100% of our research impact rated outstanding (REF2021)

Mathematics and computing are intrinsically linked. By combining them in one Master's programme you gain a robust, advanced understanding of the two disciplines, equipping you with sophisticated specialist skills and detailed technical knowledge, allowing you to excel in your chosen career.

Mathematics forms the foundations of all technology and computing, and as such, a rigorous study of the discipline provides invaluable insight and understanding into computer science. Furthermore, computer science is itself a dynamic discipline with a wide range of applications. As a result, this combined programme offers you a robust and comprehensive skill set, in-depth specialist knowledge, and fantastic career opportunities.

You will explore the theory and practice of innovative and experimental computer science, while gaining an advanced understanding of the mathematical concepts and processes behind them. The depth and breadth of knowledge and experience gained over the four years will prove to be a challenging but rewarding opportunity, placing you in the strongest position as you move forward into your chosen career.

During your first year, 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. Additionally, you will be introduced to software development and the fundamentals of computer science. These topics will allow you to gain a wealth of technical knowledge and develop key interdisciplinary skills.

In the second year, you will begin to drill down into specialist maths and computing modules, studying Human-Computer Interaction; Software Design; Linear Algebra; and Social, Ethical and Professional Issues in Computing. These core modules will ensure you gain a solid understanding of the disciplines that is applicable in the real-world. Alongside these, you will also be able to choose from a range of optional maths modules, these include: Abstract Algebra, Complex Analysis, and Real Analysis. In addition, you will bring your skills and knowledge together in a group project, which will allow you to apply what you have learnt to the real-world and gain valuable, practical experience.

For the third year, your study will be largely guided by your own interests. Compulsory modules, such as Artificial Intelligence, Languages and Compilation, and Security and Risk, will enhance and progress your computer science knowledge and provide insight into the sort of activity you will encounter in the real-world. However, the wide range of optional modules you can choose from will allow you to delve deeper into your own interests and customise the year to suit your career ambitions.

The fourth and final year of your degree will introduce a variety of advanced modules for you to choose from. You can build a strong repertoire of maths and computer science skills and knowledge, to suit your interests and goals, including: Data Mining; Galois Theory; Lie Groups and Lie Algebras; Operator Theory; and Systems Architecture and Integration. You will also benefit from our Research Methods module, which will provide you with a formal understanding of research, and allow you to gain the appropriate skills and practices. You will learn to critically reflect on your research and will gain an appreciation of the different ways that other disciplines, academic communities and industries conduct research. This will provide invaluable insight and experience for many graduate careers as well as for continuing in academia.

Discover what studying Computer Science and 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.

Mathematics and Computing are both fundamental disciplines within our modern world, providing you with the skills to tackle a wide range of exciting challenges. The combination of the abstract reasoning and analytical thinking inherent to maths, in conjunction with the practical coding and problem-solving abilities you’ll develop by studying computing will set you apart from the crowd when it comes to finding employment. Alumni from our Computer Science and Mathematics degrees have found careers in data analysis, software engineering, finance, and even higher management. Our graduates are well-paid too, with the median starting salary of graduates from our Mathematics and Computer Science degrees being £28,250 and £30,361 respectively (HESA Graduate Outcomes Survey 2023).

Here are just some of the roles that our BSc and MSci Computer Science and Mathematics students have progressed into upon graduating:

Software Engineer - Dolby Digital

Graduate Trainee – Sellafield Ltd

Cyber Security Assurance Manager – BAE Systems

Frontend Engineer – Seaquake

Software Developer – Sky

NHS Digital Graduate – NHS

Lead Data Analyst – NFU Mutual

Programmer – Quanticate

Statistical Officer – Department for Education

Statistician – AstraZeneca

Technology Associate – Goldman Sachs

Consultant - Deloitte

Lancaster University is dedicated to ensuring you not only gain a highly reputable degree, you also graduate with the 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 A*AA including A level Mathematics or Further Mathematics OR AAA 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 38 points overall with 17 points from the best 3 Higher Level subjects including 6 in Mathematics HL (either analysis and approaches or applications and interpretations)

BTEC Accepted alongside A level Mathematics grade A and Further Mathematics grade A

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.

