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 Mathematics with Statistics (Study Abroad) MSci Hons (G1GH)
Mathematics with Statistics (Study Abroad) MSci Hons  2020 Entry
UCAS Code
G1GH
Entry Year
2020
also available in 2019
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 4 Year(s)
Course Overview
Mathematics and statistics underpin all aspects of modern life: advances in science, innovations in medicine, new technology, and business. This Masters 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, problemsolving 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 problemsolving 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, 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.
In the fourth year, you will be able to widen your knowledge and skills by selecting from a pool of optional Masterslevel 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.
Entry Requirements
Grade Requirements
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
IELTS 6.0 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics and/or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
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.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
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.

Convergence and Continuity
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.

Discrete Mathematics
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.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. They will see how partial derivatives can help to understand surfaces, while repeated integrals enable them to calculate volumes. The module will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
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, temperature and wind direction. To study functions of several variables, rates of change are introduced with respect to several quantities. How to find maxima and minima will be explained. Applications include the method of least squares. Finally, various methods for solving differential equations of one variable will be investigated.

Geometry and Calculus
The main focus of this module is vectors in two and threedimensional 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.

Integration and Differentiation
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.

Linear Algebra
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.

Numbers and Relations
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
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.

Statistics
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
Year 2

Abstract Algebra
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 wayto 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
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 nonconstant 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.

Computational Mathematics
Students will gain a solid understanding of computation and computer programming within the context of maths and statistics. This module expands on five key areas:
 Programming and R
 Numerical solutions of equations
 Numerical differentiation and integration
 Monte Carlo methods
 Numerical solutions to ODEs
Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixedpoint iteration, higher dimensions, first and second derivatives, nonparametric bootstraps, and modified Euler methods.
Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation.
Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics.

Linear Algebra II
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 structurepreserving 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 II
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.

Project Skills
Project Skills is a module designed to support and develop a range of key technical and professional skills that will be valuable for all career paths. Covering five major components, this module will guide students through and explore:
 Mathematical programmes
 Scientific writing
 Communication and presentation skills
 Short projects
 Group projects
Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures.
Scientific writing, communication and presentations skills will also be developed. Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations.

Real Analysis
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 Alevel.
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 Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics II
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 likelihoodbased 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.
Core
Year 3

Study Abroad Year
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.
Core
Year 4

Algebraic Curves
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.

Assessing Financial Risk: Extreme Value Methods
Assessment of financial risk requires accurate estimates of the probability of rare events. Estimating the probability of such "extreme" events is challenging, as by nature they are sufficiently rare that there is little direct empirical evidence on which to base inference. Instead we have to extrapolate based on the past frequency of the occurrence of less extreme events. This module covers ideas from Extreme Value Theory which give a sound mathematical basis to such extrapolation, and shows practically how it can be used to give accurate assessments of financial risk in a wide range of scenarios.
In this module, students will study topics related to the understanding of special models to describe the extreme values of a financial times series, and they will learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. Extreme value models will be used to evaluate Value at Risk and gain an understanding of the impact of heavy tailed data on standard statistical diagnostic tools.

Bayesian Inference
Bayesian statistics is a framework for rational decision making using imperfect knowledge, expressed through probability distributions. Bayesian principles are applied in the fields of navigation, control, automation and artificial intelligence. The aim of decision makers is to make rational decisions that maximise some personal utility function which may represent quantities such as money which are related to the wealth of an individual.
Within the Bayesian framework, knowledge of the world, (the prior) is updated as fresh observations arrive to yield a posterior distribution which shows the revised knowledge. The evidence for the model is expressed by calculating a marginal likelihood. Future behaviour and the fit of the model are assessed using a predictive distribution. This includes sampling uncertainty and uncertainty of our knowledge.
In this module students will look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multiparameter fully conjugate models. The range of belief types that can be modelled by using mixtures of conjugate priors will be extended, and will also explore the use of nonconjugate formulations of models and use MonteCarlo integration, importance sampling and rejection sampling for calculating and simulating from these distributions.

