Ever wondered about the hidden structures that govern mathematics? Algebra is more than just equations, it's the language of symmetry and structure, underpinning subjects ranging from geometry and quantum mechanics to number theory and cryptography. The main frameworks for modern algebra are group theory and ring theory.

Group theory topics include classifying symmetries, the symmetric group, Lagrange's theorem and the first isomorphism theorem. Similarly, ring theory explores the notions of subrings, ideals, and homomorphisms in an example-driven methodology, using abstract number systems, polynomial structures, and matrices.

This module introduces the essential theory and techniques for algebra, laying a solid foundation for further study in mathematics, physics, and other related fields.

The success of Newton/Leibniz’s calculus raises the question: what happens if we replace the real numbers with the complex numbers? Afterall, their arithmetic structure is similar, and we can measure distances between points in both. You will learn how to define the derivative of a complex function as usual and explore the behaviour of functions that are complex differentiable. Everything resembles the real case, ultimately leading to the astonishing result that if a complex function can be differentiated once, it can be differentiated infinitely often and is expressed by its Taylor series. Integral calculus for complex functions opens a route towards evaluating definite integrals that cannot be reached by real variables.

Applications of these results include a proof of the fundamental theorem of algebra, which states that every non-constant complex polynomial has a root.

This module lays foundations for further studies of mathematical analysis, pure and applied.

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning.

You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem.

Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving techniques across multiple disciplines, preparing you for advanced studies while developing your analytical reasoning abilities.

Continuing with your study into real numbers, you will explore their completeness (the idea that there are no ‘gaps’, unlike in the rationals). This completeness will be used to understand the limits of sequences, convergence of series, and power series.

This framework will allow for precision when exploring continuity, differentiability, and integrability of functions of a real variable, providing an improved foundation for calculus. That will enable you to understand when it is appropriate to use calculus; for instance, in proving theorems in other areas of mathematics, such as mathematical physics, probability and number theory.

The cornerstone of mathematical analysis is the construction of proofs involving arbitrarily small numbers, so-called epsilons and deltas. You will have opportunities to practise and improve your management of these quantities, in the process developing your skills in logic, communication and problem-solving.

The nineteenth-century was a period of transformative changes in thinking and society. There was a new attention to history and the idea of historical progress, and to philosophical engagement with industrial capitalism and the rise of socialism. The period was marked by the struggle for the abolition of slavery and growing interest in its philosophical dimensions, as well as criticisms and defences of Christianity, atheism, secularism.

You will explore these issues through a diverse range of thinkers. They will vary from year to year but typical examples include:

• G. W. F. Hegel
• Ludwig Feuerbach
• Harriet Martineau
• Karl Marx
• Friedrich Nietzsche
• Frederick Douglass
• Sojourner Truth
• Arthur Schopenhauer
• Frances Power Cobbe
• Annie Besant
• Søren Kierkegaard

You will read and analyse the original texts as well as recent interpretations, and you will learn to articulate and evaluate the central arguments made by historical figures. By the end of the module, you will be able to discuss the philosophical questions raised by a range of nineteenth-century authors, present their ideas in a balanced and well-informed way, and formulate your own conclusions on the issues.

Buddhist philosophy is one of the most enduring, voluminous and influential philosophical traditions in the world. In this module you will encounter some of the most central and well-known Buddhist concepts, texts and thinkers.

In addition to analysing core ideas, such as not-self, dependent origination, emptiness and Buddha nature, you will examine themes that pervade the study of Buddhist philosophy in its various contexts, such as the relationships between teaching and practice, philosophy and literature, and religion and politics. Through reading original Buddhist texts in translation, you will cultivate skills in critical thinking and interpreting primary sources.

In addition to developing the ability to engage in informed argument about key topics in the study of Buddhist philosophy, you will also learn to be reflective about the challenges of studying philosophy from different cultural contexts and time periods.

Create a portfolio of investigative and critical writing which explores a particular philosophical topic in depth. In this module you will be guided with expert support from a contemporary philosopher to develop your philosophical and independent study skills. Through deep engagement with a narrow topic you will develop your ability to assess philosophical arguments and make independent judgements, informed by reasoning and evidence.

In this module you will engage with a text, problem, figure or body of work chosen by an academic within the philosophy team at Lancaster who is a specialist on the topic. You will work with their expert support, in groups and independently.

Project topics offered each year will be drawn from one or more of Lancaster’s many areas of expertise, such as:

• Ethics
• Metaphysics
• Political philosophy
• Applied philosophy
• Social ontology and epistemology
• Philosophy of science
• Philosophy of mind
• The history of philosophy
• Feminist philosophy
• Continental philosophy
• Global philosophy
• Comparative philosophy

The module will equip you with the skills and knowledge you need for further independent writing in your final year of study.