Computing and data drive many critical elements of modern society, directly or indirectly. It’s vital that there is a strong theoretical foundation to computer science. This module begins by examining the hard questions central to computer science and reasoning itself to prepare students for the in-depth critical thinking and discussion required at university level. Students will cover the fundamentals in logic, sets, and mathematics of vectors, matrices, and linear algebra which have practical applications in software such as computer graphics. Algorithms, abstract data types, and analysis of algorithms is introduced to allow our students to make reasoned decisions about the design of their programs. Finally, they will get the chance to investigate and apply the principles of Data Science to select, process, and analyse data, and examine the way programs and systems can be designed to efficiently support work with data and question the limits of conclusions that can be drawn from such systems.

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.

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.

Software now forms a central aspect of our lives. From the applications we run on our phones to the satellites in space, all modern technology is enabled by software. This module provides an introduction to the field of Software Development - the processes and skills associated with designing and constructing computer programs. Students are not expected to have any previous experience with the field of computing, and will study the contemporary knowledge, skills and techniques needed to develop high-quality computer software. This includes a thorough treatment of the principles of computer programming and how these principles can be applied using a range of contemporary and established languages such as Python, JavaScript and C. They will discover how programming languages can be classified and how to choose the best language for the task at hand.

Students will also investigate and apply the practical Software Engineering skills needed to ensure software is correct, robust and maintainable. These include techniques for problem analysis, design formulation, programming conventions, software commenting and documentation, testing and test case design, debugging techniques and version control.

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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The group project will give students experience in executing a project through all stages, working to the demands of a client, and practically combine and apply concepts and skills gained in other modules studied so far in their programme. Students will learn to apply their knowledge about prototyping, project planning, management, design, and user evaluation or testing strategies. Teams will deliver reports, code, and demonstrate a working system. They will also communicate their work through reports, demonstrations, and presentations.

The project content may differ from year to year, and groups may be able to select projects aligned with the School’s main themes of Software, Systems, Data and Theory, Interactions and Implications, and Cyber Security, although each theme may not be available every year. Example topic areas could be desktop application development, game programming, computer graphics, user interfaces, mobile computing, or other areas. The exact requirements of a group project will vary according to the focus of its theme; however the course structure of a group project will be the same between themes and different years. Students will receive about 30 hours of workshop contact time throughout the module, in addition to lectures, and then will be expected to work independently as a group.

To support this practical activity, two strands of lectures are delivered. One covers programming and continues the development of the students practical programming skills to allow them to confidently contribute to larger, team-based programming projects. The second covers teamwork, project management, risks, and costings so that the student has a sound base for managing collaborative projects.

Most computing systems are interactive and have people in the loop. Human-computer interaction (HCI) is concerned with all aspects of designing, building, evaluating, and studying systems that involve human interaction. From a computing perspective, students focus on enabling interaction through user interfaces, and on creating interactive systems that are usable and provide a good user experience.

The module introduces students to the foundations of HCI in understanding human behaviour, technologies for interaction, and human-centred design. Students will review human perception, cognition and action and relate these to design principles and guidelines; discuss different user interface paradigms and key technologies such as pointing; and introduce practical methods for design and evaluation with users.

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.

Students will gain the essential skills and knowledge to operate within the professional, legal and ethical frameworks of their profession. Techniques for breaking down a project into manageable parts and efficient time allocation are taught, leading to a fundamental understanding of the skills and methods required to pursue scientific inquiry and the fundamental concepts and tools for statistical analysis to measure and explain data. Exemplars and guidelines on producing concise and structured scientific reports are offered and students receive additional lectures on presentation skills, professional ethics in relation to computing and communications. Finally, lectures provide an awareness of fundamental legal aspects related to a profession in computing and communications, including intellectual property rights and patent law.

Throughout this course, students will gain a high level of awareness of subject specific skills and general competence needed to gain employment in their field. The module develops academic writing and research skills in a computing context, complimenting students’ other modules.

Software development is a collaborative and professional process, requiring far more than a single individual undertaking programming activities. This module investigates the processes, tools, techniques, and notations required to successfully undertake the development of commercial grade software.

Focussing on the key non-functional parameters of software reuse, scalability, maintainability, and extensibility, students will explore the benefits brought by the rigour associated with object-oriented, strongly typed languages (such as Java). Students will practice the concepts of composition, inheritance, polymorphism, interfaces, and traits and the commonly employed design patterns that they enable. They will also study the processes and notations associated with defining the relationship and behaviour of complex computer software systems.

