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 Mathematics BSc Hons (G100)
Mathematics BSc Hons  2020 Entry
UCAS Code
G100
Entry Year
2020
also available in 2019
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 3 Year(s)
Course Overview
Mathematics forms the foundations of all science and technology and as such is an extensive and rigorous discipline. Our programme reflects this and offers a comprehensive, detailed study through excellent teaching and development opportunities.
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 you will receive high quality teaching delivered by academics who are leaders in their field. Throughout the three years, you will develop a range of discipline specific skills and gain specialist knowledge that will prepare you for your chosen career.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, 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 final 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. You will have the chance to apply the skills and knowledge you have gained in the first two years in advanced mathematical modules such as Combinatorics, Number Theory, Hilbert Space and Metric Spaces.
MSci Mathematics
As well as offering a BSc degree, we also offer a four year MSci Mathematics degree. During the course of this programme, you have the option to graduate after three years with a BSc, or progress onto an advanced fourth year and complete an MSci. This additional year features Masterslevel modules and a dissertation.
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
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

Algebraic Curves
Students will be given a solid foundation in the basicsof algebraic geometry. They will explore how curves can be described by algebraic equations, and learn how to use abstract groups in dealing with geometrical objects (curves). The module will present applications and results of the theory of elliptic curves and provide a useful link between concepts from algebra and geometry.
Students will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally, they will learn to perform algebraic computations with elliptic curves.

Bayesian Inference
Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.
Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.

Combinatorics
Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject  with many important connections to other areas of modern mathematics  it is a very accessible one.
In this module, students will be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms) and combinatorial design theory (Latin squares and block designs). They will also explore important practical applications of the results and methods.

Commutative Algebra
Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.
They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
How wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.

Differential Equations
Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.
While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.

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.

Geometry of Curves and Surfaces
The topic of smooth curves and surfaces in threedimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.
A number of wellknown concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

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.

Groups and Symmetry
Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.
Situations will be considered in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.
Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.

Hilbert Spaces
Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.
Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.

Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.

Likelihood Inference
Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.
This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.
Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

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

Mathematical Education
Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.
Having studied mathematics for many years, students are wellplaced to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system.

Mathematical Education Placement
This module is run as a partnership between the Department of Mathematics and Statistics, the University of Cumbria and the Students’ Union’s volunteering unit.
Designed with employability in mind, this module is based on the Students’ Union’s Schools Partnership Scheme, which supports Lancaster students on 10week placements in local primary and secondary schools.
Students will have the opportunity to take part in classroom observation and assistance, the development of classroom resources, the provision of oneonone or small group support and possibly even teaching sections of lessons to the class as a whole.

Medical Statistics: study design and data analysis
The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.
Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.
They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

Metric Spaces
An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.
They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

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.

Number Theory
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the nonspecialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Representation Theory of Finite Groups
Students will cover the basics of ordinary representation theory. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between the two will be discussed.
The second part is an introduction to the ordinary character theory of finite groups, intrinsic to representation theory. Students will learn the concepts of Rmodule and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups.
They will also learn to perform computations with representations and morphisms in a selection of finite groups

Statistical Models
The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).
Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

Stochastic Processes
Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.
As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).
The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Time series analysis
Modern statistics is characterised by computerintensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.
Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.
Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.
Lastly,students will spend time looking at ChangePoint Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
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 postgraduate 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 forms the foundations of all science and technology and as such is an extensive and rigorous discipline. Our programme reflects this and offers a comprehensive, detailed study through excellent teaching and development opportunities.
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 you will receive high quality teaching delivered by academics who are leaders in their field. Throughout the three years, you will develop a range of discipline specific skills and gain specialist knowledge that will prepare you for your chosen career.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, 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 final 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. You will have the chance to apply the skills and knowledge you have gained in the first two years in advanced mathematical modules such as Combinatorics, Number Theory, Hilbert Space and Metric Spaces.
MSci Mathematics
As well as offering a BSc degree, we also offer a four year MSci Mathematics degree. During the course of this programme, you have the option to graduate after three years with a BSc, or progress onto an advanced fourth year and complete an MSci. This additional year features Masterslevel modules and a dissertation.

