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 Computer Science and Mathematics BSc Hons (GG14)
Computer Science and Mathematics BSc Hons  2019 Entry
UCAS Code
GG14
Entry Year
2019
also available in 2018
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 3 Year(s)
Course Overview
Mathematics and computing are intrinsically linked. By combining them in one programme you gain a comprehensive understanding of the fundamental and advanced aspects of the two disciplines, equipping you with the specialist skills and technical knowledge 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.
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 and advanced Linear Algebra. 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 and computing modules, these include: Abstract Algebra; Complex Analysis; Databases; and Social, Ethical and Professional Issues in Computing. 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 final year, your study will largely be guided by your own interests. Based upon what you have previously learnt, you can select from a wide range of modules, customising the third year to suit your career ambitions. These include: Artificial Intelligence, Combinatorics, Languages and Compilation, Security and Risk, and Stochastic Processes.
MSci Computer Science and Mathematics
As well as our BSc, we also offer a four year MSci Computer Science and Mathematics degree. During the course of this programme you have the option to graduate after three years with a BSc, or progress onto an advanced fourth year and complete an MSci. This additional year features Masterslevel modules and a substantial dissertation project.
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 or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

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

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

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

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. We see how partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. Students will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

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

Integration and Differentiation
Building on MATH113, this module explores the familiar topics of integration, series and differentiation, and develops them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: “Can you sum up infinitely many numbers and get a finite number?”. Students will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic equation and eigenvectors and eigenvalues.

Numbers and Relations
This module introduces the student to logic and mathematical proofs, with emphasis placed on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
We take a look at the language and structure of mathematical proofs in general, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas are then applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

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 and then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, and this underpins the skills needed for all subsequent statistical modules of the degree.
Core
Year 2

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
This module will give you the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
You’ll consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of your study will also involve looking at the concepts of length and angle with regard to vector spaces.

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 you’ll select a small number of properties which these and other examples have in common, and use them to define a group.
You’ll also consider the elementary properties of groups. It turns out that several surprisingly elegant results can be proved fairly simply! By looking at maps between groups which 'preserve structure' you’ll discover a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives us a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials, but you’ll meet several less familiar examples too.

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

Databases
During this module, students will receive a theoretical background to the design, implementation and use of database management systems, for both data designers and application developers. The module also explores the need to define the requirements of database systems, making use of the ExtendedEntity Relationship (EER) model as a technique and notation for designing the data in database management systems (DBMS). Students will investigate the mapping of the EER model into an equivalent relational model and then examine it in terms of access rights and privileges.
Over the course of the module, students will become familiar with all the relevant aspects related to information security in the design, development and use of database systems. They will also gain an understanding of how the need for DBMS has evolved over time and how they are applied in everyday scenarios. This technical knowledge will be supplemented by transferable skills in applying efficient physical storage organisation; an increased awareness of the correct processes, models and notations that can be applied to problems; and an ability to critically evaluate a range of technical ideas.

Probability II
Probability provides the theoretical basis for statistics and is of interest in its own right.
You’ll revisit basic concepts from the first year probability module, and extend these to encompass continuous random variables, investigating several important continuous probability distributions.
You’ll then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Real Analysis
In this module you’ll take a thorough look at the limits of sequences and convergence of series. You’ll learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
You’ll spend time studying the Intermediate Value Theorem and the Mean Value Theorem, and will discover that they have many applications of widely differing kinds. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next we put the notion of integration under the microscope. Once it’s properly defined (via limits), you’ll learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. You’ll also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

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.

Statistics II
Statistics is the science of understanding patterns of population behaviour from data.
In this module we approach this problem by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
You’ll focus on likelihoodbased parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and also considering linear regression techniques within the statistical modelling framework.
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.

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 you’ll 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). You’ll also explore important practical applications of the results and methods.

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.

Differential Equations
This module considers questions relating to linear ordinary differential equations. 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
This module gives you a solid foundation in the basics of algebraic geometry. You’ll explore how curves can be described by algebraic equations, and learn how to understand and use abstract groups in dealing with geometrical objects (curves).
You’ll 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 you’ll 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
This module is an introduction to smooth curves and surfaces in threedimensional space. You’ll encounter various geometrical properties of these objects, such as length, area, torsion and curvature, and will have the opportunity to explore the meaning of these quantities. You’ll use a variety of examples to calculate their values, and will use them to apply techniques from calculus and linear algebra.

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
In this module you’ll develop the knowledge of groups that you’ve gained in second year. You’ll study ‘direct products’ which are used to construct new groups, while any finite group is shown to ‘factor’ into ‘simple’ pieces. You’ll also consider situations 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.

