Computer Science and Mathematics

The group project will give students experience in executing a project through all stages, working to the demands of a client, and practically combine and apply concepts and skills gained in other modules studied so far in their programme. Students will learn to apply their knowledge about prototyping, project planning, management, design, and user evaluation or testing strategies. Teams will deliver reports, code, and demonstrate a working system. They will also communicate their work through reports, demonstrations, and presentations.

The project content may differ from year to year, and groups may be able to select projects aligned with the School’s main themes of Software, Systems, Data and Theory, Interactions and Implications, and Cyber Security, although each theme may not be available every year. Example topic areas could be desktop application development, game programming, computer graphics, user interfaces, mobile computing, or other areas. The exact requirements of a group project will vary according to the focus of its theme; however the course structure of a group project will be the same between themes and different years. Students will receive about 30 hours of workshop contact time throughout the module, in addition to lectures, and then will be expected to work independently as a group.

To support this practical activity, two strands of lectures are delivered. One covers programming and continues the development of the students practical programming skills to allow them to confidently contribute to larger, team-based programming projects. The second covers teamwork, project management, risks, and costings so that the student has a sound base for managing collaborative projects.

Most computing systems are interactive and have people in the loop. Human-computer interaction (HCI) is concerned with all aspects of designing, building, evaluating, and studying systems that involve human interaction. From a computing perspective, students focus on enabling interaction through user interfaces, and on creating interactive systems that are usable and provide a good user experience.

The module introduces students to the foundations of HCI in understanding human behaviour, technologies for interaction, and human-centred design. Students will review human perception, cognition and action and relate these to design principles and guidelines; discuss different user interface paradigms and key technologies such as pointing; and introduce practical methods for design and evaluation with users.

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

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

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 development is a collaborative and professional process, requiring far more than a single individual undertaking programming activities. This module investigates the processes, tools, techniques, and notations required to successfully undertake the development of commercial grade software.

Focussing on the key non-functional parameters of software reuse, scalability, maintainability, and extensibility, students will explore the benefits brought by the rigour associated with object-oriented, strongly typed languages (such as Java). Students will practice the concepts of composition, inheritance, polymorphism, interfaces, and traits and the commonly employed design patterns that they enable. They will also study the processes and notations associated with defining the relationship and behaviour of complex computer software systems.

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

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

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

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

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

Students will 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, fixed-point iteration, higher dimensions, first and second derivatives, non-parametric 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 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 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
• Individual 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.

A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.

Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.

Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.

Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.

The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.

Students will be given a solid foundation in the basics of 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.

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.

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 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.

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 well-known 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.

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; eye-tracking for ubiquitous computing applications; and a focus on energy aware control and sensing in home environments.

Students will conduct independent and in-depth 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.

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.

The topic of smooth curves and surfaces in three-dimensional 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 well-known 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.

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.

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.

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.

All programming languages are based on theoretical principles of formal language theory. In this module, students take a deep dive into formal languages representation and grammars, and how relate to programming language compilers and interpreters.

Students will study formal language syntax and semantics, phrase structure grammars and the Chomsky Hierarchy. They will learn how to classify languages and explore the concepts of ambiguity in Context Free grammars and its implications. In particular, they will learn about the compilation process including lexical analysis and syntactic analysis, recursive descent parsers, and semantic analysis. Finally, students get to investigate the synthesis phase, where intermediate representations, target languages, and structures lead to code generation. In the School, we blend lectures with small group lab sessions where students gain hands-on experience of applying such theory.

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.

Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.

This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.

Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?

The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

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.

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 high-dimensional 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.

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 well-placed 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.

This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.

Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models.

Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:

• Conditional expectation
• Filtrations
• Martingales
• Stopping times
• Brownian motion
• Black-Scholes formula

Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.

In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking.

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.

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.

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 non-specialist, 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.

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.

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 R-module 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

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.

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.

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.

Modern statistics is characterised by computer-intensive 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 Change-Point Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.

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.

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.

This module provides students with up-to-date 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 large-scale challenges are being addressed with current state-of-the-art 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 large-scale problems, by focusing on primer, user-generated content and crowd-sourced 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.

In this module we explore the architectural approaches, techniques and technologies that underpin today's Big Data system infrastructure and particularly large-scale enterprise systems. It is one of two complementary modules that comprise the Systems stream of the Computer Science MSc, which together provide a broad knowledge and context of systems architecture enabling students to assess new systems technologies, to know where technologies fit in the larger scheme of enterprise systems and state of the art research thinking, and to know what to read to go deeper.

The principal ethos of the module is to focus on the principles of Big Data systems, and applying those principles using state of the art technology to engineer and lead data science projects. Detailed case studies and invited industrial speakers will be used to provide supporting real-world context and a basis for interactive seminar discussions.

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 self-learning 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 trade-offs 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.

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 well-reasoned trade-offs 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 end-to-end IT infrastructure, and is also highly discussion-oriented, with frequent in-class discussion sections and problem-solving group work. The module covers a very broad range of state-of-the-art techniques and principles of modern distributed systems. These including: caching, tiering, replication, synchronisation, failure and reliability. Students will also explore real-world technologies, from interaction paradigms in distributed systems, to peer-to-peer architectures and scalable and high-performance networking and storage.

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 real-world 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 day-to-day 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 is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1-to-1 correspondence with the subgroups of G.

In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.

The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.

In the first term there will be weekly supervisor meetings and students will be guided into their in-depth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.

Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Students 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 first-hand 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.

Explore advanced topics in experimental Human-Computer 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.

Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.

The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.

The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.

In the first term they will meet their supervisor weekly and will be guided into their in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.

Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.

Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models.

Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:

• Conditional expectation
• Filtrations
• Martingales
• Stopping times
• Brownian motion
• Black-Scholes formula

Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.

In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking.

In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T

They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems.

Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.

First students will review Hilbert spaces, before spending some time studying infinite-dimensional operators, notably the unilateral shift and multiplication operators, as well as basic concepts. They will then consider the criteria for invertibility of self adjoint operators, leading to the spectral theory of such operators.

The aim of this module is to develop an analytical and axiomatic approach to the theory of probabilities.

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

In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of R-module and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.

The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.

The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

At the end of Year 3 students will fill in a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student's degree scheme and choice.

During the first term students will meet their supervisor on a weekly basis and will be guided into an in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.

Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discrete-time stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuous-time processes, in particular, Brownian motion.

Students will learn to derive basic properties of Brownian motion and explore integration with respect to it. They will also examine the derivation of Ito's formula and how to apply this to Brownian motion.

Over the course of the module, students will also learn to justify and critique the use of stochastic models for real-life applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.

In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.

Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.

This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.