Computer Science and Mathematics

Core Modules

Year 1

• Object-oriented Programming

  In this module you will develop an understanding of programming and object-orientation concepts. You will learn about program basics, control flow, data structures, objects, exceptions, and file input and output. You will consider how to solve basic programming tasks and the need for program documentation, testing, readability and modifiability.

• Calculus

  In this module, you will develop an understanding of the key concepts in Calculus, including differentiation and integration. You will learn how to factorise polynomials and separate rational functions into partial fractions, differentiate commonly occurring functions, and find definite and indefinite integrals of a variety of functions using substitution or integration by parts. You will also examine how to recognise the standard forms of first-order differential equations, and reduce other equations to these forms and solve them.

• Calculus II

  In this module you will develop an understanding of the calculus functions of more than one variable and how it may be used in areas such as geometry and optimisation. You learn how to manipulate partial derivatives, construct and manipulate line integrals, represent curves and surfaces in higher dimensions, calculate areas under a curve and volumes between surfaces, and evaluate double integrals, including the use of change of order of integration and change of coordinates.

• Introduction to Pure Mathematics

  In this module you will develop an understanding of the fundamental algebraic structures, including familiar integers and polynomial rings. You will learn how to apply Euclid's algorithm to find the greatest common divisor of two integers, and use mathematical induction to prove simple results. You will examine the use of arithmetic operations on complex numbers, extract roots of complex numbers, prove De Morgan's laws, and determine whether a given mapping is bijective.

• Linear Algebra I

  In this module you will develop an understanding of basic linear algebra, in particular the use of matrices and vectors. You will look at the basic theoretical and computational techniques of matrix theory, examining the power of vector methods and how they may be used to describe three-dimensional space. You will consider the notions of field, vector space and subspace, and learn how to calculate the determinant of an n x n matrix.

• Academic Integrity

  This module will describe the key principles of academic integrity, focusing on university assignments. Plagiarism, collusion and commissioning will be described as activities that undermine academic integrity, and the possible consequences of engaging in such activities will be described. Activities, with feedback, will provide you with opportunities to reflect and develop your understanding of academic integrity principles.

   

Year 2

• Software Engineering

  In this module you will develop an understanding of software engineering techniques and the managerial discipline required to work as part of a team. You will look at basic object-oriented concepts and consider the need for effective program documentation, testing, readability, and modifiability. You will consider the tools used to support software development, such as version controllers, debuggers, and code style checkers, and see how these are integrated into an industry-standard development environment (IDE). You will deliver a small-scale project using test-driven development.

• Algorithms and Complexity

  In this module you will develop an understanding of the design of algorithms, with a focus on time and space complexity. You will examine basic algorithms, looking at the implementation and analysis of linear search, binary search, and basic sorting, including insertion sort, selection sort, merger sort, quick sort, and heap sort. You will consider alternative data structure representations, such as binary search trees, hash tables, and binary heaps, and will gain an insight into the basics of graph algorithms.

• Team Project

  In this module you will develop an understanding of the role of the computer professional, gaining practical experience in developing medium scale software as part of a team using Scrum-based Agile development. You will apply managerial discipline and learn about the software lifecycle, team development, standard industrial software engineering, project management, use of version control, and integrated development environments (IDEs). You will see why project cost and effort is hard to estimate, and consider why project quality is hard to prescribe.

• Linear Algebra II

  In this module you will develop an understanding of vectors and matrices within the context of vector spaces, with a focus on deriving and using various decompositions of matrices, including eigenvalue decompositions and the so-called normal forms. You will learn how these abstract notions can be used to solve problems encountered in other fields of science and mathematics, such as optimisation theory.

• Probability Theory

  In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

Year 3

• All modules are optional

Optional Modules

There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.

Year 1

• All modules are core

Year 2

• Introduction to Artificial Intelligence
• Mathematical Methods for Computer Science
• Computer and Network Security

  In this module you will develop an understanding of computer and network security. You will look at software vulnerabilities, hands-on hacking-oriented attacks, memory errors, and web and network security. You will learn how to identify such vulnerabilities and consider the countermeasures that can mitigate their exploitation. You will also examine malicious software (malware) as a typical consequence of a successful software exploitation.

• Human-computer Interaction

  In this module you will develop an understanding of the principles of human-computer interaction (HCI) and the approaches that can be used to create interfaces matching users' needs and expectations. You will evaluate the usability and suitability of user interfaces, and build simple prototypes using different media and technologies. You will consider how perception and cognition influence HCI, and examine tests, such as web experiments.