Hilbert Space
In this module you’ll examine the notion of a norm, which introduces a generalized notion of ‘distance’ to the purely algebraic setting of vector spaces. You’ll learn several quite natural ways to do this, both for vectors of any dimension and for functions. You’ll then focus on the more special notion of an inner product which generalizes angles at the same time as distances.
Once we’ve established these concepts, you’ll 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 you can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, you’ll look at some of the main results of linear algebra, which generalize very nicely to linear operators between Hilbert spaces.

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.

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.
You’ll also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

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

Mathematical Education
This module is designed to give you an opportunity to consider key issues in the teaching and learning of mathematics. 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 studies Mathematics for many years, you 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. Within this course we hope to help you with this process so that as a Mathematics graduate you will be able to contribute knowledgeably to future debate about the ways in which your subject is treated within the education system.

Medical Statistics: study design and data analysis
This module aims to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, study design, causality and confounding.
You’ll 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 you’re investigating as well as the mathematical and statistical concepts underpinning inference.

Metric Spaces
This module gives an introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications. Studying this module will give you a deeper understanding of continuity as well as a basic grounding in abstract topology.
You’ll also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply your knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

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 you’ll examine the notion of a probability space through simple examples featuring both discrete and continuous sample spaces. You’ll then use random variables and the expectation to develop a probability calculus, which you can apply to achieve laws of large numbers for sums of independent random variables.
You’ll also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Representation Theory of Finite Groups
This module covers the basics of ordinary representation theory. You’ll 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. You’ll also learn to perform computations with representations and morphisms in a selection of finite groups.

Rings Fields and Polynomials
This module furthers your knowledge of commutative rings from your second year study.
You’ll be introduced to two new classes of integral domains called Euclidean domains, where you have a counterpart of the division algorithm, and unique factorization domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
You’ll also explore how wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorization of polynomials, carry over to this new setting.

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.

Statistical Models
This module explores the concept of generalized 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. 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). You’ll also become familiar with the programme R, which you’ll have the opportunity to use in weekly workshops.

Stochastic Processes
This module covers important examples of stochastic processes, and how these processes can be analysed.
As an introduction to stochastic processes you’ll 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).
You’ll then focus on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). You’ll 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 you’ll become familiar with topics from classical statistics as well as some from emerging areas.
You’ll explore time series data through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. You’ll also study time series and volatility modelling, where we’ll discuss the techniques for the analysis of such data with emphasis on financial application.
Another area you’ll focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis. Lastly you’ll spend time looking at ChangePoint Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.
Careers
Careers
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 postgraduate study at Lancaster and pursue a career in research and teaching.
Fees and Funding
Fees
We set our fees on an annual basis and the 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 programme you gain a comprehensive understanding of the fundamental and advanced aspects of the two disciplines, equipping you with the specialist skills and technical knowledge 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.
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 and advanced Linear Algebra. 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 and computing modules, these include: Abstract Algebra; Complex Analysis; Databases; and Social, Ethical and Professional Issues in Computing. 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 final year, your study will largely be guided by your own interests. Based upon what you have previously learnt, you can select from a wide range of modules, customising the third year to suit your career ambitions. These include: Artificial Intelligence, Combinatorics, Languages and Compilation, Security and Risk, and Stochastic Processes.
MSci Computer Science and Mathematics
As well as our BSc, we also offer a four year MSci Computer Science and Mathematics degree. During the course of this programme you have the option to graduate after three years with a BSc, or progress onto an advanced fourth year and complete an MSci. This additional year features Masterslevel modules and a substantial dissertation project.

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 or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk

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

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

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

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

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. We see how partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. Students will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

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

Integration and Differentiation
Building on MATH113, this module explores the familiar topics of integration, series and differentiation, and develops them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: “Can you sum up infinitely many numbers and get a finite number?”. Students will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic equation and eigenvectors and eigenvalues.

Numbers and Relations
This module introduces the student to logic and mathematical proofs, with emphasis placed on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
We take a look at the language and structure of mathematical proofs in general, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas are then applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

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 and then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, and this underpins the skills needed for all subsequent statistical modules of the degree.
Core
Year 2

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
This module will give you the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
You’ll consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of your study will also involve looking at the concepts of length and angle with regard to vector spaces.

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 you’ll select a small number of properties which these and other examples have in common, and use them to define a group.
You’ll also consider the elementary properties of groups. It turns out that several surprisingly elegant results can be proved fairly simply! By looking at maps between groups which 'preserve structure' you’ll discover a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives us a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials, but you’ll meet several less familiar examples too.

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

Databases
During this module, students will receive a theoretical background to the design, implementation and use of database management systems, for both data designers and application developers. The module also explores the need to define the requirements of database systems, making use of the ExtendedEntity Relationship (EER) model as a technique and notation for designing the data in database management systems (DBMS). Students will investigate the mapping of the EER model into an equivalent relational model and then examine it in terms of access rights and privileges.
Over the course of the module, students will become familiar with all the relevant aspects related to information security in the design, development and use of database systems. They will also gain an understanding of how the need for DBMS has evolved over time and how they are applied in everyday scenarios. This technical knowledge will be supplemented by transferable skills in applying efficient physical storage organisation; an increased awareness of the correct processes, models and notations that can be applied to problems; and an ability to critically evaluate a range of technical ideas.