The exploration of philosophical views about the mind is a highly active field on the interface between contemporary philosophy and science.

We will begin with the mind-body problem, that is, how the mental relates to the physical. We will cover key positions on this problem, such as substance dualism, Wittgensteinian ‘dissolving’ approaches, mind-brain identity theory, functionalism, and extended/embedded cognition.

We will also consider the implications of different views on the mind-body problem, such as:

• Could a computer think?
• Can we have a full understanding of how consciousness is possible?
• How can we tell if non-human animals, or even plants, have minds?

In addition, we will cover some of the problems around how we know about mental states:

• Can we be sure of our own mental states?
• How can we find out what other people are thinking or feeling?
• What are the limits of our abilities to understand beings with minds very unlike our own?

What is the role of science in enabling us to make sense of the world? In this module, we explore a series of interconnected questions that lie at the heart of the philosophy of science:

• What distinguishes science from other ways of knowing?
• Is there a scientific method?
• Is science the most reliable way to acquire knowledge?

Throughout the module, you will examine the influential ideas of key twentieth-century philosophers of science, such as Karl Popper's theory of falsifiability and Thomas Kuhn's concept of paradigm shifts. These thinkers challenge traditional views and lay the groundwork for ongoing debates.

Building on their ideas, you will explore contemporary discussions on scientific theories, the nature of scientific progress, and the relationship between science and other forms of knowledge. By the end of the module, you will have gained a deeper understanding of the proper role and limits of science.

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, beginning by looking at rings with certain properties and finding the key examples of these, continuing to describe several constructions that allow us to produce rings with properties we would like. You will conclude by discussing the applications to the areas mentioned above.

Discover key thinkers from what is known as the ‘continental’ tradition of philosophy. In different ways, these thinkers have critiqued the assumptions and methodologies of the western philosophical tradition, as well as its development in Anglo-American philosophy. The particular philosophers considered will vary from year to year, but will include thinkers who have been particularly influential (for example, Hegel, Kierkegaard, Nietzsche and Wittgenstein) as well as more recent continental thinkers (for example, Lyotard, Derrida, Levinas, Žižek, Foucault, Arendt and Beauvoir).

You will engage with original texts, as well as secondary literature, in order to understand and interpret their central arguments. The approach will be predominantly critical rather than historical, encouraging you to assess their distinctive claims, methods and approaches and their wider contribution to philosophy.

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance.

You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proofs of correctness and efficiency of algorithms.

You will gain experience in following and constructing mathematical proofs, correctly and coherently using mathematical notation, and choosing and carrying out appropriate algorithms to solve problems. The module will enable you to develop an appreciation for a range of discrete mathematical techniques.

An inner product space is a real or complex vector space, equipped with certain extra structure that formalises the geometrical notion of orthogonality. It turns out that each inner product space has an intrinsic notion of distance, allowing us to discuss convergence and completeness. Complete inner product spaces are known as Hilbert spaces.

The theory of Hilbert spaces blends linear algebra and (real) analysis. It is a natural and powerful tool for studying problems of quantitative approximation. Furthermore, it provides an abstract framework that can be applied to diverse areas of maths, from differential equations and spectral theory to quantum mechanics and stochastic processes.

This module will introduce you to the theory of Hilbert spaces and prepare for advanced study in functional analysis, approximation theory, signal processing, and statistical learning.

Knots play a fundamental role in many areas of mathematics, from pure topology and algebra through to quantum field theory and protein-folding.

Develop tools to measure knottedness, including geometrical ideas like curvature, knot invariants like the Jones polynomial, and the crucial concept of the fundamental group, which has applications in topology far beyond detecting knots.

Linear systems of differential and integral equations provide a mathematical model for a wide range of real-world devices, including communication systems, 5G networks, electrical circuits, heating systems and economic processes. Mathematical analysis of these models gives insight into the behaviour of these devices, with applications in automatic control, signal processing, wireless communications and numerous other areas.

Linear systems are considered in continuous time that reduce to a standard (A,B,C,D) state space representation. Via the Laplace transform, these are reduced further to rational transfer functions. Linear algebra enables us to classify and solve (A,B,C,D) models, while we describe their properties via diagrams in standard computer software. You will consider feedback control for linear systems, describing the rational controllers that stabilise an (A,B,C,D) system. Alongside the development of analytic methods to study linear systems, you will also gain experience in modelling real-world devices by such systems.