Clinical Trials
Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancyinduced hypertension; or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients. Treatments may be procedural, for example, surgery or methods of care.
This module combines the study of technical methodology with discussion of more general research issues. First , the relative advantages and disadvantages of different types of medical studies will be discussed. Then students will explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. They will also cover crossover trials, concepts of sample size determination, and equivalence trials. The module also includes a brief introduction to sequential trial designs and metaanalysis.

Computer Intensive Methods
Students will be introduced to Markov chain Monte Carlo methods and how to use them as a powerful technique for performing Bayesian inference on complex stochastic models.
The first part of the module looks in detail at the necessary concepts and theory for finite statespace Markov chains, before introducing analogous concepts and theory for continuous statespace Markov chains. In the second part of the course module the MetropolisHastings algorithm for sampling from a distribution known up to a constant of proportionality will be investigated.
In the third (and largest) part, students will take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. They will also examine the two most common MetropolisHastings algorithms (the random walk and the independence sampler). Examples will include hierarchical models, random effects models, and mixture models.

Environmental Epidemiology
This module focuses on the kinds of statistical methods commonly used by statisticians to investigate the relationship between risk of disease and environmental factors.
You’ll explore methods for the analysis of spatial data, including spatial pointprocess models, spatial casecontrol methods, spatially aggregated data, point source problems and geostatistics, and will spend time developing your skills performing similar analyses using the statistical package R.

Financial Mathematics
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 BlackScholes 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
 BlackScholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated riskneutral probability; performing calculations with the BlackScholes formula; and proving various steps in the derivation of the BlackScholes 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 realworld; improve their written and oral communication skills; and develop their critical thinking.

Galois Theory
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 1to1 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.

Generalised Linear Models
In this module students will learn techniques for formulating sensible models for data, enabling you to tackle problems such as the probability of success for a particular treatment, and how this depends on the patient's age, weight, blood pressure, and so on.
Students will be introduced to a large family of models, called the generalised linear models (GLMs), including the standard linear regression model as a special case, and will have the opportunity to discuss and investigate the theoretical properties of these models.
A common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters will be studies. Using the statistical package ‘R’, you’ll fit and check these models, and will produce confidence intervals and tests corresponding to questions of interest.

Hilbert Spaces
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.

Industrial Dissertation
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 indepth 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.

Lebesgue Integration
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.

Lie Groups and Lie Algebras
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.

Likelihood Inference
In this module, students will learn how to use the likelihood function to obtain and summarise information about unknown parameters. They will calculate the likelihood function for statistical models which do not assume independent identically distributed data, and will learn to evaluate point estimates and make statements about the variability of these estimates.
Using the statistical package ‘R’,computational methods will be used to calculate maximum likelihood estimates. Time will be spent developing an understanding of the interrelationships between parameters, and the concept of orthogonality. In addition to this, hypothesis tests will be performed using the generalised likelihood ratio statistic.

Longitudinal Data Analysis
Longitudinal data arise when a timesequence of measurements is made on a response variable for each of a number of subjects in an experiment or observational study. For example, a patient’s blood pressure may be measured daily following administration of one of several medical treatments for hypertension.
Typically, the practical objective of most longitudinal studies is to find out how the average value of the response varies over time, and how this average response profile is affected by different experimental treatments. This module presents an approach to the analysis of longitudinal data, based on statistical modelling and likelihood methods of parameter estimation and hypothesis testing.

Mathematics Dissertation
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 indepth 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.

Metric Spaces
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
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 infinitedimensional 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.