Optional

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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.

In this module, students will develop their ability to solve complex mathematical problems using computers. They will learn basic programming techniques, functions and syntax within Python, good programming practice, as well as how to code simple algorithms. Within this module, students will undertake a large programming project in which they will explore a mathematical topic from a range of choices, such as cryptography, error-correcting codes, or finite group theory, allowing them to explore an area of computational mathematics that interests them. Upon successful completion of the module, students will be able to write and debug functions and programs in Python, understand the benefits and limitations of using computers to solve mathematical problems, and produce a report that incorporates text, mathematics, and Python code.

In this module, students will develop their ability to solve complex mathematical problems using computers. They will learn basic programming techniques, functions and syntax within R, good programming practice, as well as how to code simple algorithms. Within this module, students will undertake a large programming project in which they will explore a mathematical modelling topic. The project itself will consolidate learning from the module, and will answer questions about real-world systems such as epidemics or gravitational bodies. Upon successful completion of the module, students will be able to write and debug functions and programs in R, understand the benefits and limitations of using computers to solve mathematical problems, and produce a report that incorporates text, mathematics, and R code.

In this module, students will develop their ability to solve complex mathematical problems using computers. They will learn basic programming techniques, functions and syntax within R, good programming practice, as well as how to code simple algorithms. Within this module, students will undertake a large programming project in which they will explore a mathematical topic from a range of choices, such as Markov chains or Monte Carlo simulations, allowing them to explore an area of computational statistics that interests them. Upon successful completion of the module, students will be able to write and debug functions and programs in R, understand the benefits and limitations of using computers to solve mathematical problems, and produce a report that incorporates text, mathematics, and R code.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the discrete mathematics and computing strands of mathematics, such as Ramsey's theorem, the birth of computer science, and Game theory. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the dynamics and analysis strands of mathematics, such as continued fractions, branching processes, and Kepler's laws. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the geometry and algebra strands of mathematics, such as symmetry in music, error-correcting codes, and counting equivalent patterns. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

In this module, students will learn the basics of producing a mathematical project, getting to grips with the fundamentals of scientific writing and working within a scientific environment. They will learn how to utilise LaTeX for presenting mathematical formulae, using it to create a report on a given mathematical topic. Students will also undertake a group presentation and deliver it to the class, as well as collaborating with peers (under the direction of a supervisor) to produce a report on an area within the modelling and probability strands of mathematics, such as Markov chains, beyond the Central Limit Theorem, and epidemic modelling. Upon successful completion of the module, students will be able to produce documents which accurately and effectively communicate scientific material, and be able to work independently under supervision and as part of a small group.

Probability provides the theoretical basis for statistics and is of interest in its own right.

Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.

Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

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.

Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.

The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.

Optional

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Students will be given a solid foundation in the basics of algebraic geometry. They will explore how curves can be described by algebraic equations, and learn how to use abstract groups in dealing with geometrical objects (curves). The module will present applications and results of the theory of elliptic curves and provide a useful link between concepts from algebra and geometry.

Students will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally, they will learn to perform algebraic computations with elliptic curves.

Students will gain an introduction to fundamental concepts in artificial intelligence and learn about current trends and issues. Topics such as Knowledge Representation and Reasoning, Decision Making (DM) and Decision Making Under Uncertainties, and Probability Theory are all explored throughout the course. Artificial Intelligence offers experience in supervised and unsupervised machine learning, neural networks and decision trees. Multivariate methods, and clustering and classification approaches are taught and there is an introduction to evolutionary algorithms, phenotypes, genotypes and fundamental genetic operators. Programming languages suitable for intelligent systems, such as Scheme and Prolog are investigated and students are made familiar with the applications of artificial intelligence.

This module sees an awareness of the requirements of artificial intelligence systems in general, and in the context of computing and communications systems. Through knowledge based, probabilistic and logical systems, the module provides students with an awareness of competing approaches and a broad grounding in artificial intelligence. Additionally they will understand and critically analyse artificial intelligence techniques used in modern computers and mobile devices.

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.

Much of contemporary statistics and AI assumes that data follows a fixed pattern for all time. However we know that change happens, and it is critically important that methods can detect and adapt to these changes. Efficient change-detection can either be a tool to recognise that something has changed in an underlying system (e.g. in medical or financial data collected through time) or an important component of a larger system that enables efficient model-fitting and prediction “in between” the changes. Lancaster University is a world-leading centre of excellence in developing methods to detect changes in data, with our methods deployed in domains such as retail, finance, medicine, space exploration, environmental science, cybersecurity and engineering.

