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Overview
Top reasons to study with us
10
10th for Mathematics
The Guardian University Guide (2025)
12
12th for Mathematics
The Complete University Guide (2025)
Explore philosophy from a global perspective
Uncover the fundamental workings of the universe and develop a high-level of reasoning through our exciting and challenging programme with a placement year. While studying mathematics and philosophy, you will gain a wealth of skills, knowledge and experience which you will have the chance to apply whilst working full-time in industries from the public and private sectors.
Maths and philosophy are both difficult to concisely define, but at their core, they are concerned with the underlying workings and meaning of the universe. Maths is the study of change, patterns, quantities, structures and space, while philosophy is concerned with fundamental problems in topics such as knowledge and reason.
Over the three years, you will be able to draw on expertise from two specialist departments: Mathematics and Statistics; and Politics, Philosophy and Religion. This is an engaging programme of study and our reputation for excellence in research means that we are able to offer high-quality teaching delivered by academics who are leaders in their field.
In Year 1, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problem-solving and quantitative reasoning skills. Our ‘Introduction to Philosophy‘ module provides the key themes in the study of philosophy. Consciously drawing on a broad range of philosophical traditions -- Continental, Analytic, and non-Western -- it aims to present a comprehensive overview of various theoretical sub-disciplines within philosophy, but also to equip you with the ability to reason and think clearly about the most fundamental questions of human existence.
In the second year, you will further develop your knowledge in complex and real analysis, and abstract and linear algebra. These highly analytical topics will complement your study of philosophy, which will really begin to expand and develop this year. You will be able to choose from a range of philosophy modules which will allow you to build a solid repertoire of philosophy knowledge and analytical skills.
To prepare you for your work placement, our Careers and Placements Team will provide advice and guidance on the skills required to create effective CVs, cover letters and applications; tips and techniques on how to make an impact at interviews and assessment centres; how to create a relevant digital profile; and how to research employers and career sectors of interest.
In third year you will undertake a 12 month placement which will allow you to apply the knowledge and skills that you’ve learnt in Years 1 and 2, and to gain invaluable experience which will make you highly employable when you graduate.
The University will use all reasonable effort to support you to find a suitable placement for your studies. While a placement role may not be available in a field or organisation that is directly related to your academic studies or career aspirations, all placement roles offer valuable experience of working at a graduate level and gaining a range of professional skills. If you are unsuccessful in securing a suitable placement for your third year, you will be able to transfer to the equivalent non-placement degree scheme and continue with your studies at Lancaster, finishing your degree after your third year.
Returning to Lancaster, your final year offers you the chance to choose from a wide range of specialist modules, allowing you to develop the programme further to suit your interests and guide you to a specific career pathway.
Formed in 1959, and based in the Department of Politics, Philosophy and Religion, the Richardson Institute is the oldest peace and conflict research centre in the UK. Since 2012 it has provided an internship programme that gives students the opportunity to work with different organisations on issues of peace and conflict.
Discover what studying Mathematics at Lancaster is like from our students and academics.
An experience for Florence
For my placement year, I moved down to Cardiff to work as a Statistician with the Welsh Government. Whilst here, I have been working as part of two different teams – the Health and Social Services team and the Post-16 Education team. Currently, we are producing official statistics with both teams, the results of which have been published publicly.
I have loved being able to put my degree to practical use. When studying maths, it is often difficult to see how what you are learning could be used outside mathematical research. However, during my placement I have used the coding skills I learnt over my first and second years on datasets to create content for statistical publications as well as statistical tests and theory to ensure I am producing accurate statistics. My placement has also helped me within my degree by giving me insight into what modules I would like to pick for my future years, as well as developing my organisational skills – something which will be very useful when I return to Lancaster!
I believe the experience that I have gained over this year will be invaluable when it comes to applying for graduate roles after university. It has helped narrow down what I might want to do after university, as statistics is an area I am really interested in! The work, as well as networking with others already in the workforce, has opened me up to so many different job possibilities that I wouldn’t have known about otherwise.
Maths and philosophy graduates are highly employable, having in-depth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including analysis and manipulation, interpretation, logical thinking, problem-solving, and reasoning, as well as adept knowledge of the disciplines. As a result, combining these two subjects opens up a range of opportunities and graduates are highly sought after.
