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 Computer Science and Mathematics MSci Hons (GG1K)
Computer Science and Mathematics MSci Hons  2019 Entry
UCAS Code
GG1K
Entry Year
2019
also available in 2018
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 4 Year(s)
Course Overview
Mathematics and computing are intrinsically linked. By combining them in one Masters programme you gain a robust, advanced understanding of the two disciplines, equipping you with sophisticated specialist skills and detailed technical knowledge, allowing you to excel in your chosen career.
Mathematics forms the foundations of all technology and computing, and as such, a rigorous study of the discipline provides invaluable insight and understanding into computer science. Furthermore, computer science is itself a dynamic discipline with a wide range of applications. As a result, this combined programme offers you a robust and comprehensive skill set, indepth specialist knowledge, and fantastic career opportunities.
You will explore the theory and practice of innovative and experimental computer science, while gaining an advanced understanding of the mathematical concepts and processes behind them. The depth and breadth of knowledge and experience gained over the four years will prove to be a challenging but rewarding opportunity, placing you in the strongest position as you move forward into your chosen career.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problemsolving and quantitative reasoning skills. Additionally, you will be introduced to software development and the fundamentals of computer science. These topics will allow you to gain a wealth of technical knowledge and develop key interdisciplinary skills.
In the second year, you will begin to drill down into specialist maths and computing modules, studying HumanComputer Interaction; Software Design; Linear Algebra; and Social, Ethical and Professional Issues in Computing. These core modules will ensure you gain a solid understanding of the disciplines that is applicable in the realworld. Alongside these, you will also be able to choose from a range of optional maths modules, these include: Abstract Algebra, Complex Analysis, and Real Analysis. In addition, you will bring your skills and knowledge together in a group project, which will allow you to apply what you have learnt to the realworld and gain valuable, practical experience.
For the third year, your study will be largely guided by your own interests. Compulsory modules, such as Artificial Intelligence, Languages and Compilation, and Security and Risk, will enhance and progress your computer science knowledge and provide insight into the sort of activity you will encounter in the realworld. However, the wide range of optional modules you can choose from will allow you to delve deeper into your own interests and customise the year to suit your career ambitions.
The fourth and final year of your degree will introduce a variety of advanced modules for you to choose from. You can build a strong repertoire of maths and computer science skills and knowledge, to suit your interests and goals, including: Data Mining; Galois Theory; Lie Groups and Lie Algebras; Operator Theory; and Systems Architecture and Integration. You will also benefit from our Research Methods module, which will provide you with a formal understanding of research, and allow you to gain the appropriate skills and practices. You will learn to critically reflect on your research and will gain an appreciation of the different ways that other disciplines, academic communities and industries conduct research. This will provide invaluable insight and experience for many graduate careers as well as for continuing in academia.
Entry Requirements
Grade Requirements
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
IELTS 6.0 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics and/or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Fundamentals of Computer Science
This module provides students with an insight into the importance and relevance of the principles of computer science. Gaining the essential knowledge needed for analysing and characterising the efficiency of algorithms and computer programs, students learn how to make the right design choice when implementing computer programs to optimise efficiency for given design parameters.
Students also study the role and characteristics of data structures, and gain an understanding of the continuing importance of classical algorithms in computer science.

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.

Software Development
Computer programming is a highly practical skill in our quickly developing world. In this module students develop the skills expected of a principled computer programmer as they learn how to write, analyse, debug, test and document computer programs. Students will be introduced to both the C and Java programming languages, two of the most widely used languages in the world. They will learn about best practice of daytoday techniques associated with software development and gain an understanding of the software development cycle. Learning about the challenges faced by software developers in addressing scalability and complexity in computer software, students will be able to work independently to develop moderately complex computer programs.

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

Computer Science Group Project
This module gives students the opportunity to build upon their skills and knowledge from Year 1 to create a realworld system in a group context. As part of a group, students will work effectively to gather system requirements; design and then implement the project; and accurately evaluating it. The module aims to increase theoretical knowledge and practical skills in prototyping, project planning, project management, management and execution, games design, systems design and testing strategies. Alongside these, students will also enhance their teamwork, problemsolving, communication, presentation and report writing skills, which will be valuable when progressing into a career.

HumanComputer Interaction
Students will learn theoretical and practical topics in HumanComputer Interaction, with lab work offering hands on experience of design, implementation and the ability to evaluate interactive systems through practical case studies. The course explores the underpinnings of human perception, usercentred design and participatory design processes, with students learning multiple design techniques. The module leads to an understanding of how internal system design impacts external user interface behaviour and highlights the importance of accessibility for all users.
By the end of the module, students will be able to successfully integrate diverse information to form a coherent understanding of HumanComputer Interaction; critically reflect on technical advancements in HCI and demonstrate the independent learning abilities needed for continual professional development and effective written and verbal skills.

