BSc (Hons) Mathematics (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

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

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

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

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

Practical computing is all about problem solving and coding those solutions as working computer programs. Techniques exist that provide structured approaches to solving problems and you will be introduced to these core transferable skills via Computational Thinking - “the thought processes involved in formulating a problem and expressing its solution(s) in such a way that a computer (either human or machine) can effectively carry out”: algorithms, abstraction, decomposition, generalisation, handling common problems, dealing with complexity. Coding as a creative skill and technique for applied problem solving is taught in the context of two comparative languages: JavaScript and Python. You wi

Using Python, this module develops your foundational computer programming skills, in the context of heuristics for business decision-making and optimisation. It begins with basic computing concepts, data structures and algorithms, which helps develop logical and abstract thinking. By incorporating heuristics into the Python learning process, you will enhance your understanding of the language.

We introduce you to the fundamental nature of Physics and teach you key skills in the use of experiment and uncertainty, units, and dimensional analysis. You will study topics such as Newton’s laws of motion, rotation of rigid bodies and the gravitational force. You will also be introduced to more advanced concepts such as special relativity and Lagrangian mechanics. You will apply some of these concepts to astronomical problems such as determining escape speed, and the motion of satellites and planetary orbits. You will learn about some exotic phenomena like black holes and dark matter.

Explore electricity, magnetism, thermodynamics, and quantum physics, providing a strong foundation in classical and modern physics. You will learn about electric and magnetic fields and forces through developing an understanding of Maxwell’s equations and their application. You will study topics in thermodynamics including heat transfer and ideal gases. You will be introduced to quantum mechanics, by examining atomic models, wave-particle duality, and the Schrödinger equation. Through problem-solving and conceptual understanding, you will develop analytical skills applicable in physics, engineering, and research. You will learn to apply mathematical models, understand physical principles, an

The module begins with an overview of business analytics, focusing on developing your intuition about randomness and uncertainty in business. It introduces various business analytics techniques that take uncertainty into account. You will examine case studies illustrating real-life situations, enhancing your understanding of the importance of recognising uncertainty, which is omnipresent in data and in decision-making.

This module provides a comprehensive introduction to macroeconomics, which involves the study of economics at an aggregate level. We will cover various topics, including national income analysis, monetary theory, business cycles, inflation, unemployment, and the great macroeconomic debates. The module provides the foundations for further study in Economics. Throughout the module, we will develop essential theoretical concepts and demonstrate how they apply to real-world situations. The module is self-contained and can be taken by students with no prior knowledge of macroeconomics. It takes a more mathematical approach to the subject than Foundations of Macroeconomics.

You will receive a thorough introduction to microeconomics, which is the analysis of Economics at the level of the individual or firm. The topics you will cover include the theory of demand and supply, costs and pricing under various forms of market structure, and welfare economics. The module lays the groundwork for further study in Economics. In addition to developing key theoretical concepts, we will illustrate how these concepts can be applied to real-world examples. The module is self-contained and is suitable for students without prior knowledge of the subject. This module provides a more mathematical treatment of microeconomics than Foundations of Microeconomics.

You have the option to study a language which we teach at three different levels depending on your ability. We offer beginners, progressing and becoming independent in: Chinese French German Italian Spanish By the end of the year, you’ll be able to engage with basic everyday life situations such as describing your environment, express preferences and discuss past events or future plans in simple terms. In seminars you will cover a range of oral, aural, written, and reading skills in an integrated way that embraces techniques of linguistic mediation and the plurilingual contexts of each language. The study of the cultural, social and historical context is embedded in the language learning, un

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

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

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

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.

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

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

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

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

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.

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

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.

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

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

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

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

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

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

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

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