The starting salary for many of these roles is highly competitive, and popular career options include:
Actuary
Data Analyst
Investment Analyst
Research Scientist
System Developer
Teacher
A degree in this discipline can also be useful for roles such as Barrister, Local Government Office, Psychotherapist, Stockbroker, and many more.
Alternatively, you may wish to undertake postgraduate study at Lancaster and pursue a career in research and teaching.
Lancaster University is dedicated to ensuring you not only gain a highly reputable degree, but that you also graduate with relevant life and work based skills. We are unique in that every student is eligible to participate in The Lancaster Award which offers you the opportunity to complete key activities such as work experience, employability/career development, campus community and social development. Visit our Employability section for full details.
Skills for your future
A degree in mathematics will provide you with both a specialist and transferable skill set sought after by employers across a wide range of sectors.
Our alumni stories
Listen to our Mathematical Sciences alumni as they tell us how studying at Lancaster helped to prepare them for their future careers within mathematics.
Entry requirements
Grade Requirements
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
IELTS 6.0 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL (either analysis and approaches or applications and interpretations)
BTEC May be accepted alongside A level Mathematics and Further Mathematics
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Delivered in partnership with INTO Lancaster University, our one-year tailored foundation pathways are designed to improve your subject knowledge and English language skills to the level required by a range of Lancaster University degrees. Visit the INTO Lancaster University website for more details and a list of eligible degrees you can progress onto.
Contextual admissions
Contextual admissions could help you gain a place at university if you have faced additional challenges during your education which might have impacted your results. Visit our contextual admissions page to find out about how this works and whether you could be eligible.
Course structure
Lancaster University offers a range of programmes, some of which follow a structured study programme, and some which offer the chance for you to devise a more flexible programme to complement your main specialism.
Information contained on the website with respect to modules is correct at the time of publication, and the University will make every reasonable effort to offer modules as advertised. In some cases changes may be necessary and may result in some combinations being unavailable, for example as a result of student feedback, timetabling, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes and new research. Not all optional modules are available every year.
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.
This course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure and temperature, and the direction of the wind. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.
The main focus of this module is vectors in two and three-dimensional space. Starting with the definition of vectors, students will discover some applications to finding equations of lines and planes, then they will consider some different ways of describing curves and surfaces via equations or parameters. Partial differentiation will be used to determine tangent lines and planes, and integration will be used to calculate the length of a curve.
In the second half of the course, the functions of several variables will be studied. When attempting to calculate an integral over one variable, one variable is often substituted for another more convenient one; here students will see the equivalent technique for a double integral, where they will have to substitute two variables simultaneously. They will also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, the module will explain how to calculate the areas of various surfaces and the volumes of various solids.
This module introduces students to key themes in the study of philosophy. Consciously drawing on a broad range of philosophical traditions -- Continental, Analytic, and non-Western -- it aims to present a comprehensive overview of various theoretical sub-disciplines within philosophy, but also to equip students with the ability to reason and think clearly about the most fundamental questions of human existence. The course, though designed as an introduction to the advanced degree-level study of philosophy, will also function as a self-standing introduction to philosophy suitable for those seeking to broaden their understanding of philosophy as it has been practiced throughout various traditions.
The module will involve the study of European and non-European sources, and areas of study will typically include:
1. Epistemology: the study of the nature of knowledge, belief, and the mind's ability to apprehend the world.
2. Metaphysics: the study of the nature of matter, causation, freedom, and being.
3. Phenomenology: the study of the nature and structure of consciousness.
4. Philosophy of Religion: the study of the nature and existence of God and of religious faith.
5. Philosophy of Mind: the study of the nature of mind and the mental.
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.
The student will learn how to express a linear transformation of the real Euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.
The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.
Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.
Core
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This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here, students will select a small number of properties which these and other examples have in common, and use them to define a group.
They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.
Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.
The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.
Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.
They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.
A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.
Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.
Optional
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The second half of the 18th Century was a time of fierce debate between the schools of idealism, empiricism, and criticism that extended to the nature of subjectivity and the status of nature itself. This course examines key texts from Hume and Kant, two of the greatest modern philosophers, which all confront the new realities of the modern scientific method. The course will focus on the relationship between knowledge and the natural world and evolution of subjectivity and its grounding of psychology.
