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 Mathematics (Study Abroad) MSci Hons (G103)
Mathematics (Study Abroad) MSci Hons  2019 Entry
UCAS Code
G103
Entry Year
2019
also available in 2018
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 4 Year(s)
Course Overview
Mathematics forms the foundations of all science and technology and as such is an extensive and rigorous discipline. Our Study Abroad programme reflects this, offering you comprehensive, Masterslevel teaching, while allowing you to experience a new culture and society overseas.
Maths is difficult to concisely define, but at its core it is the study of change, patterns, quantities, structures and space. This engaging programme, and our reputation for excellence in research, means that we are able to offer highquality teaching delivered by academics who are leaders in their field.
Our four year MSci Mathematics (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 statistical 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 and Further Mathematics
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
This module provides the student with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. We introduce examples of functions and their graphs, and techniques for building new functions from old. We then consider the notion of a limit and introduce the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and be introduced to 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
Students are introduced to the basic ideas and notations involved in describing sets and their functions. The 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, we can say that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? We develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
Rather than counting objects, we might be interested in connections between them, 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. We see how partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. Students 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, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

Geometry and Calculus
The main focus of this course is vectors in two and threedimensional space. We start off with the definition of vectors and we see some applications to finding equations of lines and planes. We then consider some different ways of describing curves and surfaces via equations or parameters, and we use partial differentiation to determine tangent lines and planes, as well as using integration to calculate the length of a curve.
In the second half of the course, we study functions of several variables. When attempting to calculate an integral over one variable, we often substitute one variable for another more convenient one; here we will see the equivalent technique for a double integral, where we have to substitute two variables simultaneously. We also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, we discuss how to calculate the areas of various surfaces and the volumes of various solids.

Integration and Differentiation
Building on MATH113, this module explores the familiar topics of integration, series and differentiation, and develops 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?”. Students 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 equation and eigenvectors and eigenvalues.

Numbers and Relations
This module introduces the student to logic and mathematical proofs, with emphasis placed 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.
We take a look at the language and structure of mathematical proofs in general, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas are then 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 and 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, and this underpins 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 you’ll select a small number of properties which these and other examples have in common, and use them to define a group.
You’ll also consider the elementary properties of groups. It turns out that several surprisingly elegant results can be proved fairly simply! By looking at maps between groups which 'preserve structure' you’ll discover a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives us a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials, but you’ll meet several less familiar examples too.

Complex Analysis
Complex Analysis has its origins in differential calculus and the study of polynomial equations.
In this module you’ll consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. You’ll use integral calculus of complex functions to find elegant and important results, including the fundamental theorem of algebra, and you’ll also use classical theorems to evaluate real integrals.
The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Linear Algebra II
This module will give you the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
You’ll 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 your 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.
You’ll revisit basic concepts from the first year probability module, and extend these to encompass continuous random variables, investigating several important continuous probability distributions.
You’ll 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.

Real Analysis
In this module you’ll take a thorough look at the limits of sequences and convergence of series. You’ll learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
You’ll spend time studying the Intermediate Value Theorem and the Mean Value Theorem, and will discover that they have many applications of widely differing kinds. 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 we put the notion of integration under the microscope. Once it’s properly defined (via limits), you’ll learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. You’ll 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 this module we approach this problem by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
You’ll focus on likelihoodbased parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and also considering linear regression techniques within the statistical modelling framework.
Core
Year 3

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.

Graph Theory
The study of graphs  mathematical objects used to model pairwise relations between objects  is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.

Linear Systems
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics courses and provide opportunities for further study. The theory of Linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The course shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables us to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilize a (A,B,C,D) system.