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.

Social, Ethical and Professional Issues in Computing
Students will gain the essential skills and knowledge to operate within the professional, legal and ethical frameworks of their profession. Techniques for breaking down a project into manageable parts and efficient time allocation are taught, leading to a fundamental understanding of the skills and methods required to pursue scientific inquiry and the fundamental concepts and tools for statistical analysis to measure and explain data. Exemplars and guidelines on producing concise and structured scientific reports are offered and students receive additional lectures on presentation skills, professional ethics in relation to computing and communications. Finally, lectures provide an awareness of fundamental legal aspects related to a profession in computing and communications, including intellectual property rights and patent law.
Throughout this course, students will gain a high level of awareness of subject specific skills and general competence needed to gain employment in their field. The module develops academic writing and research skills in a computing context, complimenting students’ other modules.

Software Design
Software Design offers the opportunity to gain an understanding of the importance of software architecture design, different styles of architecture and the meaning of quality attributes for software design such as maintainability, performance and scalability. Students will gain knowledge of systematic approaches to developing software design using a set of graphical models. The design process involved in developing several modes of the system at different levels of abstraction is explained and they will be introduced to object oriented design with UML.
Throughout the module, students will appreciate the broader context of the role of computer science in the workplace, and the key role it plays in implementing software. The course also looks at understanding the meaning of quality attributes for software design as well as architectural models for specific software systems. Students will gain an insight into the main quality attributes for deciding classes. Students will be able to interpret and construct UML models of software and implement a design expressed as a UML mode as well as understanding how to use various design patterns to address certain problems.
Core

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.

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.
Optional
Year 3

Artificial Intelligence
Students will gain an introduction to fundamental concepts in artificial intelligence and learn about current trends and issues. Topics such as Knowledge Representation and Reasoning, Decision Making (DM) and Decision Making Under Uncertainties, and Probability Theory are all explored throughout the course. Artificial Intelligence offers experience in supervised and unsupervised machine learning, neural networks and decision trees. Multivariate methods, and clustering and classification approaches are taught and there is an introduction to evolutionary algorithms, phenotypes, genotypes and fundamental genetic operators. Programming languages suitable for intelligent systems, such as Scheme and Prolog are investigated and students are made familiar with the applications of artificial intelligence.
This module sees an awareness of the requirements of artificial intelligence systems in general, and in the context of computing and communications systems. Through knowledge based, probabilistic and logical systems, the module provides students with an awareness of competing approaches and a broad grounding in artificial intelligence. Additionally they will understand and critically analyse artificial intelligence techniques used in modern computers and mobile devices.

Computer Science Seminars
Students will be exposed to a range of current computer science related topics from different subject areas. The areas covered come from our different thematic strands and will include: natural language engineering; policy based network resilience; eyetracking for ubiquitous computing applications; and a focus on energy aware control and sensing in home environments.
Students will conduct independent and indepth research into an advanced topic of computing or communications, reflecting current topical and research issues. During the course of the module, students will analyse, structure, summarise, document and present findings in front of a large group. They will gain topical knowledge and skills related to the subject areas of the seminars, and will learn with and from their peers. The module will enable students to produce a detailed document describing their research findings, present technically intricate issues in a coherent manner, and discuss and defend their position on a specific topic within a seminar group.

Languages and Compilation
Providing an introduction to formal languages, grammars, automata and how these concepts relate to programming in terms of compilers and the compilation process, students will learn about syntax and semantics, phrase structure grammars and the Chomsky Hierarchy as well as processes such as derivation and parsing. The module focuses on grammar equivalence and ambiguity in context free grammars and its implications. There is exploration of the relationship between languages and abstract machines. Students are presented with the concept of computation alongside Turing’s thesis, alternative models of computation and applications of abstract machine representations. There are further introductions to the compilation process including lexical analysis and syntactic analysis.
By the end of this module, students will understand the relation of programming languages and the theory of formal languages. They will possess an essential understanding of the compilation process for a highlevel programming language. Students are encouraged to engage with theoretical aspects of computer science to compliment practical skills in other parts of their degree. There are links to other disciplines such as linguistics, and the course explains the challenges of compilation in the context of software development and computer science.

Security and Risk
Covering a range of topics, including asset identification and assessment, threat analysis and management tools and frameworks, students will become familiar with attack lifecycle and processes, as well as risk management and assessment processes, tools and frameworks. The module covers mitigation strategies and the most appropriate mitigation technologies and offers knowledge on assurance frameworks and disaster recovery planning. There is also an opportunity to learn about infrastructure design and implementation technologies and attack tree and software design evaluation.
Students will gain an understanding of the different ways in which an IT professional can make effective decisions when securing an IT infrastructure. The course will make them aware of the tools, frameworks and models that can be used to identify assets, threats and risks, before selecting the most appropriate strategies to manage the exposure that IT infrastructure faces in the light of this analysis. The module builds on their skills with a practical examination of the mechanisms by which IT infrastructures are attacked.
Core

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.