This module offers a critical introduction to Chinese philosophy by focusing on its diversity and contemporary relevance. The module introduces the main schools of thought that emerged during the classical era (one of the most vibrant periods of Chinese philosophy), including Daoism, Confucianism, Legalism, Mohism, and the school of names.
Students will be introduced to the main concepts used in classical Chinese philosophy as well as the central issues debated by classical Chinese philosophers, such as whether human nature is good or bad, whether one should engage in society or retrieve from it in order to live a fulfilled life, whether humans are at the centre of the world or simply a part of it, and whether language is enhancing human potential or limiting it, to name but a few examples.
Can we know what it is like to be a fish? Why do we spontaneously try to save a child who is in danger? Is a white horse a horse? Should we rejoice in non-action, engage in politics, or do both at the same time? Is there such a thing as feminist philosophy in ancient China? These are some of the issues that will be discussed during the term.
This module will offer an introduction to feminist philosophy by addressing the question of what feminist philosophy is and providing an overview of some important debates in feminist philosophy including the debate on the concept “woman” and the distinction between sex and gender, the literature on intersectionality and the relationship between gender-based and other forms of oppression, and feminist thinking on care, marriage and the family. These issues are among the most important ones in contemporary feminist philosophy as well as being issues that occupy a prominent place in public debate. Further, discussing them will allow us to explore different traditions and approaches to feminist philosophy, analysing contributions of analytic and continental feminist philosophers, of liberal feminist philosophers and those critical of the liberal tradition, and centring feminist perspectives that are often marginalised.
This module considers a range of issues currently being debated by political philosophers and political theorists. Specific topics may change slightly, but the current plan is cover the following, with attention to questions of freedom and justice throughout:
Business corporations and employment
Racism and sexism
Democracy
Climate politics
Structural injustice and sweatshop labour
Public health and state interventions
Studying this module should improve students’ knowledge and understanding of some key issues in metaphysics as determined by the syllabus. This focuses primarily on some issues concerning space and time, the nature of physical objects and persons, and some key philosophical distinctions. Studying this module should also enable them to see connections between various philosophical issues that should be of value to them with regard to other philosophy modules that they are studying.
This course explores ideas central to any understanding of politics. It focuses on two related themes: Equality, and Community. In the course we will explore the thought of thinkers who are associated with these ideas of equality and community (Rousseau, Marx, the Fabians, and Rawls). By the end of the course, you will have an understanding of the key ideas of the thinkers under review and be able to assess the contribution that these thinkers have made to our wider understanding of politics. You will also be able to recognise the relevance of these thinkers to our current political debates, and be able to employ their ideas within those debates. Additionally, you will be able to evaluate the key features of an argument, be confident to express your own views, and evaluate the responses of others.
Moral philosophy is the systematic theoretical study of morality or ethical life: what we ought to do, what we ought to be, what has value or is good. This module engages in this practice by critical investigation of some of the following topics, debates, and figures: value and valuing; personhood/selfhood; practical reason; moral psychology; freedom, agency, and responsibility; utilitarianism and its critics; virtue ethics and its critics; deontology and its critics; contractarianism and its critics; the nature of the good life; the source and nature of rights; the nature of justice; major recent and contemporary figures such as Bernard Williams, Martha Nussbaum, Peter Railton, Christine Korsgaard, Philippa Foot, Allan Gibbard, Simon Blackburn; major historical figures such as Aristotle, David Hume, Immanuel Kant, John Stuart Mill, G. E. Moore.
This course covers nineteenth-century philosophy, a crucial period in several ways: there was a new attention to history and the relation between philosophy and history; there was the rise of socialism and its impact on philosophy; and there were philosophical criticisms of Christianity, which were met by explicit defences of Christianity by some philosophers. We explore these issues through the work of six figures in nineteenth-century German and British philosophy: Hegel, Feuerbach, and Marx; Nietzsche, Cobbe, and Besant.