Principles of Epidemiology
This module focuses on the basic principles of epidemiology, including its methodology and application to prevention and control of disease.The concepts and strategies used in epidemiologic studies will be examined, and students will come to appreciate a historical and general overview of epidemiology and related strategies for study design. They will also develop their knowledge of how to apply basic epidemiologic methods, and will gain confidence in assessing the validity of epidemiologic studies with respect to their design and inferences. Most inference will be likelihood based, although the emphasis is on conceptual considerations and interpretation.

Probability Theory
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.

Project Skills
This module gives students the opportunity to enhance their project skills, including both subjectrelated and transferable skills such as oral communication and scientific writing. They will work on mathematical document preparation and presentation, and with a statistical software package.
The module will revisit LaTeX and R, which will be used to prepare mathematical documents; text and mathematical symbols, displayed formulae, numbering, environments, lists, page and document layout, sections and table of contents, and tables and figures. In addition to this, students will undertake a short project, investigating a mathematical or statistical topic, and a group project, involving a group investigation of a mathematical or statistical topic under the direction of a supervisor.

Representation Theory of Finite Groups
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of Rmodule 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.

Statistics Dissertation
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 indepth 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 for Finance
Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discretetime stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuoustime 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 reallife applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

Stochastic Processes
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.

Topology and Fractals
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.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.
Careers
Careers
Maths graduates are very versatile, having indepth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including data analysis and manipulation, logical thinking, problemsolving and quantitative reasoning, as well as adept knowledge of the discipline. As a result, mathematicians are sought after in a range of industries, such as business and finance, defence, education, infrastructure and power, and IT and communications.
The starting salary for many maths graduate roles is highly competitive, and career options include:
 Actuary
 Data Analyst
 Investment Analyst
 Research Scientist
 System Developer
 Teacher
A degree in this discipline can also be useful for roles such as Finance Manager, Insurance Underwriter, Meteorologist, Quantity Surveyor, Software Developer, and many more.
Alternatively, you may wish to undertake PhD level study at Lancaster and pursue a career in research and teaching.
Fees and Funding
Fees
We set our fees on an annual basis and the 2020/21 entry fees have not yet been set.
Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.

Course Overview
Course Overview
Mathematics and statistics underpin all aspects of modern life: advances in science, innovations in medicine, new technology, and business. This Masters 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, problemsolving 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 problemsolving 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, 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.
In the fourth year, you will be able to widen your knowledge and skills by selecting from a pool of optional Masterslevel 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.

Entry Requirements
Entry Requirements
Grade Requirements
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
IELTS 6.0 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics and/or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
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.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk

Course Structure
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
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.

Convergence and Continuity
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.

Discrete Mathematics
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.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. They will see how partial derivatives can help to understand surfaces, while repeated integrals enable them to calculate volumes. The module will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
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, temperature and wind direction. To study functions of several variables, rates of change are introduced with respect to several quantities. How to find maxima and minima will be explained. Applications include the method of least squares. Finally, various methods for solving differential equations of one variable will be investigated.

Geometry and Calculus
The main focus of this module is vectors in two and threedimensional 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.

Integration and Differentiation
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.

Linear Algebra
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.

Numbers and Relations
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
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.

Statistics
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
Year 2

Abstract Algebra
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 wayto 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
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 nonconstant 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.

Computational Mathematics
Students will gain a solid understanding of computation and computer programming within the context of maths and statistics. This module expands on five key areas:
 Programming and R
 Numerical solutions of equations
 Numerical differentiation and integration
 Monte Carlo methods
 Numerical solutions to ODEs
Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixedpoint iteration, higher dimensions, first and second derivatives, nonparametric bootstraps, and modified Euler methods.
Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation.
Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics.

Linear Algebra II
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 structurepreserving 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 II
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.

Project Skills
Project Skills is a module designed to support and develop a range of key technical and professional skills that will be valuable for all career paths. Covering five major components, this module will guide students through and explore:
 Mathematical programmes
 Scientific writing
 Communication and presentation skills
 Short projects
 Group projects
Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures.
Scientific writing, communication and presentations skills will also be developed. Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations.