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.

Elliptic 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.

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 Space
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 module 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.

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

Rings Fields and Polynomials
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.

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
Year 4

Research Methods
Students will gain a formal understanding of research and will develop the ability to critically reflect on research approaches and practices in the field of computing. Research Methods will also encourage an appreciation of the different ways that other disciplines, academic communities and industries all conduct research. There will be an opportunity to plan a research project and develop a convincing study design to address a challenge or problem. This module explores ethical and data management issues associated with research as well as research and innovation practices in industry.
The module covers the fundamentals of research such as sampling and design, before considering strategies and research methods. Furthermore, the module offers greater insight into research design, such as how to structure and frame research studies, choosing a research strategy and selecting the best research method. Students will learn about ethical issues in research and approval processes before understanding the opportunities and expectations from their industrial placements.
Core

Advanced HumanComputer Interaction 1
Explore advanced topics in experimental HumanComputer Interaction (HCI), such as understanding users and their requirements, investigating design spaces and prototyping and developing innovative interaction techniques. The module offers increased experience in HCI literature and design methods both with and without users, as well as practical experience of using supporting tools. Students will learn about modelling techniques and design space techniques as part of the module.
Upon completion of the module, students will have the knowledge to conduct experimental HCI research and have the motivation, experience and tools for understanding users and their requirements for interaction. The module helps to develop scientific writing skills and analytical thinking and prepares students for further postgraduate study, or for a successful career in IT or computing.

Advanced HumanComputer Interaction 2
This module covers advanced topics in experimental HumanComputer Interaction (HCI) with an emphasis on experimental design, evaluation methodologies, statistical analysis and result interpretation. Whilst engaging with a number of key topics, students will be asked to explore the evaluation process. Students will learn to recognise when HCI is required and which forms of evaluation is necessary in a given situation, for example making appropriate selections from systems vs user and qualitative vs quantitative evaluations.
Practical sessions will enable students to develop skills with statistical analysis packages. They will also receive guidance on the application of appropriate tests and result reporting.

Applied Data Mining
This module provides students with uptodate information on current applications of data in both industry and research. Expanding on the module ‘Fundamentals of Data’, students will gain a more detailed level of understanding about how data is processed and applied on a large scale across a variety of different areas.
Students will develop knowledge in different areas of science and will recognise their relation to big data, in addition to understanding how largescale challenges are being addressed with current stateoftheart techniques. The module will provide recommendations on the Social Web and their roots in social network theory and analysis, in addition their adaption and extension to largescale problems, by focusing on primer, usergenerated content and crowdsourced data, social networks (theories, analysis), recommendation (collaborative filtering, content recommendation challenges, and friend recommendation/link prediction).
On completion of this module, students will be able to create scalable solutions to problems involving data from the semantic, social and scientific web, in addition to abilities gained in processing networks and performing of network analysis in order to identify key factors in information flow.

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

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

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

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

Data Mining
Students are provided with a comprehensive coverage of the problems related to data representation, manipulation and processing in terms of extracting information from data, including big data. They will apply their working understanding to the data primer, data processing and classification. They will also enhance their familiarity with dynamic data space partitioning, using evolving, clustering and data clouds, and monitoring the quality of the selflearning system online.
Students will also gain the ability to develop software scripts that implement advanced data representation and processing, and demonstrate their impact on performance. In addition, they will develop a working knowledge in listing, explaining and generalising the tradeoffs of performance, as well as the complexity in designing practical solutions for problems of data representation and processing in terms of storage, time and computing power.

Elements of Distributed Systems
This module will equip students with the ability to develop and apply a deep understanding of fundamental principles, techniques and technologies that underpin today's global IT infrastructure. They will learn to assess new systems technologies, to know where technologies fit in a comprehensive schema, and to know what to read in order to develop a deeper level of understanding. Students will focus on the properties of system components, and will become familiar with the strengths, weaknesses, scalability and bottlenecks of systems components. This will enable them to make intelligent and wellreasoned tradeoffs between fundamental building blocks of distributed systems in today’s IT infrastructure.
This is a highly practical module, in which students build a major system representative of an endtoend IT infrastructure, and is also highly discussionoriented, with frequent inclass discussion sections and problemsolving group work. The module covers a very broad range of stateoftheart techniques and principles of modern distributed systems. These including: caching, tiering, replication, synchronisation, failure and reliability. Students will also explore realworld technologies, from interaction paradigms in distributed systems, to peertopeer architectures and scalable and highperformance networking and storage.