• Applications of Vector Calculus

  In this module the vector calculus methods are applied to a variety of problems in the physical sciences, with a focus on electromagnetism and optics. On completion of the module, the student should be able to calculate relevant physical quantities such as field strengths, forces, and energy distributions in static as well as dynamical electromagnetic systems and be able to treat mathematically the interactions between moving electrical charges, magnets and optical fields.

• Statistical Methods

  In this module you will develop an understanding of statistical modelling, becoming familiar with the theory and the application of linear models. You will learn how to use the classic simple linear regression model and its generalisations for modelling dependence between variables. You will examine how to apply non-parametric methods, such as the Wilxocon and Kolmogorov-Smirnov goodness-of-fit tests, and learn to use the R open source software package.

• Rings and Factorisation

  In this module you will develop an understanding of ring theory and how this area of algebra can be used to address the problem of factorising integers into primes. You will look at how these ideas can be extended to develop notions of 'prime factorisation' for other mathematical objects, such as polynomials. You will investigate the structure of explicit rings and learn how to recognise and construct ring homomorphisms and quotients. You will examine the Gaussian integers as an example of a Euclidean ring, Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

• Real Analysis

  In this module you will develop an understanding of the convergence of series. You will look at the Weierstrass definition of a limit and use standard tests to investigate the convergence of commonly occuring series. You will consider the power series of standard functions, and analyse the Intermediate Value and Mean Value Theorems. You will also examine the properties of the Riemann integral.

• Probability Theory

  In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

• Ordinary Differential Equation

  In this module you will develop an understanding of the concepts arising when the boundary conditions of a differential equation involves two points. You will look at eigenvalues and eigenfunctions in trigonometric differential equations, and determine the Fourier series for a periodic function. You will learn how to manipulate the Dirac delta-function and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

• Further Linear Algebra and Modules

  In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

• Complex Variable

  In this module you will develop an understanding of the basic complex variable theory. You will look at the definitions of continuity and differentiability of a complex valued function at a point, and how Cauchy-Riemann equations can be applied. You will examine how to use a power series to define the complex expontential function, and how to obtain Taylor series of rational and other functions of standard type, determining zeros and poles of given functions. You will also consider how to use Cauchy's Residue Theorem to evaulate real integrals.

Year 3

• Half Unit Project
• Full Unit Project
• Computational Finance
• Intelligent Agents and Multi-agent Systems
• Machine Learning
• Semantic Web
• Data Visualisation and Exploratory Analysis
• Advanced Data Communications
• Concurrent and Parallel Programming
• Interconnected Devices
• Applications of Cryptography

  In this module you will develop an understanding of the fundamentals behind cryptography and how it is deployed in real systems. You will look at a range of security services that can be provided by cryptography and the mechanisms behind them, such as symmetric and public-key encryption, hash functions, MACs, digital signatures and authentication protocols. You will consider the architecture of security systems using cryptography, including key management, implementation issues, cryptographic standards and crypto politics, and examine real-world applications such as 3G, EMV, and SSL/TLS.

• Critical Infrastructure Security

  The module covers key areas of cyber security with the critical national (information) infrastructure forming its background. Fault and attack models for information systems and cyber-physical systems are covered in the form of multiple techniques including variants of attack trees allowing probabilistic attack and defence refinements. The module covers models of large-scale networks and their robustness properties to both random failures and particularly to deliberate attacks and discusses how key elements of the CNI such as the Internet but also other infrastructure sectors such as power and transport sectors can be captured by such models. The security of cyber-physical systems and particularly industrial control systems is another major component of the module, including case studies of attacks by state actors and analyses of control system protocols as well as properties peculiar to CPS.

• Digital Forensics
• Malicious Software
• Smart Cards, RFIDs and Embedded Systems Security
• Human-computer Interaction
• IT Project Management
• Software Language Engineering
• Bioinformatics
• Compilers and Code Generation
• Computational Optimisation
• Digital Audio and Applications
• Functional Programming and Applications
• Mathematics Project

  You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 7,000 words in length, and give a ten-minute presentation outlining your findings.

• Number Theory

  In this module you will develop an understanding of how prime numbers are the building blocks of the integers 0, ±1, ±2, … You will look at how simple equations using integers can be solved, and examine whether a number like 2017 should be written as a sum of two integer squares. You will also see how Number Theory can be used in other areas such as Cryptography, Computer Science and Field Theory.

• Quantum Theory 1

  In this module you will develop an understanding of quantum theory, and the development of the field to explain the behaviour of particles at the atomic level. You will look at the mathematical foundations of the theory, including the Schrodinger equation. You will examine how the theory is applied to one and three dimensional systems, including the hydrogen atom, and see how a probabilistic theory is required to interpret what is measured.