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
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

Algebraic Curves
Students will be given a solid foundation in the basicsof algebraic geometry. They will explore how curves can be described by algebraic equations, and learn how to use abstract groups in dealing with geometrical objects (curves). The module will present applications and results of the theory of elliptic curves and provide a useful link between concepts from algebra and geometry.
Students will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally, they will learn to perform algebraic computations with elliptic curves.

Bayesian Inference
Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.
Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.

Combinatorics
Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject  with many important connections to other areas of modern mathematics  it is a very accessible one.
In this module, students will be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms) and combinatorial design theory (Latin squares and block designs). They will also explore important practical applications of the results and methods.

Commutative Algebra
Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.
They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
How wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.

Differential Equations
Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.
While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.

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.

Geometry of Curves and Surfaces
The topic of smooth curves and surfaces in threedimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.
A number of wellknown concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

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.

Groups and Symmetry
Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.
Situations will be considered in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.
Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.

Hilbert Spaces
Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.
Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.

Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.

Likelihood Inference
Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.
This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.
Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

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

Mathematical Education
Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.
Having studied mathematics for many years, students are wellplaced to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system.

Mathematical Education Placement
This module is run as a partnership between the Department of Mathematics and Statistics, the University of Cumbria and the Students’ Union’s volunteering unit.
Designed with employability in mind, this module is based on the Students’ Union’s Schools Partnership Scheme, which supports Lancaster students on 10week placements in local primary and secondary schools.
Students will have the opportunity to take part in classroom observation and assistance, the development of classroom resources, the provision of oneonone or small group support and possibly even teaching sections of lessons to the class as a whole.

Medical Statistics: study design and data analysis
The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.
Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.
They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.

Metric Spaces
An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.
They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

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.

Number Theory
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the nonspecialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Representation Theory of Finite Groups
Students will cover the basics of ordinary representation theory. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between the two will be discussed.
The second part is an introduction to the ordinary character theory of finite groups, intrinsic to representation theory. Students will learn the concepts of Rmodule and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups.
They will also learn to perform computations with representations and morphisms in a selection of finite groups

Statistical Models
The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).
Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.

Stochastic Processes
Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.
As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).
The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.

Time series analysis
Modern statistics is characterised by computerintensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.
Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.
Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.
Lastly,students will spend time looking at ChangePoint Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
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 postgraduate 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 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.
The Institute of Mathematics and its Applications is a chartered professional body for mathematicians in the UK. Accreditation means that our degrees demonstrate both a high level of competency and professionalism in the area of mathematics.
Download the latest Mathematics & Statistics brochure, featuring all of the information you could need. The includes detailed course information about each of our courses.
Ella's story
"Lancaster University instantly attracted me after looking around various universities in England, this was because I liked the size of the campus and the fact that it is a top ten university. I enjoy workshops as they provide me with an opportunity to openly discuss questions asked in the lecture with my peers and tutor. After I have completed my degree, I want to become an actuary as it is a profession which is predominantly numericalbased and is the type of role I aspire to be in to put my skills into practice."

Which A Levels should I be studying?
Our minimum requirement for A Level applicants is that you should be studying at least three A Levels including A Level Mathematics. We encourage but do not require, you to study A Level Further Mathematics as well  our standard offer is slightly lower for those taking A Level Further Mathematics. If it is not possible for you to study A Level Further Mathematics, think about studying AS Level Further Mathematics  the style of mathematics in Further Mathematics, especially Further Pure Mathematics, is similar to university mathematics and will be excellent preparation for further study.
We do not usually accept vocational qualifications like BTEC or Cambridge Technicals unless you are also studying A Level Mathematics and A Level Further Mathematics. If this applies to you, then we would treat your vocational qualification as equivalent to a third A Level.