Elliptic Curves
This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.
Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

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

Financial Mathematics
This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.
Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and BlackScholes models.
Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:
 Conditional expectation
 Filtrations
 Martingales
 Stopping times
 Brownian motion
 BlackScholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated riskneutral probability; performing calculations with the BlackScholes formula; and proving various steps in the derivation of the BlackScholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.
In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the realworld; improve their written and oral communication skills; and develop their critical thinking.

Fourth Year Project (Computer Science)
The Fourth Year Project will focus on a significant specification, design, implementation and/or evaluation project at the suitable level for an MSci qualification. The project sees students tackle a realworld problem by applying their knowledge in computer science. The project is usually achieved in conjunction with an industry placement; however, it can be completed at the University. Suggestions made by industry will be vetted by a team of academics to ensure appropriate depth, and if no suitable project with industry can be found, one will be provided by academic staff.
Weekly guidance is given from a member of academic staff from the department to ensure the necessary level of academic content and rigour is being maintained. There are also Business Development mentors in the Knowledge Business Centre (KBC) to provide students with an insight into the daytoday expectations and responsibilities of working with industry. The Fourth Year Project is designed to challenge students and develop their existing knowledge, understanding and skills from their undergraduate degree to produce a significant piece of academically rigorous project work.

Galois Theory
Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1to1 correspondence with the subgroups of G.
In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

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

Hilbert Space
Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.
The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

Industrial Dissertation
At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.
In the first term there will be weekly supervisor meetings and students will be guided into their indepth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Industrial Placement(s)
Students complete a 10 week industrial placement in the Lent term of their 4th year. The University has a range of businesses from SMEs to large corporates for students to be placed in. There are no taught elements in this module, but students have access to an academic supervisor to guide and assist them during the placement. Placements are assigned to students in the Michaelmas term.
Students will gain firsthand experience of working in a contemporary ICT related environment, developing an appreciation and understanding of professional practices and codes of conduct in industrial, commercial and professional settings. Companies will set tasks that are related to students’ knowledge and experience gained throughout their degree, allowing them to apply it in a professional setting. Placements are offered by a variety of companies with different topics. Through an initial matching and application process, we ensure the biggest possible overlap between student interest and company requirements.

Lebesgue Integration
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Lie Groups and Lie Algebras
The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.
The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

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

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

Mathematics Dissertation
At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.
In the first term they will meet their supervisor weekly and will be guided into their indepth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

Metric Spaces
In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T
They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Operator Theory
Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems. Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.
First, students will review Hilbert spaces, before spending some time studying infinitedimensional operators, notably the unilateral shift and multiplication operators, aswell as basic concepts. They will then consider the criteria for invertibilityof selfadjoint operators, leading to the spectral theory of such operators.

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

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

Representation Theory of Finite Groups
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of Rmodule and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.
The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.
The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

Statistics Dissertation
At the end of Year Three, students will fill in a form stating their mathematical or statistical interests and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student’s degree scheme and personal choice.
During the first term students will meet their supervisor weekly and will be guided into an in depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

Stochastic Calculus for Finance
Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discretetime stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuoustime processes, in particular, Brownian motion.
Students will learn to derive basic properties of Brownian motion and explore integration with respect to it. They will also examine the derivation of Ito's formula and how to apply this to Brownian motion.
Over the course of the module, students will also learn to justify and critique the use of stochastic models for reallife applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

Stochastic Processes
This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.
In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.

Systems Architecture and Integration
Introducing a range of architectural approaches, techniques and technologies that underpin today’s global IT infrastructure, this module combines with other modules to form the systems stream of the programme. It is designed to enhance students’ knowledge of how building blocks are composed to create systems of systems.
Students will gain a detailed understanding and an ability to critique contemporary systems’ architecture in terms of scalability, resilience, performance and other shortcomings.
The principal ethos of this module is to focus on the principles, emergent properties and the application of systems elements as used in largescale and high performance systems. Detailed studies and invited industrial speakers will be used to provide supporting realworld context and a basis for seminar discussions. Students are also offered ‘handson’ measurementbased coursework that focuses on the scalability of a significant technology.

Topology and Fractals
Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.
This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.
Careers
Careers
Both computer science and maths graduates are highly sought after individually, but by combining the two disciplines a wide range of opportunities become available. Many of our graduates go on to work in professional software and systems development environments or in technology, computing, financial services or management roles. Our graduates can expect a competitive starting salary in careers such as:
 Business Analyst
 Data Scientist
 Information Systems Manager
 Multimedia Programmer
 Quantitative Analyst
 Risk Analyst
 Software Engineer
Alternatively, you may wish to undertake PhDlevel study at Lancaster and pursue a career in research and teaching.
Fees and Funding
Fees
We set our fees on an annual basis and the 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU  Overseas 

£9,250  £18,890 
Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.