Real Analysis
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 Alevel.
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 Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics II
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 likelihoodbased 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.
Core
Year 3

Study Abroad Year
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.
Core
Year 4

Algebraic Curves
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.

Assessing Financial Risk: Extreme Value Methods
Assessment of financial risk requires accurate estimates of the probability of rare events. Estimating the probability of such "extreme" events is challenging, as by nature they are sufficiently rare that there is little direct empirical evidence on which to base inference. Instead we have to extrapolate based on the past frequency of the occurrence of less extreme events. This module covers ideas from Extreme Value Theory which give a sound mathematical basis to such extrapolation, and shows practically how it can be used to give accurate assessments of financial risk in a wide range of scenarios.
In this module, students will study topics related to the understanding of special models to describe the extreme values of a financial times series, and they will learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. Extreme value models will be used to evaluate Value at Risk and gain an understanding of the impact of heavy tailed data on standard statistical diagnostic tools.

Bayesian Inference
Bayesian statistics is a framework for rational decision making using imperfect knowledge, expressed through probability distributions. Bayesian principles are applied in the fields of navigation, control, automation and artificial intelligence. The aim of decision makers is to make rational decisions that maximise some personal utility function which may represent quantities such as money which are related to the wealth of an individual.
Within the Bayesian framework, knowledge of the world, (the prior) is updated as fresh observations arrive to yield a posterior distribution which shows the revised knowledge. The evidence for the model is expressed by calculating a marginal likelihood. Future behaviour and the fit of the model are assessed using a predictive distribution. This includes sampling uncertainty and uncertainty of our knowledge.
In this module students will look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multiparameter fully conjugate models. The range of belief types that can be modelled by using mixtures of conjugate priors will be extended, and will also explore the use of nonconjugate formulations of models and use MonteCarlo integration, importance sampling and rejection sampling for calculating and simulating from these distributions.

Clinical Trials
Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancyinduced hypertension; or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients. Treatments may be procedural, for example, surgery or methods of care.
This module combines the study of technical methodology with discussion of more general research issues. First , the relative advantages and disadvantages of different types of medical studies will be discussed. Then students will explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. They will also cover crossover trials, concepts of sample size determination, and equivalence trials. The module also includes a brief introduction to sequential trial designs and metaanalysis.

Computer Intensive Methods
Students will be introduced to Markov chain Monte Carlo methods and how to use them as a powerful technique for performing Bayesian inference on complex stochastic models.
The first part of the module looks in detail at the necessary concepts and theory for finite statespace Markov chains, before introducing analogous concepts and theory for continuous statespace Markov chains. In the second part of the course module the MetropolisHastings algorithm for sampling from a distribution known up to a constant of proportionality will be investigated.
In the third (and largest) part, students will take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. They will also examine the two most common MetropolisHastings algorithms (the random walk and the independence sampler). Examples will include hierarchical models, random effects models, and mixture models.

Environmental Epidemiology
This module focuses on the kinds of statistical methods commonly used by statisticians to investigate the relationship between risk of disease and environmental factors.
You’ll explore methods for the analysis of spatial data, including spatial pointprocess models, spatial casecontrol methods, spatially aggregated data, point source problems and geostatistics, and will spend time developing your skills performing similar analyses using the statistical package R.

Financial Mathematics
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 BlackScholes 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
 BlackScholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated riskneutral probability; performing calculations with the BlackScholes formula; and proving various steps in the derivation of the BlackScholes 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 realworld; improve their written and oral communication skills; and develop their critical thinking.

Galois Theory
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 1to1 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.

Generalised Linear Models
In this module students will learn techniques for formulating sensible models for data, enabling you to tackle problems such as the probability of success for a particular treatment, and how this depends on the patient's age, weight, blood pressure, and so on.
Students will be introduced to a large family of models, called the generalised linear models (GLMs), including the standard linear regression model as a special case, and will have the opportunity to discuss and investigate the theoretical properties of these models.
A common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters will be studies. Using the statistical package ‘R’, you’ll fit and check these models, and will produce confidence intervals and tests corresponding to questions of interest.