What about STEP?
The Sixth Term Examination Papers in Mathematics are a set of three papers, set by Cambridge Assessment and sat in June, which are designed to really test your problemsolving skills. Each paper lasts three hours and contains thirteen relatively long questions, all of which are optional  you answer as many of them as you wish, up to a maximum of six questions. You may enter any combination of the three papers  those applying to Cambridge will usually sit Paper 2 and Paper 3, while those applying to Warwick or Bath will usually sit Paper 1 plus, optionally, one or both of the other papers.
We value the way that STEP develops advanced problemsolving skills and all our standard offers include an alternative, slightly lower offer that includes a pass (Grade 3) in any STEP. Sitting STEP is optional, however, and if you choose not to sit STEP, this won't harm your chances of receiving an offer from us. If you sit STEP and do not pass, please do not worry  we will treat you exactly the same as if you hadn't sat STEP at all.

What about the Test of Maths for University Admission?
The Test of Maths for University Admission is a test, set by Cambridge Assessment and sat in November. Like STEP, the Test of Maths for University Admission is designed to test your problemsolving and readiness for universitylevel mathematics, but the format is very different  it is made up of two multiplechoice question papers, each lasting 75 minutes. Results are available at the end of November  you will receive a grade ranging from 1.0 to 9.0.
A strong performance in the Test of Maths for University Admission is very impressive. If you sit the Test and perform well, achieving a score above 4.5, you may well receive a lower offer from us. Sitting the Test is optional, however, and if you choose not to sit the Test, this won't harm your chances of receiving an offer from us. Indeed:
 you do not need to tell us that you're sitting the Test;
 if you sit the test, you do not need to tell us your result; and
 a poor result will not damage your chances  we will treat you the same as if you hadn't sat the Test at all.
If you inform us of your Test result before receiving an offer from us and have achieved a score of 4.5 or higher, we will take this into account when we decide on your application, and we may well make you a lower offer than usual.
If you inform us of your Test result after receiving an offer from us and have achieved a score of 4.5 or higher, we will consider your result, and may well replace your existing offer with a lower one.