Course Overview
Course Overview
Mathematics and computing are intrinsically linked. By combining them in one Masters programme you gain a robust, advanced understanding of the two disciplines, equipping you with sophisticated specialist skills and detailed technical knowledge, allowing you to excel in your chosen career.
Mathematics forms the foundations of all technology and computing, and as such, a rigorous study of the discipline provides invaluable insight and understanding into computer science. Furthermore, computer science is itself a dynamic discipline with a wide range of applications. As a result, this combined programme offers you a robust and comprehensive skill set, indepth specialist knowledge, and fantastic career opportunities.
You will explore the theory and practice of innovative and experimental computer science, while gaining an advanced understanding of the mathematical concepts and processes behind them. The depth and breadth of knowledge and experience gained over the four years will prove to be a challenging but rewarding opportunity, placing you in the strongest position as you move forward into your chosen career.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problemsolving and quantitative reasoning skills. Additionally, you will be introduced to software development and the fundamentals of computer science. These topics will allow you to gain a wealth of technical knowledge and develop key interdisciplinary skills.
In the second year, you will begin to drill down into specialist maths and computing modules, studying HumanComputer Interaction; Software Design; Linear Algebra; and Social, Ethical and Professional Issues in Computing. These core modules will ensure you gain a solid understanding of the disciplines that is applicable in the realworld. Alongside these, you will also be able to choose from a range of optional maths modules, these include: Abstract Algebra, Complex Analysis, and Real Analysis. In addition, you will bring your skills and knowledge together in a group project, which will allow you to apply what you have learnt to the realworld and gain valuable, practical experience.
For the third year, your study will be largely guided by your own interests. Compulsory modules, such as Artificial Intelligence, Languages and Compilation, and Security and Risk, will enhance and progress your computer science knowledge and provide insight into the sort of activity you will encounter in the realworld. However, the wide range of optional modules you can choose from will allow you to delve deeper into your own interests and customise the year to suit your career ambitions.
The fourth and final year of your degree will introduce a variety of advanced modules for you to choose from. You can build a strong repertoire of maths and computer science skills and knowledge, to suit your interests and goals, including: Data Mining; Galois Theory; Lie Groups and Lie Algebras; Operator Theory; and Systems Architecture and Integration. You will also benefit from our Research Methods module, which will provide you with a formal understanding of research, and allow you to gain the appropriate skills and practices. You will learn to critically reflect on your research and will gain an appreciation of the different ways that other disciplines, academic communities and industries conduct research. This will provide invaluable insight and experience for many graduate careers as well as for continuing in academia.

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

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

Calculus
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Fundamentals of Computer Science
This module provides students with an insight into the importance and relevance of the principles of computer science. Gaining the essential knowledge needed for analysing and characterising the efficiency of algorithms and computer programs, students learn how to make the right design choice when implementing computer programs to optimise efficiency for given design parameters.
Students also study the role and characteristics of data structures, and gain an understanding of the continuing importance of classical algorithms in computer science.

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.

Software Development
Computer programming is a highly practical skill in our quickly developing world. In this module students develop the skills expected of a principled computer programmer as they learn how to write, analyse, debug, test and document computer programs. Students will be introduced to both the C and Java programming languages, two of the most widely used languages in the world. They will learn about best practice of daytoday techniques associated with software development and gain an understanding of the software development cycle. Learning about the challenges faced by software developers in addressing scalability and complexity in computer software, students will be able to work independently to develop moderately complex computer programs.

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

Computer Science Group Project
This module gives students the opportunity to build upon their skills and knowledge from Year 1 to create a realworld system in a group context. As part of a group, students will work effectively to gather system requirements; design and then implement the project; and accurately evaluating it. The module aims to increase theoretical knowledge and practical skills in prototyping, project planning, project management, management and execution, games design, systems design and testing strategies. Alongside these, students will also enhance their teamwork, problemsolving, communication, presentation and report writing skills, which will be valuable when progressing into a career.

HumanComputer Interaction
Students will learn theoretical and practical topics in HumanComputer Interaction, with lab work offering hands on experience of design, implementation and the ability to evaluate interactive systems through practical case studies. The course explores the underpinnings of human perception, usercentred design and participatory design processes, with students learning multiple design techniques. The module leads to an understanding of how internal system design impacts external user interface behaviour and highlights the importance of accessibility for all users.
By the end of the module, students will be able to successfully integrate diverse information to form a coherent understanding of HumanComputer Interaction; critically reflect on technical advancements in HCI and demonstrate the independent learning abilities needed for continual professional development and effective written and verbal skills.

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.