Hilbert Spaces
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.

Industrial Dissertation
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 indepth 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.

Lebesgue Integration
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.

Lie Groups and Lie Algebras
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.

Likelihood Inference
In this module, students will learn how to use the likelihood function to obtain and summarise information about unknown parameters. They will calculate the likelihood function for statistical models which do not assume independent identically distributed data, and will learn to evaluate point estimates and make statements about the variability of these estimates.
Using the statistical package ‘R’,computational methods will be used to calculate maximum likelihood estimates. Time will be spent developing an understanding of the interrelationships between parameters, and the concept of orthogonality. In addition to this, hypothesis tests will be performed using the generalised likelihood ratio statistic.

Longitudinal Data Analysis
Longitudinal data arise when a timesequence of measurements is made on a response variable for each of a number of subjects in an experiment or observational study. For example, a patient’s blood pressure may be measured daily following administration of one of several medical treatments for hypertension.
Typically, the practical objective of most longitudinal studies is to find out how the average value of the response varies over time, and how this average response profile is affected by different experimental treatments. This module presents an approach to the analysis of longitudinal data, based on statistical modelling and likelihood methods of parameter estimation and hypothesis testing.

Mathematics Dissertation
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 indepth 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.

Metric Spaces
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
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 infinitedimensional 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.

Principles of Epidemiology
This module focuses on the basic principles of epidemiology, including its methodology and application to prevention and control of disease.The concepts and strategies used in epidemiologic studies will be examined, and students will come to appreciate a historical and general overview of epidemiology and related strategies for study design. They will also develop their knowledge of how to apply basic epidemiologic methods, and will gain confidence in assessing the validity of epidemiologic studies with respect to their design and inferences. Most inference will be likelihood based, although the emphasis is on conceptual considerations and interpretation.

Probability Theory
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.

Project Skills
This module gives students the opportunity to enhance their project skills, including both subjectrelated and transferable skills such as oral communication and scientific writing. They will work on mathematical document preparation and presentation, and with a statistical software package.
The module will revisit LaTeX and R, which will be used to prepare mathematical documents; text and mathematical symbols, displayed formulae, numbering, environments, lists, page and document layout, sections and table of contents, and tables and figures. In addition to this, students will undertake a short project, investigating a mathematical or statistical topic, and a group project, involving a group investigation of a mathematical or statistical topic under the direction of a supervisor.

Representation Theory of Finite Groups
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of Rmodule 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.

Statistics Dissertation
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 indepth 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 for Finance
Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discretetime stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuoustime 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 reallife applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

Stochastic Processes
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.

Topology and Fractals
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.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.

Calculus

Careers
Careers
Maths graduates are very versatile, having indepth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including data analysis and manipulation, logical thinking, problemsolving and quantitative reasoning, as well as adept knowledge of the discipline. As a result, mathematicians are sought after in a range of industries, such as business and finance, defence, education, infrastructure and power, and IT and communications.
The starting salary for many maths graduate roles is highly competitive, and career options include:
 Actuary
 Data Analyst
 Investment Analyst
 Research Scientist
 System Developer
 Teacher
A degree in this discipline can also be useful for roles such as Finance Manager, Insurance Underwriter, Meteorologist, Quantity Surveyor, Software Developer, and many more.
Alternatively, you may wish to undertake PhD level study at Lancaster and pursue a career in research and teaching.

Fees and Funding
Fees and Funding
Fees
We set our fees on an annual basis and the 2020/21 entry fees have not yet been set.
Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.
Graduate Statistician status is awarded by the Royal Statistical Society to successful graduates of accredited degrees.
Completion of this programme can lead to recommendation for exemption from some examinations with the Institute and Faculty of Actuaries.
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