Our standard offer
The standard offers for courses administered by our Department are outlined in the table below.
Programme of Study UCAS STEP/TMUA Including Maths at Grade A Including Maths and Further Maths at least one at Grade A Mathematics BSc
Mathematics BSc (Placement Year)
Mathematics MSci
Mathematics MSci (Study Abroad)
Statistics BSc
Statistics BSc (Placement Year)
Statistics MSci
Statistics MSci (Study Abroad)
Mathematics with Statistics BSc
Mathematics with Statistics BSc (Placement Year)
Mathematics with Statistics MSci
Mathematics with Statistics MSci (Study Abroad)
Financial Mathematics BSc
Financial Mathematics BSc (Industry)
Financial Mathematics MSci
Computer Science and Mathematics BSc
Computer Science and Mathematics MSciG100
G102
G101
G103
G300
G302
G303
G301
G1G3
GCG3
G1GJ
G1GH
GN13
GN1J
GN1H
GG14
GG1KNot taken AAA AAB Mathematics BSc
Mathematics BSc (Placement Year)
Mathematics MSci
Mathematics MSci (Study Abroad)
Statistics BSc
Statistics BSc (Placement Year)
Statistics MSci
Statistics MSci (Study Abroad)
Mathematics with Statistics BSc
Mathematics with Statistics BSc (Placement Year)
Mathematics with Statistics MSci
Mathematics with Statistics MSci (Study Abroad)
Financial Mathematics BSc
Financial Mathematics BSc (Industry)
Financial Mathematics MSci
Computer Science and Mathematics BSc
Computer Science and Mathematics MSciG100
G102
G101
G103
G300
G302
G303
G301
G1G3
GCG3
G1GJ
G1GH
GN13
GN1J
GN1H
GG14
GG1KSTEP Level 3 +
or TMUA 4.5+AAB ABB Mathematics and Philosophy BA GV15 Not taken AAB ABB Mathematics and Philosophy BA GV15 STEP Level 3 +
or TMUA 4.5+ABB ABB French Studies and Mathematics BA
German Studies and Mathematics BA
Spanish Studies and Mathematics BAGR11
GR12
GR14N/A AAB
(Maths Grade A, Language Grade B)ABB
(Maths Grade A, Language Grade B)Accounting, Finance and Mathematics BSc
Accounting, Finance and Mathematics BSc (Industry)
Economics and Mathematics BScNG41
NG42
GL11N/A AAB including Maths or Further Maths Grade A BSc Theoretical Physics with Mathematics F3GC N/A AAB including Maths and Physics MSci Theoretical Physics with Mathematics
MSci Theoretical Physics with Mathematics (Study Abroad)F3G1
F3G5N/A AAA including Maths and Physics
Our Courses
Our courses offer academic excellence. We have 3 singlehonours degrees and 10 combinedhonours degrees. We have focused our courses in the first two years, so you can choose from a wide range of third and fourthyear topics. Our academic research shapes our courses with the latest developments.
Friendly and Collaborative
The department has around 50 academic staff and 500 undergraduate students. Throughout your course, you will have weekly workshops and marked assignments. You will have opportunity to discuss each topic and get feedback from expert tutors. We also offer onetoone sessions during lecturers' office hours, so you soon get to know us and vice versa.
Supportive Environment
You will receive the support you need to achieve your full academic potential. We are a friendly department and foster a supportive learning environment. You will have an academic tutor who will assist you throughout your degree. We also encourage you to attend our weekly Maths Café, lead by fellow students.
Study Abroad
You can spend your second year studying at one of our partner universities. These partner Universities span the globe: in Europe, the USA, New Zealand and Australia. Where you choose to study at is up to you, although places are subject to availability.
Similar Courses

Mathematics and Statistics
 Accounting, Finance and Mathematics BSc Hons: NG41
 Accounting, Finance and Mathematics (Industry) BSc Hons: NG42
 Computer Science and Mathematics BSc Hons: GG14
 Computer Science and Mathematics MSci Hons: GG1K
 Computer Science and Mathematics (Placement Year) BSc Hons: GG1L
 Economics and Mathematics BSc Hons: GL11
 Economics and Mathematics (Industry) BSc Hons: GL12
 Financial Mathematics BSc Hons: GN13
 Financial Mathematics MSci Hons: GN1H
 Financial Mathematics (Industry) BSc Hons: GN1J
 French Studies and Mathematics BA Hons: GR11
 German Studies and Mathematics BA Hons: GR12
 Mathematics MSci Hons: G101
 Mathematics (Placement Year) BSc Hons: G102
 Mathematics (Study Abroad) MSci Hons: G103
 Mathematics and Philosophy BA Hons: GV15
 Mathematics with Statistics BSc Hons: G1G3
 Mathematics with Statistics MSci Hons: G1GJ
 Mathematics with Statistics (Placement Year) BSc Hons: GCG3
 Mathematics with Statistics (Study Abroad) MSci Hons: G1GH
 Mathematics, Operational Research, Statistics and Economics (MORSE) BSc Hons: GLN0
 Mathematics, Operational Research, Statistics and Economics (MORSE) (Industry) BSc Hons: GLN1
 Spanish Studies and Mathematics BA Hons: GR14
 Theoretical Physics with Mathematics BSc Hons: F3GC
 Theoretical Physics with Mathematics MSci Hons: F3G1
 Theoretical Physics with Mathematics (Study Abroad) MSci Hons: F3G5
Undergraduate Open Days
Visit us at one of our summer open days, and find out why Lancaster is the place for you.
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