The module commences by looking at classical methods of encryption, discussing their advantages, disadvantages and efficiency. You will also investigate statistical attacks on these methods of encryption and the need for better methods.

After this, you will explore modern methods of encryption that are used in the real-world and rely on the robustness of modular arithmetic. While most encryption methods are still considered secure, you will review potential attacks on these systems (e.g. factorisation algorithms) and situations where bad key generation or implementation has occurred.

Production of a big enough quantum computer renders the above schemes useless. Therefore, you will dive into a short introduction to post-quantum cryptography, including the production of next-gen cryptographic schemes considered to be impenetrable to both classical and quantum computers. You will also explore the theory of lattices and see how these can be used to produce new schemes that may be quantum secure (e.g. NTRU).

A metric space consists of a set, whose elements are called points, and a notion of distance between points governed by three simple rules, abstracted from basic properties of Pythagorean distance in the Euclidean plane. In examples, ‘points’ may be functions where uniformity of convergence can be captured, or binary sequences with applications in computer science, or even subsets of a Euclidean space delivering fractal sets as limits.

Topology goes further, abstracting the notions of continuity and convergence, rendering a teacup and doughnut indistinguishable. A topological space equips each of its ‘points’ with its so-called ‘neighbourhoods’. The few simple principles governing these unlock a robust theory that now pervades the mathematical sciences and theoretical physics.

You will learn the fundamental concepts of completeness, total boundedness for metric spaces, compactness, and the Hausdorff property and metrisability for topological spaces.

In this module you will consider some of the most fundamental questions of existence and ethics. What are you? That is, what is the self? A representation, a subject of experience, a bundle of experiences, an agent, an organism, a person?

We will take a curated path through these interconnected questions, and work together to understand, develop, and communicate answers to questions such as:

• Is the self one thing or many? Or is it nothing, because there is no self?
• Which kinds of things have selves or might be considered to be persons? Only humans, or also non-human animals, artificial intelligences, aliens, corporations, nation-states?
• Is the self found or made? And what does that mean for how we should live?
• Which things have moral standing or rights or responsibilities?
• How do you stay the same through time and change?
• Is self-knowledge valuable?
• What is the good life for a person?

From public philosophy articles online to science communications, from funding bids to policy advisory notes, academic philosophers regularly engage with, inform and persuade audiences outside the field of academic philosophy. To do so, they need to provide compelling, clearly-stated arguments; understand their target audience; and tailor their material to the audience they seek to impact.

You will develop and implement these practical philosophical skills on this module. Choosing from amongst the wide range of philosophical specialisms at Lancaster, and working closely with your academic supervisor to develop a question and relevant reading materials, you will first develop your own philosophical claim or perspective on a topic.

You will then participate in a series of structured workshops where you’ll develop and practice skills in writing for diverse public audiences, present and discuss your ideas and drafts with peers, and work towards the completion of a portfolio of pieces of public-facing philosophy. This module is an opportunity for you to take the philosophical skills and content you have learned over the course of the degree and use them to communicate important ideas beyond the field of academic philosophy.

Together, we create a social world. This world is made up of:

• Groups and institutions such as nations, corporations and educational institutions
• Social products and structures such as money, marriage and class
• Collectively produced outcomes such as global climate change, wars and revolutions, and viral hits and memes

It is a world created by us through language, beliefs and our combined actions and choices, but – at the same time – it cannot easily be changed or ignored and has enormous power over our lives and reality.

In this module you will investigate how exactly our social world is created, sustained and changed, and the ethical and political impacts of this for our lives, and those of others. You will take part in some of contemporary philosophy’s newest and most lively debates in the fields of social ontology, social epistemology and collective ethics, and have the chance to make your own contributions to these new and still-developing fields of research.

You will leave the module with a greater understanding of the complex metaphysical, epistemological and ethical challenges that our social world compels us to address.

Examine key philosophical questions raised by warfighting, from ancient traditions to contemporary debates, and explore some of the central dilemmas faced by soldiers, governments, and non-combatant groups.

You will learn about the ethics of fighting and killing within diverse Just War and critical traditions as well as political, jurisprudential and experiential dimensions of war.

This module will allow you to build on your past studies to develop your knowledge of philosophy and enhance your critical evaluation and argumentative skills by addressing questions such as:

• 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?
• And who has the epistemic authority to speak about war?

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 to representations of directed graphs. We see how Jordan Normal Form in linear algebra is a first theorem of representation theory and methods for complete classification of the building blocks when complete reducibility fails, culminating in connections to Lie theory.