This module examines some theoretical issues involved in gaining knowledge about human societies. We will look at the role of theories and models in economics and political science, the special nature of "social institutions," and whether economic and political knowledge can be separated from value-judgments:
Rational choice theory and models based on it
Social norms and cooperation
John Searle’s theory of “institutional facts”
The nature of money and different accounts of power
Whether values can or should be kept out of economics and political science
Some ways in which states and markets are related
This course considers philosophical issues that arise in connection with the sciences. It will consider what scientific method is, how science relates to the rest of knowledge, whether it provides an ideal model for rational inquiry in general, and whether we should think of science as describing reality.
In the first few weeks we will consider traditional accounts of scientific method and theory-testing, and then examine philosophical challenges to the status of science as a rational form of enquiry. We give particular consideration to three of the most important twentieth-century philosophers of science: Popper, Kuhn, and Feyerabend. Next we will consider whether and in what sense we should be confident that our best current scientific theories are accurate descriptions of reality.
It is not assumed that students have an extensive knowledge of science: the relevant scientific concepts will be presented in a simple and accessible way, and there will be no maths.
Knowledge is an essential aspect of our social lives. This module focuses on a range of real world social, ethical and political problems involving knowledge. Topics include: problems of epistemic injustice (where people are not believed because of identity prejudice); whether virtues of open-mindedness might provide a solution to epistemic injustice. A proper understanding of the ethics and politics of knowledge requires us to examine both doubt and ignorance. We consider whether systemic racism is sustained by an active kind of “wilful” ignorance. We explore how powerful corporations seek to deliberately engineer doubt to further their interests. We examine political deception and the idea of “Post-truth” politics. In the final section we turn to the limits of seeking knowledge and how to balance the interests that states and corporations have in knowing personal information, against our interests in keeping such information private.
This module aims (a) to provide students an introduction to issues within the philosophy and politics of higher education, and (b) to help students to reflect about their own position in, and aims while at the university. During the modules, students will consider key questions regarding the aims of university education and its political context and history, as well as dedicate time to thinking about how their own studies fit into those aims and context, and what they wish to achieve during their undergraduate degrees. Twice weekly lectures will be supplemented by seminars, but also by regular ‘keynotes’ in which those working within and alongside higher education will present their own views and approach to higher education. Invited figures may include senior figures from within Lancaster University, representatives of UCU, the Student Union, and local stakeholders.
This module will introduce students to some of the most well-known women philosophers from ancient India, China, and Greece. The content of the module will be philosophical sources that were either: authored by women, include views, voices, and/or characters that claim to represent a woman’s perspective, or that are explicitly about women. In addition to reading this source material, the module will also develop the awareness of and skills to address some of the unique challenges of studying women philosophers, particularly in contexts where it is not clear if women composed the sources attributed to them.
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Students will spend their third year working on a graduate level placement to gain valuable experience in an industry or sector that they might aspire to work in once they graduate. They will be supported by the Faculty Placements Team and the Careers Team in securing their own placement.
Optional
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This module focuses on selected topics in Applied Philosophy. Applied Philosophy involves the application of philosophical methods and knowledge to a range of problems that face institutions, professions, policymakers and regulatory bodies. Further questions arise about the nature and limits of applied philosophy.
Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.
Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.
This module will trace the development of Buddhist thought, from its emergence in India in the fifth century BCE, through its development across Asia, particularly China, but also Tibet and Japan. It will also look at how Buddhist philosophy has been received by Western philosophers from the 19th century. In addition to tracking and analysing key concepts, such as not-self, dependent origination, emptiness, and Buddha nature, this module will examine themes that pervade Buddhist philosophy in its various contexts, such as the relationships between teaching and practice, philosophy and literature, and religion and politics. Although the regional emphasis will change depending on the lecturer, this module will offer students an in-depth exploration of one of the most enduring, voluminous, and influential philosophical traditions of the world.
Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject - with many important connections to other areas of modern mathematics - it is a very accessible one.
In this module, students will be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms) and combinatorial design theory (Latin squares and block designs). They will also explore important practical applications of the results and methods.
Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.
They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
How well-known concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.
The module will look at philosophical issues that arise out of Darwin’s theory of evolution. These include questions about how best to understand the theory of evolution, and questions about what evolution implies for our view of the world, and in particular of ourselves. The course breaks down into three broad areas:
Different ways to understand the theory of evolution, e.g., Is evolution, as some would have us believe, all about genes? Is natural selection the only important factor in evolution?