Probability II
Probability provides the theoretical basis for statistics and is of interest in its own right.
You’ll revisit basic concepts from the first year probability module, and extend these to encompass continuous random variables, investigating several important continuous probability distributions.
You’ll then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Real Analysis
In this module you’ll take a thorough look at the limits of sequences and convergence of series. You’ll learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
You’ll spend time studying the Intermediate Value Theorem and the Mean Value Theorem, and will discover that they have many applications of widely differing kinds. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next we put the notion of integration under the microscope. Once it’s properly defined (via limits), you’ll learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. You’ll also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

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.

Statistics II
Statistics is the science of understanding patterns of population behaviour from data.
In this module we approach this problem by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
You’ll focus on likelihoodbased parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and also considering linear regression techniques within the statistical modelling framework.
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.

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 you’ll 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). You’ll also explore important practical applications of the results and methods.

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.

Differential Equations
This module considers questions relating to linear ordinary differential equations. 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
This module gives you a solid foundation in the basics of algebraic geometry. You’ll explore how curves can be described by algebraic equations, and learn how to understand and use abstract groups in dealing with geometrical objects (curves).
You’ll 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 you’ll 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
This module is an introduction to smooth curves and surfaces in threedimensional space. You’ll encounter various geometrical properties of these objects, such as length, area, torsion and curvature, and will have the opportunity to explore the meaning of these quantities. You’ll use a variety of examples to calculate their values, and will use them to apply techniques from calculus and linear algebra.

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
In this module you’ll develop the knowledge of groups that you’ve gained in second year. You’ll study ‘direct products’ which are used to construct new groups, while any finite group is shown to ‘factor’ into ‘simple’ pieces. You’ll also consider situations 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.

Hilbert Space
In this module you’ll examine the notion of a norm, which introduces a generalized notion of ‘distance’ to the purely algebraic setting of vector spaces. You’ll learn several quite natural ways to do this, both for vectors of any dimension and for functions. You’ll then focus on the more special notion of an inner product which generalizes angles at the same time as distances.
Once we’ve established these concepts, you’ll 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 you can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, you’ll look at some of the main results of linear algebra, which generalize very nicely to linear operators between Hilbert spaces.

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.

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.
You’ll also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

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

Mathematical Education
This module is designed to give you an opportunity to consider key issues in the teaching and learning of mathematics. 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 studies Mathematics for many years, you 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. Within this course we hope to help you with this process so that as a Mathematics graduate you will be able to contribute knowledgeably to future debate about the ways in which your subject is treated within the education system.

Medical Statistics: study design and data analysis
This module aims to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, study design, causality and confounding.
You’ll 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 you’re investigating as well as the mathematical and statistical concepts underpinning inference.

Metric Spaces
This module gives an introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications. Studying this module will give you a deeper understanding of continuity as well as a basic grounding in abstract topology.
You’ll also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply your knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

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 you’ll examine the notion of a probability space through simple examples featuring both discrete and continuous sample spaces. You’ll then use random variables and the expectation to develop a probability calculus, which you can apply to achieve laws of large numbers for sums of independent random variables.
You’ll also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Representation Theory of Finite Groups
This module covers the basics of ordinary representation theory. You’ll 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. You’ll also learn to perform computations with representations and morphisms in a selection of finite groups.

Rings Fields and Polynomials
This module furthers your knowledge of commutative rings from your second year study.
You’ll be introduced to two new classes of integral domains called Euclidean domains, where you have a counterpart of the division algorithm, and unique factorization domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
You’ll also explore how wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorization of polynomials, carry over to this new setting.

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.

Statistical Models
This module explores the concept of generalized 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. 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). You’ll also become familiar with the programme R, which you’ll have the opportunity to use in weekly workshops.

Stochastic Processes
This module covers important examples of stochastic processes, and how these processes can be analysed.
As an introduction to stochastic processes you’ll 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).
You’ll then focus on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). You’ll 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 you’ll become familiar with topics from classical statistics as well as some from emerging areas.
You’ll explore time series data through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. You’ll also study time series and volatility modelling, where we’ll discuss the techniques for the analysis of such data with emphasis on financial application.
Another area you’ll focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis. Lastly you’ll spend time looking at ChangePoint Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.

Calculus

Careers
Careers
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 postgraduate study at Lancaster and pursue a career in research and teaching.

Fees and Funding
Fees and Funding
Fees
We set our fees on an annual basis and the 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
14 hours
Typical time in lectures, seminars and similar per week during term time
45%
Average assessment by coursework
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