and time. As you progress, you’ll be encouraged to critically engage with current research and understand how physical geographers interpret and contribute to this growing body of knowledge. By the end, you’ll be better equipped to identify how glaciology can inform environmental management and benefit both society and the natural environment.

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

Soils are one of the most important, yet often overlooked, resources on the planet. Almost all our food relies on soil - it is the largest active store of carbon. The water we drink is stored and filtered through soils and they support over half of the planet’s biodiversity. In this module you will explore how soils are formed and the underpinning chemical, physical and biological processes that support the major biogeochemical cycles and life itself. We go on to explore some of the threats to soils, including soil erosion and salinisation and what we might do to mitigate them and then consider how soils are linked to the wider environment. You will be taught through a combination of lecture

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.

Problem solving, gathering relevant information, writing, presenting and working as part of a group are essential in almost every professional career. You will develop and practice these skills as you undertake both individual and group-based, industry-linked and MARS-related projects. You will also integrate prose, mathematics, computer code and its output into a clear, well-structured and appropriately linked Markdown document - a key skill in reproducibly documenting your research.

Differential equations are fundamental to mathematical modelling, with countless 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 explore their forms in diverse geometries, noting the importance of boundary conditions. Applications wi

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

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

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

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

From the complicated behaviours of ecosystems to the rapid spread of wildfires, nonlinear dynamics govern many fascinating phenomena around us. They underpin vital technological applications, such as the functioning of chemical reactors and the design of efficient transport networks. Nonlinear systems exhibit a rich variety of behaviours including sudden transitions, chaos and self-organised pattern-formation. This module will combine theoretical insights with computational approaches to explore the nonlinear world in which we live, introducing key ideas in the geometric theory of dynamical systems and nonlinear partial differential equations. You will uncover how simple rules give rise to i

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

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

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

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

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

Optimisation is the hidden engine behind the remarkable success of modern AI. Training an AI model, whether a simple regression or a state-of-the-art Transformer architecture, ultimately boils down to minimising a loss function that encodes both the training data and the neural architecture. The optimisation algorithms that make this possible are not only efficient in practice, but also mathematically elegant and broadly applicable across science, engineering and economics. You will develop the mathematical foundations of optimisation and see how they translate into practice. We will begin with convexity and smoothness of functions before introducing core optimisation schemes and key theoret