Conceptual issues relating to biology, e.g., How do we define ‘function’? Is there one right way to classify living things
Implications of Darwinism for understanding human nature, e.g., Does the fact that we have evolved affect ow we should see human nature? Why are evolutionary theories of human nature so controversial? Does Darwinism have any implications for moral questions?
Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.
While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.
PPR.399 provides an opportunity for students to choose a topic related to some aspect of Politics and International Relations, Philosophy and Religious Studies which particularly interests them, and to pursue it in depth. The topic may be related to work that is being done on a formally taught course, or it may be less directly linked to course work. The intention is that students will develop their research skills, and their ability to work at length under their own direction.
Students write a dissertation of 9,000-10,000 words. They are expected to start thinking seriously about the dissertation towards the end of the Lent term of their second year, and to submit a provisional topic by the end of that term. Work should be well advanced by Christmas in the third year. The completed dissertation must be submitted at the start of Summer Term in the third year. To help students prepare for work on the dissertation, there will be an introductory talk on topics relating to doing one's own research and planning and writing a dissertation. A course handout will be available setting out in more detail the requirements for the dissertation and giving full details of lectures, supervision arrangements and assessment.
The aim of this module is to allow students to pursue independent in-depth studies of a topic of their choice, within the scope of their scheme of study. The topic will be formulated in dialogue with one or more external collaborator(s) and may be related to work that is being done on a formally taught course, or it may be less directly linked to course work. Students will develop their employability and research skills, and their ability to work independently at length under their own direction with input from external collaborators and an academic supervisor. The external collaboration will enhance students’ ability to reflect on the impact of academic work. One option is to incorporate work done through the Richardson Institute Internship Programme, but students may also discuss other forms of collaboration with their supervisor.
Students are expected to start thinking seriously about the dissertation towards the end of the Lent term of the second year, and to submit a provisional topic by the end of that term. Work should begin during the Summer term of the second year and a draft plan must be approved by the end of the Summer term. Work should be well advanced by Christmas in the third year. The completed dissertation must be submitted at the start of Summer Term in the third year. To help students prepare for work on the dissertation, there will be an introductory talk on topics relating to doing one’s own research and planning and writing a dissertation. A course handout will be available setting out in more detail the requirements for the dissertation and giving full details of lectures, supervision arrangements and assessment.
What moral obligations do we have towards future generations – to people who are yet to be born, and to merely possible people whose existence (or non-existence) depends on how we decide to act now?
In this module, we explore this question in detail by examining both a series of case studies and some of the main concepts and theories that philosophers use when thinking about these issues.Questions considered normally include:
Is there a moral obligation to refrain from having children (e.g. for environmental reasons) and what measures may governments take to encourage or enforce population control?
Should we use selection techniques to minimise the incidence of genetic disorders and disabilities in future populations?
Should parents be allowed to use these techniques to determine the characteristics of their future children? How should we weigh quality against quantity of life?
Would a world with a relatively small number of ‘happier’ people be preferable to one with many more ‘less happy’ ones?
The topic of smooth curves and surfaces in three-dimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.
A number of well-known concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.
The study of graphs - mathematical objects used to model pairwise relations between objects - is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.
Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.
Situations will be considered in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.
Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.
This course considers conceptual questions around 'health' and 'disease' (and related concepts of 'disability', 'normality', 'medicine', 'treatment') and explores how these relate to issues of health policy. We start by considering concepts of health and disease:
Does whether a condition is a disease depend purely on matters of biological fact?
Does a condition also have to be harmful to count as pathological?
Is there any distinction that can be drawn between mental and physical disorders?
Is it justified to treat people with mental disorders differently, e.g. in involuntary treatment?
Should psychopaths who commit horrible crimes be considered to suffer from a disorder, or are they evil?
What does it mean to say that someone is ‘normal’?
Many critics worry about medicalisation, and think that ever more conditions are coming to be considered diseases. Is this true, and does it matter?
We’ll also consider conceptual issues connected to treatment. ‘Evidence Based Medicine’ aims to employ treatments that have been shown to work. But, how can it be determined whether a treatment works? What should the aims of therapy be? What is the distinction between medicines and other drugs?
Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.
Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.