• Electromagnetism

  In this module you will develop an understanding of the elementary ideas of electromagnetism. You will learn how to calculate electric fields and electric potentials from given fixed charge distributions and how to calculate magnetic fields and vector potentials from given steady current distributions. You will examine the magnetic effects of currents, including electromagnetic induction and displacement currents, and analyse the Biot-Savart law and Ampere's law. You will examine Maxwell's equations, and the properties of electromagnetic waves in free space, as well as electric and magnetic dipoles and the electromagnetism of matter.

• Non-Linear Dynamic Systems

  In this module you will develop an understanding of non-linear dynamical systems. You will investigate whether the behaviour of a non-linear system can be predicted from the corresponding linear system and see how dynamical systems can be used to analyse mechanisms such as the spread of disease, the stability of the universe, and the evolution of economic systems. You will gain an insight into the 'secrets' of the non-linear world and the appearance of chaos.

• Statistical Inference

  In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the Neyman-Pearson framework, likelihood ratio tests, and decision theory.

• Time Series Analysis

  In this module you will develop an understanding of some of the descriptive methods and theoretical techniques that are used to analyse time series. You will look at the standard theory around several prototype classes of time series models and learn how to apply appropriate methods of times series analysis and forecasting to a given set of data using Minitab, a statistical computing package. You will examine inferential and associated algorithmic aspects of time-series modelling and simulate time series based on several prototype classes.

• Channels

  In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

• Mathematics of Financial Markets

  In this module you will develop an understanding of how financial markets operate, with a focus on the ideas of risk and return and how they can be measured. You will look at the random behaviour of the stock market, Markowitz portfolio optimisation theory, the Capital Asset Pricing Model, the Binomial model, and the Black-Scholes formula for the pricing of options.

• Combinatorics

  In this module you will develop an understanding of some of the techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

• Cipher Systems

  In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

• Applications of Field Theory

  In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X²⁰¹⁷=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

• Dynamics of Real Fluids

  In this module you will develop an understanding of how the theory of ideal fluids can be used to explain everyday phenomena in the world around us, such as how sound travels, how waves travel over the surface of a lake, and why golden syrup (or volcanic lava) flows differently from water. You will look at the essential features of compressible flow and consider basic vector analysis techniques.

• Mathematics in the Classroom

  In this module you will develop an understanding of a range of methods for teaching children up to A-level standard. You will act act as a role model for pupils, devising appropriate ways to convey the principles and concepts of mathematics. You will spend one session a week in a local school, taking responsibility for preparing lesson plans, putting together relevant learning aids, and delivering some of the classes. You will work with a specific teacher, who will act as a trainer and mentor, gaining valuable transferable skills.

• Computational Number Theory

  In this module you will develop an understanding of a range methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind state-of-the art factorisation methods. You will also look at how to analyse the complexity of fundamental number-theoretic algorithms.

• Principles of Algorithm Design

  In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worst-case analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

• Markov Chains and Applications

  In this module you will develop an understanding of the probabilistic methods used to model systems with uncertain behaviour. You will look at the structure and concepts of discrete and continuous time Markov chains with countable stable space, and consider the methods of conditional expectation. You will learn how to generate functions, and construct a probability model for a variety of problems.

• Quantum Information Theory

  In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the no-cloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

• Financial Mathematics I

  This module shows how mathematics is used by those who work in securities markets. The topics in the module are chosen to demonstrate important applications, and may include the basic asset pricing models, the general behaviour of markets, the theory of derivative securities and elements of portfolio theory.

• Financial Mathematics II

  This module continues the study of financial mathematics begun in Financial Mathematics I. In this module, students will develop a further understanding of the role of mathematics in securities markets. The topics in the module may include models of interest rates, elements of portfolio theory and models for financial returns.

• Error Correcting Codes

  In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

• Public Key Cryptography

  In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cryptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

• Groups and Group Actions

  In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

• Topology

  In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.

• Group Theory

  In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. Groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory.

• Introduction to Optimisation

  Optimisation problems usually seek to maximise or minimise some function subject to a number of constraints; for example, how products should be shipped from factories to shops so as to minimise cost, or finding the shortest route a delivery driver should take. Linear programming is concerned with solving optimisation problems whose requirements are represented by linear relationships. This module introduces different types of optimisation problems and the appropriate approaches for solving each type of problem.

• Game Theory

  This module will analyse ways in which mathematics can be applied to situations that involve making strategic decisions, and will apply game theory to a wide variety of situations including to examples from business and economics.