Multivariate Statistics in Machine Learning
Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.
An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise highdimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.
Core
Year 4

Assessing Financial Risk: Extreme Value Methods
In this module you’ll study topics related to the understanding of special models to describe the extreme values of a financial times series, and you’ll learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. You’ll also learn to use extreme value models 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 you’ll look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multiparameter fully conjugate models. You’ll extend the range of belief types that can be modelled by using mixtures of conjugate priors, 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 we’ll discuss the relative advantages and disadvantages of different types of medical studies. We’ll then explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. We’ll 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
In this module you’ll 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 you’ll investigate the MetropolisHastings algorithm for sampling from a distribution known up to a constant of proportionality.
In the third (and largest) part, you’ll take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. You’ll 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.

Elliptic Curves
In this module you’ll have the opportunity to learn the basics of algebraic geometry.
You’ll 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). You’ll also study the notions and the main results pertaining to elliptic curves, and will investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

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 you’ll 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 you’ll 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.
You’ll 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.
You’ll also study a common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters. 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 Space
In this module you’ll have the opportunity to learn about Hilbert space, consolidating your understanding of linear algebra and enabling you to study applications of Hilbert space such as quantum mechanics and stochastic processes.
You’ll learn 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 you’ll fill in a form stating your mathematical or statistical interests and based on that you 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 your degree scheme and your choice.
During the first term you’ll meet your supervisor weekly and will be guided into your indepth study of a specific topic. During the second term you’ll have to produce a written dissertation on what you have learned and give an oral presentation. You will hand in your dissertation in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on your oral presentation, and 20% on the initiative and effort that you 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
In this module you’ll construct Lebesgue measure on the line, extending the idea of the length of an interval. You’ll 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, you will explore properties of countable sets, open sets and algebraic numbers.
You’ll 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.

Lie Groups and Lie Algebras
In this module you'll be introduced to the theory of Lie groups and Lie algebras. You'll also explore the relationship between the two, and 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. You'll 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 course will give you a taste of modern research in pure mathematics.

Likelihood Inference
In this module you’ll learn how to use the likelihood function to obtain and summarise information about unknown parameters. You’ll 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’, you’ll use computational methods to calculate maximum likelihood estimates. You’ll spend time developing an understanding of the interrelationships between parameters, and the concept of orthogonality, and will perform hypothesis tests 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 3 you’ll fill in a form stating your mathematical or statistical interests and based on that you 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 your degree scheme and your choice.
During the first term you’ll meet your supervisor weekly and will be guided into your indepth study of a specific topic. During the second term you’ll have to produce a written dissertation on what you have learned and give an oral presentation. You will hand in your dissertation in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on your oral presentation, and 20% on the initiative and effort that you 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.

Metric Spaces
In this module you’ll construct Lebesgue measure on the line, extending the idea of the length of an interval. You’ll 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, you will explore properties of countable sets, open sets and algebraic numbers.
You’ll 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 you’ll review Hilbert spaces, before spending some time studying infinitedimensional operators, notably the unilateral shift and multiplication operators, as well as basic concepts. You’ll 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.
You’ll examine the concepts and strategies used in epidemiologic studies, and will develop an understanding of the role of epidemiology in preventive medicine and disease investigation. You’ll also develop your knowledge of basic epidemiologic methods and how to apply them, and will develop confidence in assessing the validity of epidemiologic studies with respect to their design and inferences.

Probability Theory
The aim of this course is to develop an analytical and axiomatic approach to the theory of probabilities.
You’ll consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. You’ll 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. You’ll also use the characteristic function to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.

Project Skills
This module gives you the opportunity to enhance your project skills, including both subjectrelated and transferable skills. You’ll work on mathematical document preparation and presentation, scientific writing, and working with a statistical software package.
You’ll revisit LaTeX and R, and work on your oral communication skills, scientific writing, a written group project, and a group presentation.

Representation Theory of Finite Groups
In this module you’ll learn the basics of ordinary representation theory.
You’ll 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. You’ll also develop your skills in performing computations with representations and morphisms in a selection of finite groups.