This special subject focuses on feminist philosophy and in particular the study of women and feminists in the history of philosophy, using nineteenth-century British philosophy as a case study. The course provides an in-depth understanding of debates around women in the history of philosophy, the relation between feminism and women, and how to research and study historical women philosophers who until recently have been omitted from the canon. This will provide important transferable skills in doing research in the digital world, including working with digital archives and historical journals. The course will allow students to undertake a sustained piece of independent research on a historical essay of their choice by a woman philosopher from nineteenth-century Britain. Students taking this course will not merely be learning about philosophy as done by others; they will be doing cutting-edge philosophical research themselves.
This module will examine key sources in the history of Indian political philosophy from ancient times to the present. We will begin by looking at the most influential political sources from ancient India, including the inscriptions of King Ashoka and the Arthashastra. Some of the questions we will be asking are how the ideas in these texts speak to modern debates about secularism, pluralism, and civil religion. We will then turn our attention to the modern period, reading the political thought of figures such as Gandhi, Ambedkar, and Ashis Nandy. We will look at how these and other modern political thinkers draw from premodern Indian traditions, as well as how they engage with and critique Western political ideas from Indian perspectives.
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.
Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.
This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.
Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.
The module provides an introduction to formal logic together with an examination of various philosophical issues that arise out of it. The syllabus includes a study of the languages of propositional and quantificational logic, how to formalize key logical concepts within them, and how to prove elementary results using formal techniques. Additional topics include identity, definite descriptions, modal logic and its philosophical significance, and some criticisms of classical logic.
Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.
Having studied mathematics for many years, students are well-placed to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system.
This module is run as a partnership between the Lancaster University's School of Mathematical Sciences, the University of Cumbria and the Students’ Union’s volunteering unit.
Designed with employability in mind, this module is based on the Students’ Union’s Schools Partnership Scheme, which supports Lancaster students on 10-week placements in local primary and secondary schools.
Students will have the opportunity to take part in classroom observation and assistance, the development of classroom resources, the provision of one-on-one or small group support and possibly even teaching sections of lessons to the class as a whole.
Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.
An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise high-dimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.
This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.
Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models.
Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:
Conditional expectation
Filtrations
Martingales
Stopping times
Brownian motion
Black-Scholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.
In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking.
An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.
They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.
This module will address central issues in contemporary ethical (including meta-ethical), legal and political philosophy, and will allow a systematic critical exploration of the connections between ideas and arguments in each of the three areas of the subject.
Topics covered will include some of the following: modern theory of moral motivation, value theory, contractualism, the 'moral problem'; responsibility and criminal liability, the justification of punishment, the proper scope of the law; democratic theory, egalitarianism, justice, nationalism, multiculturalism, liberty and human rights.
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the non-specialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.
This course will examine some of the core philosophical questions raised by warfare and conflict. We will look at the ethics of war and killing, but also at more neglected philosophical issues in this area, and non-Western approaches as well as classic texts in the Western tradition.
We will do so by examining some of the central dilemmas faced by soldiers, policy makers and non-combatants, in the form of a weekly question for discussion. These questions include: Can war be beautiful? When, if ever, should we go to war? What counts as legitimate action in war? What, if anything, do we owe to our enemies? Is soldiering a good life? What does technological development mean for warfare? What should a responsible citizen do when their country is, or looks about to be, at war? Who has the epistemic authority to speak about war? Is war always tragic?
This module will introduce students to the thought of Sigmund Freud, the founder of psychoanalysis. As well as providing a firm grounding in Freud's own thought, it will also raise questions about the implications of Freud's thought for philosophy. To this end, it will examine Paul Ricoeur's pioneering work on Freud and philosophy and will also look at the work of subsequent thinkers who have explored the ramifications of Freud's work for philosophy, especially Jacques Lacan, Michel de Certeau and Slavoj Žižek. The module will raise questions about the extent to which philosophy should respond to some of the insights of psychoanalysis.
This module introduces central issues, problems and theories in philosophical aesthetics by critically examining a number of central topics including: the nature of aesthetic experience; the objectivity of aesthetic judgement; emotional responses to fiction; the moral and cognitive value of art; the aesthetic value of nature. In addition to central philosophical discussions, various findings from empirical psychology and neuroscience will also be used. Although examples from all of the arts will be employed throughout the course, the emphasis will be on the wider issues just listed, and not exclusively focussed on art. That is, aesthetics will be explored as an important area of the philosophy of value in general.