Social, Ethical and Professional Issues in Computing
Students will gain the essential skills and knowledge to operate within the professional, legal and ethical frameworks of their profession. Techniques for breaking down a project into manageable parts and efficient time allocation are taught, leading to a fundamental understanding of the skills and methods required to pursue scientific inquiry and the fundamental concepts and tools for statistical analysis to measure and explain data. Exemplars and guidelines on producing concise and structured scientific reports are offered and students receive additional lectures on presentation skills, professional ethics in relation to computing and communications. Finally, lectures provide an awareness of fundamental legal aspects related to a profession in computing and communications, including intellectual property rights and patent law.
Throughout this course, students will gain a high level of awareness of subject specific skills and general competence needed to gain employment in their field. The module develops academic writing and research skills in a computing context, complimenting students’ other modules.

Software Design
Software Design offers the opportunity to gain an understanding of the importance of software architecture design, different styles of architecture and the meaning of quality attributes for software design such as maintainability, performance and scalability. Students will gain knowledge of systematic approaches to developing software design using a set of graphical models. The design process involved in developing several modes of the system at different levels of abstraction is explained and they will be introduced to object oriented design with UML.
Throughout the module, students will appreciate the broader context of the role of computer science in the workplace, and the key role it plays in implementing software. The course also looks at understanding the meaning of quality attributes for software design as well as architectural models for specific software systems. Students will gain an insight into the main quality attributes for deciding classes. Students will be able to interpret and construct UML models of software and implement a design expressed as a UML mode as well as understanding how to use various design patterns to address certain problems.
Core

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.

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.
Optional
Year 3

Artificial Intelligence
Students will gain an introduction to fundamental concepts in artificial intelligence and learn about current trends and issues. Topics such as Knowledge Representation and Reasoning, Decision Making (DM) and Decision Making Under Uncertainties, and Probability Theory are all explored throughout the course. Artificial Intelligence offers experience in supervised and unsupervised machine learning, neural networks and decision trees. Multivariate methods, and clustering and classification approaches are taught and there is an introduction to evolutionary algorithms, phenotypes, genotypes and fundamental genetic operators. Programming languages suitable for intelligent systems, such as Scheme and Prolog are investigated and students are made familiar with the applications of artificial intelligence.
This module sees an awareness of the requirements of artificial intelligence systems in general, and in the context of computing and communications systems. Through knowledge based, probabilistic and logical systems, the module provides students with an awareness of competing approaches and a broad grounding in artificial intelligence. Additionally they will understand and critically analyse artificial intelligence techniques used in modern computers and mobile devices.

Computer Science Seminars
Students will be exposed to a range of current computer science related topics from different subject areas. The areas covered come from our different thematic strands and will include: natural language engineering; policy based network resilience; eyetracking for ubiquitous computing applications; and a focus on energy aware control and sensing in home environments.
Students will conduct independent and indepth research into an advanced topic of computing or communications, reflecting current topical and research issues. During the course of the module, students will analyse, structure, summarise, document and present findings in front of a large group. They will gain topical knowledge and skills related to the subject areas of the seminars, and will learn with and from their peers. The module will enable students to produce a detailed document describing their research findings, present technically intricate issues in a coherent manner, and discuss and defend their position on a specific topic within a seminar group.

Languages and Compilation
Providing an introduction to formal languages, grammars, automata and how these concepts relate to programming in terms of compilers and the compilation process, students will learn about syntax and semantics, phrase structure grammars and the Chomsky Hierarchy as well as processes such as derivation and parsing. The module focuses on grammar equivalence and ambiguity in context free grammars and its implications. There is exploration of the relationship between languages and abstract machines. Students are presented with the concept of computation alongside Turing’s thesis, alternative models of computation and applications of abstract machine representations. There are further introductions to the compilation process including lexical analysis and syntactic analysis.
By the end of this module, students will understand the relation of programming languages and the theory of formal languages. They will possess an essential understanding of the compilation process for a highlevel programming language. Students are encouraged to engage with theoretical aspects of computer science to compliment practical skills in other parts of their degree. There are links to other disciplines such as linguistics, and the course explains the challenges of compilation in the context of software development and computer science.

Security and Risk
Covering a range of topics, including asset identification and assessment, threat analysis and management tools and frameworks, students will become familiar with attack lifecycle and processes, as well as risk management and assessment processes, tools and frameworks. The module covers mitigation strategies and the most appropriate mitigation technologies and offers knowledge on assurance frameworks and disaster recovery planning. There is also an opportunity to learn about infrastructure design and implementation technologies and attack tree and software design evaluation.
Students will gain an understanding of the different ways in which an IT professional can make effective decisions when securing an IT infrastructure. The course will make them aware of the tools, frameworks and models that can be used to identify assets, threats and risks, before selecting the most appropriate strategies to manage the exposure that IT infrastructure faces in the light of this analysis. The module builds on their skills with a practical examination of the mechanisms by which IT infrastructures are attacked.
Core

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.