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 you’ll learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. You’ll 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. You’ll then 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 R2.
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 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU  Overseas 

£9,250  £18,890 
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 forms the foundations of all science and technology and as such is an extensive and rigorous discipline. Our Study Abroad programme reflects this, offering you comprehensive, Masterslevel teaching, while allowing you to experience a new culture and society overseas.
Maths is difficult to concisely define, but at its core it is the study of change, patterns, quantities, structures and space. This engaging programme, and our reputation for excellence in research, means that we are able to offer highquality teaching delivered by academics who are leaders in their field.
Our four year MSci Mathematics (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 statistical 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 and Further Mathematics
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
This module provides the student with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. We introduce examples of functions and their graphs, and techniques for building new functions from old. We then consider the notion of a limit and introduce the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and be introduced to 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
Students are introduced to the basic ideas and notations involved in describing sets and their functions. The 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, we can say that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? We develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
Rather than counting objects, we might be interested in connections between them, 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. We see how partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. Students 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, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

Geometry and Calculus
The main focus of this course is vectors in two and threedimensional space. We start off with the definition of vectors and we see some applications to finding equations of lines and planes. We then consider some different ways of describing curves and surfaces via equations or parameters, and we use partial differentiation to determine tangent lines and planes, as well as using integration to calculate the length of a curve.
In the second half of the course, we study functions of several variables. When attempting to calculate an integral over one variable, we often substitute one variable for another more convenient one; here we will see the equivalent technique for a double integral, where we have to substitute two variables simultaneously. We also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, we discuss how to calculate the areas of various surfaces and the volumes of various solids.

Integration and Differentiation
Building on MATH113, this module explores the familiar topics of integration, series and differentiation, and develops 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?”. Students 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 equation and eigenvectors and eigenvalues.

Numbers and Relations
This module introduces the student to logic and mathematical proofs, with emphasis placed 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.
We take a look at the language and structure of mathematical proofs in general, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas are then 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 and 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, and this underpins 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 you’ll select a small number of properties which these and other examples have in common, and use them to define a group.
You’ll also consider the elementary properties of groups. It turns out that several surprisingly elegant results can be proved fairly simply! By looking at maps between groups which 'preserve structure' you’ll discover a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives us a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials, but you’ll meet several less familiar examples too.

Complex Analysis
Complex Analysis has its origins in differential calculus and the study of polynomial equations.
In this module you’ll consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. You’ll use integral calculus of complex functions to find elegant and important results, including the fundamental theorem of algebra, and you’ll also use classical theorems to evaluate real integrals.
The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Linear Algebra II
This module will give you the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
You’ll 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 your 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.
You’ll revisit basic concepts from the first year probability module, and extend these to encompass continuous random variables, investigating several important continuous probability distributions.
You’ll 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.

Real Analysis
In this module you’ll take a thorough look at the limits of sequences and convergence of series. You’ll learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
You’ll spend time studying the Intermediate Value Theorem and the Mean Value Theorem, and will discover that they have many applications of widely differing kinds. 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 we put the notion of integration under the microscope. Once it’s properly defined (via limits), you’ll learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. You’ll 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 this module we approach this problem by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
You’ll focus on likelihoodbased parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and also considering linear regression techniques within the statistical modelling framework.
Core
Year 3

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.

Graph Theory
The study of graphs  mathematical objects used to model pairwise relations between objects  is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.

Linear Systems
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics courses and provide opportunities for further study. The theory of Linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The course shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables us to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilize a (A,B,C,D) system.

Multivariate Statistics in Machine Learning
Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.
An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise highdimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.
Core
Year 4

Assessing Financial Risk: Extreme Value Methods
In this module you’ll study topics related to the understanding of special models to describe the extreme values of a financial times series, and you’ll learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. You’ll also learn to use extreme value models 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 you’ll look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multiparameter fully conjugate models. You’ll extend the range of belief types that can be modelled by using mixtures of conjugate priors, 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 we’ll discuss the relative advantages and disadvantages of different types of medical studies. We’ll then explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. We’ll 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
In this module you’ll 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 you’ll investigate the MetropolisHastings algorithm for sampling from a distribution known up to a constant of proportionality.
In the third (and largest) part, you’ll take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. You’ll 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.