This module considers key philosophical issues in the sciences of human mind, behaviour and social structures, such as psychology, psychiatry, sociology, economics and history. Topics to be considered may include the status of reason-based explanations of human behaviour, the legitimacy of psychoanalytic explanations, the understanding of other societies, individualism versus collectivism in social explanation, and the scientific status of social models based on postulates of rational choice.
This course examines central themes in the liberal branch of contemporary Anglo-American analytic political philosophy. The liberal positions on justice, liberty, equality, the state, power, rights and utility are all explored. The approach is philosophical rather than applied; its focus is on the ideas of liberal politics: how individual liberty can be maximised while not harming others; how an individual philosophical position can guide political determinants of a society and places the developments of liberal ideas in their appropriate historical contexts. The course also examines the connection between the ideas of liberalism and the idea of democracy to explore the philosophical tensions between the two and how these might be resolved. The course is a survey of major topics and concepts in Anglo-American liberal political ideas. The syllabus will include the following topics: questions about justice; visions of the state; negative and positive liberty; equality, utility and rights; toleration and multiculturalism; neutrality and the market.
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.
This module critically explores a range of key topics in the ethics and politics of communication. In the first half of the course, we begin by an introduction to some basic concepts in linguistics and philosophy of language – especially to do with the practical side of communication. We then focus on (a) how certain kinds of communication can bring about ethical change (e.g. making something permissible); (b) upon whether lying and other kinds of deception are permissible, and if so, when. In the second half we turn to some broadly political issues: whether political lying is justified in a way that everyday lying is not. We consider three domains where freedom of communication is both important and contentious: freedom of speech, freedom of the press, and freedom on social media, including the challenges posed by “content moderation”.
In the Twentieth Century, Western philosophy underwent a number of fundamental “turns” — the linguistic turn, the phenomenological turn, the postmodern turn. Some of these changes were viewed as “revolutions” in philosophy. At the extreme end, there were even arguments that Western philosophy, as conceived since Plato, was finished. In this module we explore some of these key transformations. We consider the “linguistic” turn, and the formation of “analytic philosophy” at the turn of the C20. One central figure of this linguistic turn is Ludwig Wittgenstein. But Wittgenstein shifts from being at the centre of analytic philosophy to arguing that philosophy is finished. At the same time, philosophy also undergoes a phenomenological turn. We focus on how this leads, via Sartre, to a revival of existentialism. The contrasts between French philosophy and English-speaking philosophy become even more pronounced in the final third of the C20, with post-structuralism and post-modernist philosophy viewed by the “analytic” philosophy community as not even being a kind of philosophy. We assess the roots of, and justification of, this “analytic/continental” divide.
Fees and funding
Our annual tuition fee is set for a 12-month session, starting in the October of your year of study.
There may be extra costs related to your course for items such as books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation, you may need to pay a subscription to a professional body for some chosen careers.
Specific additional costs for studying at Lancaster are listed below.
College fees
Lancaster is proud to be one of only a handful of UK universities to have a collegiate system. Every student belongs to a college, and all students pay a small college membership fee which supports the running of college events and activities. Students on some distance-learning courses are not liable to pay a college fee.
For students starting in 2025, the fee is £40 for undergraduates and research students and £15 for students on one-year courses.
Computer equipment and internet access
To support your studies, you will also require access to a computer, along with reliable internet access. You will be able to access a range of software and services from a Windows, Mac, Chromebook or Linux device. For certain degree programmes, you may need a specific device, or we may provide you with a laptop and appropriate software - details of which will be available on relevant programme pages. A dedicated IT support helpdesk is available in the event of any problems.
The University provides limited financial support to assist students who do not have the required IT equipment or broadband support in place.
Study abroad courses
In addition to travel and accommodation costs, while you are studying abroad, you will need to have a passport and, depending on the country, there may be other costs such as travel documents (e.g. VISA or work permit) and any tests and vaccines that are required at the time of travel. Some countries may require proof of funds.