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.

Elliptic 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.

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 Space
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 module 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.

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

Rings Fields and Polynomials
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.

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
Year 4

Research Methods
Students will gain a formal understanding of research and will develop the ability to critically reflect on research approaches and practices in the field of computing. Research Methods will also encourage an appreciation of the different ways that other disciplines, academic communities and industries all conduct research. There will be an opportunity to plan a research project and develop a convincing study design to address a challenge or problem. This module explores ethical and data management issues associated with research as well as research and innovation practices in industry.
The module covers the fundamentals of research such as sampling and design, before considering strategies and research methods. Furthermore, the module offers greater insight into research design, such as how to structure and frame research studies, choosing a research strategy and selecting the best research method. Students will learn about ethical issues in research and approval processes before understanding the opportunities and expectations from their industrial placements.
Core

Advanced HumanComputer Interaction 1
Explore advanced topics in experimental HumanComputer Interaction (HCI), such as understanding users and their requirements, investigating design spaces and prototyping and developing innovative interaction techniques. The module offers increased experience in HCI literature and design methods both with and without users, as well as practical experience of using supporting tools. Students will learn about modelling techniques and design space techniques as part of the module.
Upon completion of the module, students will have the knowledge to conduct experimental HCI research and have the motivation, experience and tools for understanding users and their requirements for interaction. The module helps to develop scientific writing skills and analytical thinking and prepares students for further postgraduate study, or for a successful career in IT or computing.

Advanced HumanComputer Interaction 2
This module covers advanced topics in experimental HumanComputer Interaction (HCI) with an emphasis on experimental design, evaluation methodologies, statistical analysis and result interpretation. Whilst engaging with a number of key topics, students will be asked to explore the evaluation process. Students will learn to recognise when HCI is required and which forms of evaluation is necessary in a given situation, for example making appropriate selections from systems vs user and qualitative vs quantitative evaluations.
Practical sessions will enable students to develop skills with statistical analysis packages. They will also receive guidance on the application of appropriate tests and result reporting.

Applied Data Mining
This module provides students with uptodate information on current applications of data in both industry and research. Expanding on the module ‘Fundamentals of Data’, students will gain a more detailed level of understanding about how data is processed and applied on a large scale across a variety of different areas.
Students will develop knowledge in different areas of science and will recognise their relation to big data, in addition to understanding how largescale challenges are being addressed with current stateoftheart techniques. The module will provide recommendations on the Social Web and their roots in social network theory and analysis, in addition their adaption and extension to largescale problems, by focusing on primer, usergenerated content and crowdsourced data, social networks (theories, analysis), recommendation (collaborative filtering, content recommendation challenges, and friend recommendation/link prediction).
On completion of this module, students will be able to create scalable solutions to problems involving data from the semantic, social and scientific web, in addition to abilities gained in processing networks and performing of network analysis in order to identify key factors in information flow.

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

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

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

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

Data Mining
Students are provided with a comprehensive coverage of the problems related to data representation, manipulation and processing in terms of extracting information from data, including big data. They will apply their working understanding to the data primer, data processing and classification. They will also enhance their familiarity with dynamic data space partitioning, using evolving, clustering and data clouds, and monitoring the quality of the selflearning system online.
Students will also gain the ability to develop software scripts that implement advanced data representation and processing, and demonstrate their impact on performance. In addition, they will develop a working knowledge in listing, explaining and generalising the tradeoffs of performance, as well as the complexity in designing practical solutions for problems of data representation and processing in terms of storage, time and computing power.

Elements of Distributed Systems
This module will equip students with the ability to develop and apply a deep understanding of fundamental principles, techniques and technologies that underpin today's global IT infrastructure. They will learn to assess new systems technologies, to know where technologies fit in a comprehensive schema, and to know what to read in order to develop a deeper level of understanding. Students will focus on the properties of system components, and will become familiar with the strengths, weaknesses, scalability and bottlenecks of systems components. This will enable them to make intelligent and wellreasoned tradeoffs between fundamental building blocks of distributed systems in today’s IT infrastructure.
This is a highly practical module, in which students build a major system representative of an endtoend IT infrastructure, and is also highly discussionoriented, with frequent inclass discussion sections and problemsolving group work. The module covers a very broad range of stateoftheart techniques and principles of modern distributed systems. These including: caching, tiering, replication, synchronisation, failure and reliability. Students will also explore realworld technologies, from interaction paradigms in distributed systems, to peertopeer architectures and scalable and highperformance networking and storage.

Elliptic Curves
This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.
Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

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

Financial Mathematics
This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.
Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and BlackScholes models.
Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:
 Conditional expectation
 Filtrations
 Martingales
 Stopping times
 Brownian motion
 BlackScholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated riskneutral probability; performing calculations with the BlackScholes formula; and proving various steps in the derivation of the BlackScholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.
In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the realworld; improve their written and oral communication skills; and develop their critical thinking.