Elliptic Curves
In this module you’ll have the opportunity to learn the basics of algebraic geometry.
You’ll 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). You’ll also study the notions and the main results pertaining to elliptic curves, and will investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

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 you’ll 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 you’ll 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.
You’ll 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.
You’ll also study a common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters. 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 Space
In this module you’ll have the opportunity to learn about Hilbert space, consolidating your understanding of linear algebra and enabling you to study applications of Hilbert space such as quantum mechanics and stochastic processes.
You’ll learn 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 you’ll fill in a form stating your mathematical or statistical interests and based on that you 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 your degree scheme and your choice.
During the first term you’ll meet your supervisor weekly and will be guided into your indepth study of a specific topic. During the second term you’ll have to produce a written dissertation on what you have learned and give an oral presentation. You will hand in your dissertation in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on your oral presentation, and 20% on the initiative and effort that you 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
In this module you’ll construct Lebesgue measure on the line, extending the idea of the length of an interval. You’ll 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, you will explore properties of countable sets, open sets and algebraic numbers.
You’ll 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.

Lie Groups and Lie Algebras
In this module you'll be introduced to the theory of Lie groups and Lie algebras. You'll also explore the relationship between the two, and 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. You'll 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 course will give you a taste of modern research in pure mathematics.

Likelihood Inference
In this module you’ll learn how to use the likelihood function to obtain and summarise information about unknown parameters. You’ll 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’, you’ll use computational methods to calculate maximum likelihood estimates. You’ll spend time developing an understanding of the interrelationships between parameters, and the concept of orthogonality, and will perform hypothesis tests 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 3 you’ll fill in a form stating your mathematical or statistical interests and based on that you 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 your degree scheme and your choice.
During the first term you’ll meet your supervisor weekly and will be guided into your indepth study of a specific topic. During the second term you’ll have to produce a written dissertation on what you have learned and give an oral presentation. You will hand in your dissertation in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on your oral presentation, and 20% on the initiative and effort that you 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.

Metric Spaces
In this module you’ll construct Lebesgue measure on the line, extending the idea of the length of an interval. You’ll 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, you will explore properties of countable sets, open sets and algebraic numbers.
You’ll 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 you’ll review Hilbert spaces, before spending some time studying infinitedimensional operators, notably the unilateral shift and multiplication operators, as well as basic concepts. You’ll 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.
You’ll examine the concepts and strategies used in epidemiologic studies, and will develop an understanding of the role of epidemiology in preventive medicine and disease investigation. You’ll also develop your knowledge of basic epidemiologic methods and how to apply them, and will develop confidence in assessing the validity of epidemiologic studies with respect to their design and inferences.

Probability Theory
The aim of this course is to develop an analytical and axiomatic approach to the theory of probabilities.
You’ll consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. You’ll 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. You’ll also use the characteristic function to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.

Project Skills
This module gives you the opportunity to enhance your project skills, including both subjectrelated and transferable skills. You’ll work on mathematical document preparation and presentation, scientific writing, and working with a statistical software package.
You’ll revisit LaTeX and R, and work on your oral communication skills, scientific writing, a written group project, and a group presentation.

Representation Theory of Finite Groups
In this module you’ll learn the basics of ordinary representation theory.
You’ll 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. You’ll also develop your skills in performing computations with representations and morphisms in a selection of finite groups.

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 you’ll learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. You’ll 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. You’ll then 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 R2.
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 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU Overseas £9,250 £18,890 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.
The Department
Graduate Statistician status is awarded by the Royal Statistical Society to successful graduates who complete the required number of Part II statistics modules.
Completion of this programme can lead to recommendation for exemption from some examinations with the Institute and Faculty of Actuaries.
8.7 hours
Typical time in lectures, seminars and similar per week during term time
50%
Average assessment by coursework
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