This module introduces students to changepoint detection motivated by its applicability in a variety of different settings. Changepoints are sudden, and often unexpected shifts in the behaviour of a physical, biological, industrial or financial process. Students will understand the importance of identifying these shifts, and the module will cover the fundamental concepts within changepoint detection, the key approaches to At Most One Change (AMOC), and multiple changepoint identification. Students will explore key challenges around model selection concepts and techniques through a variety of models used in real-world applications, including changes in mean, variance, and regression. Successful completion of this module will see students be able to understand the importance of changepoint methodology, communicate technical ideas, critically evaluate approaches to solve problems, and make conclusions based on evidence in relation to a real-world problem.

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.

Students will be exposed to a range of current computer science related topics from different subject areas. The areas covered come from our different thematic strands and will include: natural language engineering; policy based network resilience; eye-tracking for ubiquitous computing applications; and a focus on energy aware control and sensing in home environments.

Students will conduct independent and in-depth research into an advanced topic of computing or communications, reflecting current topical and research issues. During the course of the module, students will analyse, structure, summarise, document and present findings in front of a large group. They will gain topical knowledge and skills related to the subject areas of the seminars, and will learn with and from their peers. The module will enable students to produce a detailed document describing their research findings, present technically intricate issues in a coherent manner, and discuss and defend their position on a specific topic within a seminar group.

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.

This module introduces students to two classes of continuous-time, dynamic compartmental models that describe the time evolution of natural systems such as epidemics, biological reactions in a cell, and the interactions between populations of predators and prey in an environment: Markov jump processes (MJPs) and ordinary differential equations (ODEs).

Students will examine model formulation in numerous real-world scenarios, delve into some tractable and partially-tractable examples, and use numerical simulation as a general method for visualising and classifying behaviour. Students will also examine stationary distributions and large n limits in order to further their understanding of MJPs, and find and examine stationary points and their stability to get to grips with ODEs, allowing students to justify their use within a variety of real-world processes. Upon successful completion of this module, students will be able to construct a mathematical model using real-world evidence.

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.

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.

All programming languages are based on theoretical principles of formal language theory. In this module, students take a deep dive into formal languages representation and grammars, and how relate to programming language compilers and interpreters.

Students will study formal language syntax and semantics, phrase structure grammars and the Chomsky Hierarchy. They will learn how to classify languages and explore the concepts of ambiguity in Context Free grammars and its implications. In particular, they will learn about the compilation process including lexical analysis and syntactic analysis, recursive descent parsers, and semantic analysis. Finally, students get to investigate the synthesis phase, where intermediate representations, target languages, and structures lead to code generation. In the School, we blend lectures with small group lab sessions where students gain hands-on experience of applying such theory.

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 aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.

In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.

Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.

The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a (A,B,C,D) system.

Cryptography is both a vital and thriving area of research, whose central aim is to provide robust methods of keeping sensitive data secure, validated and authenticated. This is clearly a highly impactful area, especially in an age where much of our sensitive data is being stored online. Without solid security protocols in place, hackers can freely access your data and use it for criminal gain (e.g. stealing money from your online banking, accessing/modifying your sensitive health data, selling your personal data on the cyber black market or using it to commit fraud).

This module provides an introduction to mathematical cryptography, introducing students to a range of methods of encryption, from the more classical forms (such as Caesar Shift, Vigenère and Enigma), to more contemporary methods of encrypting data (eg. RSA, El Gamal and Knapsack). During this module, students will consider the advantages, disadvantages, and efficiency of the different cryptographical approaches. They will also explore examples of potential attacks on these hard problems (e.g. Trial Division, Fermat’s method, Pollard rho, Dixon, Baby-step giant-step, Index calculus), and demonstrate attacks in situations where bad key generation or implementation has occurred.

The module will culminate in a brief introduction to Post-Quantum Cryptography, the theory of lattices, tough problems, and how these lead to schemes that are considered impenetrable to both classical and quantum computers. Successful completion of this module will see students be able to formulate real-world problems in mathematical formality, and recognise the applications of cryptography to other mathematical fields, such as Algebra and Number Theory.

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.

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.

The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.

Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.

