100% of our research impact rated outstanding (REF2021)

Mathematics and statistics underpin all aspects of modern life: advances in science, innovations in medicine, new technology, and business. This Master's programme provides a comprehensive, advanced study, while allowing you to experience a new culture overseas.

Mathematics and statistics allow us to understand our world at the most fundamental level. Studying these subjects provides us with information in its purest form, giving us answers to problems from all aspects of life. You can learn to study and predict patterns and changes in everything, including human behaviour, plant growth, chemical reactions, and stocks and shares.

Our four year MSci Mathematics with Statistics (Study Abroad) pathway gives you the exciting opportunity to experience a new culture and society by spending a year studying overseas. You will also gain a deeper specialisation in mathematics, and the chance to develop your research skills by undertaking a dissertation or industry research project.

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.

In the second year, you will further develop your knowledge in analysis, algebra, probability and statistics. You will also be introduced to Computational Mathematics, exploring the theory and application of computation and numerical problem-solving methods. While studying these topics, you will complete our Project Skills module, which provides you with the chance to enhance your research and employment skills through an individual and group project. Additionally, you will gain experience of scientific writing, and you will practise using statistical software such as R and LaTeX.

Your third year will be spent at one of our partner institutions in North America, Australia and New Zealand where you will experience a new culture and increase your academic network. Following your rigorous study in the first two years, the third year offers a wide range of specialist optional modules, allowing you to develop and drive the programme to suit your interests and guide you to a specific career pathway.

Lancaster University will make reasonable endeavours to place students at an approved overseas partner university that offers appropriate modules which contribute credit to your Lancaster degree. Occasionally places overseas may not be available for all students who want to study abroad or the place at the partner university may be withdrawn if core modules are unavailable. If you are not offered a place to study overseas, you will be able to transfer to the equivalent standard degree scheme and would complete your studies at Lancaster.

Lancaster University cannot accept responsibility for any financial aspects of the year or term abroad.

In the fourth year, you will be able to widen your knowledge and skills by selecting from a pool of optional Master's-level modules. Some of these will be familiar from third year, while others will be new to this year. These advanced modules allow you to exercise what you have learnt during the programme, while expanding and evolving your skill set further.

You will also complete a major research project in either pure maths or statistics, guided by your interests and supervised by an active researcher, or undertaken as part of a collaborative industry project. This will cement your learning, provide you with valuable experience and position you for a career in maths or academia and research.

Lancaster University will make reasonable endeavours to place students at an approved overseas partner university that offers appropriate modules which contribute credit to your Lancaster degree. Occasionally places overseas may not be available for all students who want to study abroad or the place at the partner university may be withdrawn if core modules are unavailable. If you are not offered a place to study overseas, you will be able to transfer to the equivalent standard degree scheme and would complete your studies at Lancaster.

Lancaster University cannot accept responsibility for any financial aspects of the year or term abroad.

Discover what studying Mathematics and Statistics 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.

Successful graduates who have completed the required modules are eligible to apply to the Royal Statistical Society for accreditation.

The Institute of Mathematics and its Applications is a chartered professional body for mathematicians in the UK. Accreditation means that our degrees demonstrate both a high level of competency and professionalism in the area of mathematics.

Mathematics is fundamental to the way our modern world works and is an essential tool to solving many of the challenges we face on a day-to-day basis. The problem-solving and analytical thinking skills you’ll gain from a degree in mathematics will be applicable in almost every industry and sector, making you a highly in-demand employee. From business and finance to technology, IT and health, there are a wealth of opportunities open to graduates of mathematics. Many are grounded within mathematical principles (including data analysis and programming), but there are also less traditionally mathematical roles that rely on skills gained during your undergraduate degree, such as positions within higher management, engineering, and teaching. Our graduates are well-paid too, with the median starting salary of graduates from our Mathematics degrees being £28,250 (HESA Graduate Outcomes Survey 2023).

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

Actuarial Analyst – Just Group Plc

Analytics Engineer – Thread

Statistical Officer - HMRC

Data Analyst – William Hill

Finance Modelling Analyst – KPMG

Trial Statistician – Liverpool Clinical Trials

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.

Lancaster University offers a range of programmes, some of which follow a structured study programme, and some which offer the chance for you to devise a more flexible programme to complement your main specialism.

Information contained on the website with respect to modules is correct at the time of publication, and the University will make every reasonable effort to offer modules as advertised. In some cases changes may be necessary and may result in some combinations being unavailable, for example as a result of student feedback, timetabling, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes and new research. Not all optional modules are available every year.

Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.

The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.

Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.

In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.

The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

This course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.

In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure and temperature, and the direction of the wind. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

The main focus of this module is vectors in two and three-dimensional space. Starting with the definition of vectors, students will discover some applications to finding equations of lines and planes, then they will consider some different ways of describing curves and surfaces via equations or parameters. Partial differentiation will be used to determine tangent lines and planes, and integration will be used to calculate the length of a curve.

In the second half of the course, the functions of several variables will be studied. When attempting to calculate an integral over one variable, one variable is often substituted for another more convenient one; here students will see the equivalent technique for a double integral, where they will have to substitute two variables simultaneously. They will also investigate some methods for finding maxima and minima of a function subject to certain conditions.

Finally, the module will explain how to calculate the areas of various surfaces and the volumes of various solids.

Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.

The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.

The student will learn how to express a linear transformation of the real Euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.

The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.

The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.

The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.

Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.

Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.

To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.

The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.

Core

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This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here, students will select a small number of properties which these and other examples have in common, and use them to define a group.

They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.

Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.

Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.

The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.

They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.

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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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.

Core

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Students on the Study Abroad scheme spend their third year studying at one of our partner universities. They are matched to a suitable institution so they can make the most out of this exciting opportunity. During the year abroad students will study all the compulsory topics that must be covered in the third year of the degree, but they will also have the benefit of choosing topics that are not offered at Lancaster. A personal tutor will help them develop their study programme and will keep in touch whilst they are studying abroad.

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.

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 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.