Placement and industry year courses
In addition to possible commuting costs during your placement, you may need to buy clothing that is suitable for your workplace and you may have accommodation costs. Depending on the employer and your job, you may have other costs such as copies of personal documents required by your employer for example.
The fee that you pay will depend on whether you are considered to be a home or international student. Read more about how we assign your fee status.
Home fees are subject to annual review, and may be liable to rise each year in line with UK government policy. International fees (including EU) are reviewed annually and are not fixed for the duration of your studies. Read more about fees in subsequent years.
We will charge tuition fees to Home undergraduate students on full-year study abroad/work placements in line with the maximum amounts permitted by the Department for Education. The current maximum levels are:
Students studying abroad for a year: 15% of the standard tuition fee
Students taking a work placement for a year: 20% of the standard tuition fee
International students on full-year study abroad/work placements will be charged the same percentages as the standard International fee.
Please note that the maximum levels chargeable in future years may be subject to changes in Government policy.
Scholarships and bursaries
You will be automatically considered for our main scholarships and bursaries when you apply, so there's nothing extra that you need to do.
You may be eligible for the following funding opportunities, depending on your fee status:
Unfortunately no scholarships and bursaries match your selection, but there are more listed on scholarships and bursaries page.
Scheme
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We also have other, more specialised scholarships and bursaries - such as those for students from specific countries.
Lancaster has been the perfect place for me. The campus feels like its own little world and the sense of community has been a really key part of my experience at Lancaster. You can find your place in colleges, liberation forums, and societies – there really is somewhere for everyone.
The way that the Mathematics course is structured at Lancaster means that by the end of first year every student is caught up to the same level so you don’t have to worry about being behind if you studied different qualifications at school. Then second year builds on that foundation to give a breadth of teaching across pure maths, statistics, and mathematical methods so that you can study what interests you in third year knowing that you have a strong basis to work from.
I decided to do a placement year, and I spent 14 months working for NHS England as a data analyst in the performance analysis team. I had the opportunity to work on official statistics that were discussed on the news and used by Number 10, the CEO of the NHS, and the general public. I was able to use the coding skills I learned in my degree to improve processes within my team which significantly increased efficiency and reduced errors.
I absolutely loved working in a sector that I feel passionately about and now know that data is the career I want to work in after I graduate. My placement experience helped me choose third-year modules that will be relevant to the graduate jobs I plan to apply for and the assessments I did during my placement year have helped me reflect on what sort of jobs I want to apply for.
We ensure that our students receive the support that they need in order to achieve their full academic potential. We are a friendly department and foster a highly supportive learning environment.
You will be assigned a tutor, meeting in the first week of the first term and once per term after that. Your tutor is available for on-demand, one-to-one consultation, and to discuss personal development. This includes assistance with module choices, monitoring of progress, support with career aspirations and provision of references, as well as providing information regarding other services available throughout the University.
We look at the representation of different genders, minorities and identities and look to encourage diversity within the department and the University. Students can become involved in helping us to identify issues.
Maths Café
We hold a Maths Café event every Monday in Fylde Common Room from 11:00 – 13:00. These are hosted by students in their third and fourth year, and provide help and support to undergraduate students in all years. The Maths Café is organised by the Maths and Stats Society.
Student Learning Advisor
The Faculty of Science and Technology's Student Learning Advisor offers free consultations to help you improve your scientific writing and help your coursework to achieve its full potential.
The information on this site relates primarily to 2025/2026 entry to the University and every effort has been taken to ensure the information is correct at the time of publication.
The University will use all reasonable effort to deliver the courses as described, but the University reserves the right to make changes to advertised courses. In exceptional circumstances that are beyond the University’s reasonable control (Force Majeure Events), we may need to amend the programmes and provision advertised. In this event, the University will take reasonable steps to minimise the disruption to your studies. If a course is withdrawn or if there are any fundamental changes to your course, we will give you reasonable notice and you will be entitled to request that you are considered for an alternative course or withdraw your application. You are advised to revisit our website for up-to-date course information before you submit your application.
More information on limits to the University’s liability can be found in our legal information.
Our Students’ Charter
We believe in the importance of a strong and productive partnership between our students and staff. In order to ensure your time at Lancaster is a positive experience we have worked with the Students’ Union to articulate this relationship and the standards to which the University and its students aspire. View our Charter and other policies.
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