They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

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.

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 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.

Students will cover the basics of ordinary representation theory. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between the two will be discussed.

The second part is an introduction to the ordinary character theory of finite groups, intrinsic to representation theory. Students will learn the concepts of R-module and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups.

They will also learn to perform computations with representations and morphisms in a selection of finite groups

Covering a range of topics, including asset identification and assessment, threat analysis and management tools and frameworks, students will become familiar with attack lifecycle and processes, as well as risk management and assessment processes, tools and frameworks. The module covers mitigation strategies and the most appropriate mitigation technologies and offers knowledge on assurance frameworks and disaster recovery planning. There is also an opportunity to learn about infrastructure design and implementation technologies and attack tree and software design evaluation.

Students will gain an understanding of the different ways in which an IT professional can make effective decisions when securing an IT infrastructure. The course will make them aware of the tools, frameworks and models that can be used to identify assets, threats and risks, before selecting the most appropriate strategies to manage the exposure that IT infrastructure faces in the light of this analysis. The module builds on their skills with a practical examination of the mechanisms by which IT infrastructures are attacked.

The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).

Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.

As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).

The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Modern statistics is characterised by computer-intensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.

Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.

Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.

Lastly,students will spend time looking at Change-Point Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.

Core

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Students will gain a formal understanding of research and will develop the ability to critically reflect on research approaches and practices in the field of computing. Research Methods will also encourage an appreciation of the different ways that other disciplines, academic communities and industries all conduct research. There will be an opportunity to plan a research project and develop a convincing study design to address a challenge or problem. This module explores ethical and data management issues associated with research as well as research and innovation practices in industry.

The module covers the fundamentals of research such as sampling and design, before considering strategies and research methods. Furthermore, the module offers greater insight into research design, such as how to structure and frame research studies, choosing a research strategy and selecting the best research method. Students will learn about ethical issues in research and approval processes before understanding the opportunities and expectations from their industrial placements.

Optional

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This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.

Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

This module provides students with up-to-date information on current applications of data in both industry and research. Expanding on the module ‘Fundamentals of Data’, students will gain a more detailed level of understanding about how data is processed and applied on a large scale across a variety of different areas.

Students will develop knowledge in different areas of science and will recognise their relation to big data, in addition to understanding how large-scale challenges are being addressed with current state-of-the-art techniques. The module will provide recommendations on the Social Web and their roots in social network theory and analysis, in addition their adaption and extension to large-scale problems, by focusing on primer, user-generated content and crowd-sourced data, social networks (theories, analysis), recommendation (collaborative filtering, content recommendation challenges, and friend recommendation/link prediction).

On completion of this module, students will be able to create scalable solutions to problems involving data from the semantic, social and scientific web, in addition to abilities gained in processing networks and performing of network analysis in order to identify key factors in information flow.

In this module we explore the architectural approaches, techniques and technologies that underpin today's Big Data system infrastructure and particularly large-scale enterprise systems. It is one of two complementary modules that comprise the Systems stream of the Computer Science MSc, which together provide a broad knowledge and context of systems architecture enabling students to assess new systems technologies, to know where technologies fit in the larger scheme of enterprise systems and state of the art research thinking, and to know what to read to go deeper.

The principal ethos of the module is to focus on the principles of Big Data systems, and applying those principles using state of the art technology to engineer and lead data science projects. Detailed case studies and invited industrial speakers will be used to provide supporting real-world context and a basis for interactive seminar discussions.

Students are provided with a comprehensive coverage of the problems related to data representation, manipulation and processing in terms of extracting information from data, including big data. They will apply their working understanding to the data primer, data processing and classification. They will also enhance their familiarity with dynamic data space partitioning, using evolving, clustering and data clouds, and monitoring the quality of the self-learning system online.

Students will also gain the ability to develop software scripts that implement advanced data representation and processing, and demonstrate their impact on performance. In addition, they will develop a working knowledge in listing, explaining and generalising the trade-offs of performance, as well as the complexity in designing practical solutions for problems of data representation and processing in terms of storage, time and computing power.

This module will equip students with the ability to develop and apply a deep understanding of fundamental principles, techniques and technologies that underpin today's global IT infrastructure. They will learn to assess new systems technologies, to know where technologies fit in a comprehensive schema, and to know what to read in order to develop a deeper level of understanding. Students will focus on the properties of system components, and will become familiar with the strengths, weaknesses, scalability and bottlenecks of systems components. This will enable them to make intelligent and well-reasoned trade-offs between fundamental building blocks of distributed systems in today’s IT infrastructure.