Fourth Year Project (Computer Science)
The Fourth Year Project will focus on a significant specification, design, implementation and/or evaluation project at the suitable level for an MSci qualification. The project sees students tackle a realworld problem by applying their knowledge in computer science. The project is usually achieved in conjunction with an industry placement; however, it can be completed at the University. Suggestions made by industry will be vetted by a team of academics to ensure appropriate depth, and if no suitable project with industry can be found, one will be provided by academic staff.
Weekly guidance is given from a member of academic staff from the department to ensure the necessary level of academic content and rigour is being maintained. There are also Business Development mentors in the Knowledge Business Centre (KBC) to provide students with an insight into the daytoday expectations and responsibilities of working with industry. The Fourth Year Project is designed to challenge students and develop their existing knowledge, understanding and skills from their undergraduate degree to produce a significant piece of academically rigorous project work.

Galois Theory
Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1to1 correspondence with the subgroups of G.
In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

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

Hilbert Space
Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.
The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

Industrial Dissertation
At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.
In the first term there will be weekly supervisor meetings and students will be guided into their indepth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Industrial Placement(s)
Students complete a 10 week industrial placement in the Lent term of their 4th year. The University has a range of businesses from SMEs to large corporates for students to be placed in. There are no taught elements in this module, but students have access to an academic supervisor to guide and assist them during the placement. Placements are assigned to students in the Michaelmas term.
Students will gain firsthand experience of working in a contemporary ICT related environment, developing an appreciation and understanding of professional practices and codes of conduct in industrial, commercial and professional settings. Companies will set tasks that are related to students’ knowledge and experience gained throughout their degree, allowing them to apply it in a professional setting. Placements are offered by a variety of companies with different topics. Through an initial matching and application process, we ensure the biggest possible overlap between student interest and company requirements.

Lebesgue Integration
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Lie Groups and Lie Algebras
The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.
The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

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

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

Mathematics Dissertation
At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.
In the first term they will meet their supervisor weekly and will be guided into their indepth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

Metric Spaces
In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T
They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Operator Theory
Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems. Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.
First, students will review Hilbert spaces, before spending some time studying infinitedimensional operators, notably the unilateral shift and multiplication operators, aswell as basic concepts. They will then consider the criteria for invertibilityof selfadjoint operators, leading to the spectral theory of such operators.

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

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

Representation Theory of Finite Groups
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of Rmodule and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.
The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.
The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

Statistics Dissertation
At the end of Year Three, students will fill in a form stating their mathematical or statistical interests and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student’s degree scheme and personal choice.
During the first term students will meet their supervisor weekly and will be guided into an in depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

Stochastic Calculus for Finance
Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discretetime stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuoustime processes, in particular, Brownian motion.
Students will learn to derive basic properties of Brownian motion and explore integration with respect to it. They will also examine the derivation of Ito's formula and how to apply this to Brownian motion.
Over the course of the module, students will also learn to justify and critique the use of stochastic models for reallife applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

Stochastic Processes
This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.
In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.

Systems Architecture and Integration
Introducing a range of architectural approaches, techniques and technologies that underpin today’s global IT infrastructure, this module combines with other modules to form the systems stream of the programme. It is designed to enhance students’ knowledge of how building blocks are composed to create systems of systems.
Students will gain a detailed understanding and an ability to critique contemporary systems’ architecture in terms of scalability, resilience, performance and other shortcomings.
The principal ethos of this module is to focus on the principles, emergent properties and the application of systems elements as used in largescale and high performance systems. Detailed studies and invited industrial speakers will be used to provide supporting realworld context and a basis for seminar discussions. Students are also offered ‘handson’ measurementbased coursework that focuses on the scalability of a significant technology.

Topology and Fractals
Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.
This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.

Calculus

Careers
Careers
Both computer science and maths graduates are highly sought after individually, but by combining the two disciplines a wide range of opportunities become available. Many of our graduates go on to work in professional software and systems development environments or in technology, computing, financial services or management roles. Our graduates can expect a competitive starting salary in careers such as:
 Business Analyst
 Data Scientist
 Information Systems Manager
 Multimedia Programmer
 Quantitative Analyst
 Risk Analyst
 Software Engineer
Alternatively, you may wish to undertake PhDlevel study at Lancaster and pursue a career in research and teaching.

Fees and Funding
Fees and Funding
Fees
We set our fees on an annual basis and the 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU Overseas £9,250 £18,890 Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.
The Department
Undergraduate Open Days
Booking is now open for Lancaster University's summer 2018 open days. Reserve your place
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11.5 hours
Typical time in lectures, seminars and similar per week during term time
47%
Average assessment by coursework