This is a highly practical module, in which students build a major system representative of an end-to-end IT infrastructure, and is also highly discussion-oriented, with frequent in-class discussion sections and problem-solving group work. The module covers a very broad range of state-of-the-art techniques and principles of modern distributed systems. These including: caching, tiering, replication, synchronisation, failure and reliability. Students will also explore real-world technologies, from interaction paradigms in distributed systems, to peer-to-peer architectures and scalable and high-performance networking and storage.

The Fourth Year Project will focus on a significant specification, design, implementation and/or evaluation project at the suitable level for an MSci qualification. The project sees students tackle a real-world problem by applying their knowledge in computer science. The project is usually achieved in conjunction with an industry placement; however, it can be completed at the University. Suggestions made by industry will be vetted by a team of academics to ensure appropriate depth, and if no suitable project with industry can be found, one will be provided by academic staff.

Weekly guidance is given from a member of academic staff from the department to ensure the necessary level of academic content and rigour is being maintained. There are also Business Development mentors in the Knowledge Business Centre (KBC) to provide students with an insight into the day-to-day expectations and responsibilities of working with industry. The Fourth Year Project is designed to challenge students and develop their existing knowledge, understanding and skills from their undergraduate degree to produce a significant piece of academically rigorous project work.

Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1-to-1 correspondence with the subgroups of G.

In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.

The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.

In the first term there will be weekly supervisor meetings and students will be guided into their in-depth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.

Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Students complete a 10 week industrial placement in the Lent term of their 4th year. The University has a range of businesses from SMEs to large corporates for students to be placed in. There are no taught elements in this module, but students have access to an academic supervisor to guide and assist them during the placement. Placements are assigned to students in the Michaelmas term.

Students will gain first-hand experience of working in a contemporary ICT related environment, developing an appreciation and understanding of professional practices and codes of conduct in industrial, commercial and professional settings. Companies will set tasks that are related to students’ knowledge and experience gained throughout their degree, allowing them to apply it in a professional setting. Placements are offered by a variety of companies with different topics. Through an initial matching and application process, we ensure the biggest possible overlap between student interest and company requirements.

Explore advanced topics in experimental Human-Computer Interaction (HCI), such as understanding users and their requirements, investigating design spaces and prototyping and developing innovative interaction techniques. The module offers increased experience in HCI literature and design methods both with and without users, as well as practical experience of using supporting tools. Students will learn about modelling techniques and design space techniques as part of the module.

Upon completion of the module, students will have the knowledge to conduct experimental HCI research and have the motivation, experience and tools for understanding users and their requirements for interaction. The module helps to develop scientific writing skills and analytical thinking and prepares students for further postgraduate study, or for a successful career in IT or computing.

Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.

The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.

The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.

In the first term they will meet their supervisor weekly and will be guided into their in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.

Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

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.

In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T

They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems.

Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.

First students will review Hilbert spaces, before spending some time studying infinite-dimensional operators, notably the unilateral shift and multiplication operators, as well as basic concepts. They will then consider the criteria for invertibility of self adjoint operators, leading to the spectral theory of such operators.

The aim of this module is to develop an analytical and axiomatic approach to the theory of probabilities.

Students will consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. They will then use random variables and the expectation to develop a probability calculus, which is applied to achieve laws of large numbers for sums of independent random variables. Finally the characteristic function will be used to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.

In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of R-module and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.

The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.

The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

At the end of Year 3 students will fill in a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student's degree scheme and choice.

During the first term students will meet their supervisor on a weekly basis and will be guided into an in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.

Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discrete-time stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuous-time processes, in particular, Brownian motion.

Students will learn to derive basic properties of Brownian motion and explore integration with respect to it. They will also examine the derivation of Ito's formula and how to apply this to Brownian motion.

Over the course of the module, students will also learn to justify and critique the use of stochastic models for real-life applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.

In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.

Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.

This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.

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.

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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.

Undergraduate open days 2024

Our summer and autumn open days will give you Lancaster University in a day. Visit campus and put yourself in the picture.

Our historic city is student-friendly and home to a diverse and welcoming community. Beyond the city you'll find a stunning coastline and the picturesque Lake District.