Mathematics with Languages BSc (Hons)
 Course length: 3/4 years
 UCAS code: G19R
 Year of entry: 2018
 Typical offer: Alevel : ABB / IB : 33 / BTEC : Applications considered
Honours Select
×This programme offers Honours Select combinations.
Honours Select 100
×This programme is available through Honours Select as a Single Honours (100%).
Honours Select 75
×This programme is available through Honours Select as a Major (75%).
Honours Select 50
×This programme is available through Honours Select as a Joint Honours (50%).
Honours Select 25
×This programme is available through Honours Select as a Minor (25%).
Study abroad
×This programme offers study abroad opportunities.
Year in China
×This programme offers the opportunity to spend a Year in China.
Accredited
×This programme is accredited.
Module details
Year One Compulsory Modules
Calculus I (MATH101)
Level 1 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims 1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.
2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.
3. To introduce the notions of sequences and series and of their convergence.
Learning Outcomes differentiate and integrate a wide range of functions;
sketch graphs and solve problems involving optimisation and mensuration
understand the notions of sequence and series and apply a range of tests to determine if a series is convergent
Calculus II (MATH102)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims · To discuss local behaviour of functions using Taylor’s theorem.
· To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals.
Learning Outcomes use Taylor series to obtain local approximations to functions;
obtain partial derivaties and use them in several applications such as, error analysis, stationary points change of variables
evaluate double integrals using Cartesian and Polar Coordinates
Introduction to Linear Algebra (MATH103)
Level 1 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims  To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.
 To develop geometrical intuition in 2 and 3 dimensions.
 To introduce students to the concept of subspace in a concrete situation.
 To provide a foundation for the study of linear problems both within mathematics and in other subjects.
Learning Outcomes manipulate complex numbers and solve simple equations involving them
solve arbitrary systems of linear equations
understand and use matrix arithmetic, including the computation of matrix inverses
compute and use determinants
understand and use vector methods in the geometry of 2 and 3 dimensions
calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics
Numbers, Groups and Codes (MATH142)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims · To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.
· To give an appreciation of the utility and power of group theory as the study of symmetries.
· To introduce publickey cryptosystems as used in the transmission of confidential data, and also errorcorrecting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.
Learning Outcomes Use the division algorithm to construct the greatest common divisor of a pair of positive integers;
Solve linear congruences and find the inverse of an integer modulo a given integer;
Code and decode messages using the publickey method
Manipulate permutations with confidence
Decide when a given set is a group under a specified operation and give formal axiomatic proofs;
Understand the concept of a subgroup and use Lagrange''s theorem;
Understand the concept of a group action, an orbit and a stabiliser subgroup
Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;
Newtonian Mechanics (MATH122)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.
Learning Outcomes:
After completing the module students should be able to analyse real world problems
involving:
 the motions of bodies under simple force systems
 conservation laws for momentum and energy
 rigid body dynamics using centre of mass,
angular momentum and moments of inertiaLearning Outcomes
After completing the module students should be able to analyse
realworld problems involving:the motions of bodies under simple force systems
conservation laws for momentum and energy
rigid body dynamics using centre of mass, angular momentum and moments
oscillation, vibration, resonance
Introduction to Statistics (MATH162)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims To introduce topics in Statistics and to describe and discuss basic statistical methods.
To describe the scope of the application of these methods.
Learning Outcomes to describe statistical data;
to use the Binomial, Poisson, Exponential and Normal distributions;
to perform simple goodnessoffit tests
to use the package Minitab to present data, and to make statistical analysis
Year One Optional Modules
Mathematical It Skills (MATH111)
Level 1 Credit level 15 Semester First Semester Exam:Coursework weighting 0:100 Aims To acquire key mathematicsspecific computer skills.
To reinforce mathematics as a practical discipline by active experience and experimentation, using the computer as a tool.
To illustrate and amplify mathematical concepts and techniques.
To initiate and develop modelling skills.
To initiate and develop problem solving, group work and report writing skills.

To develop employability skills
Learning Outcomes After completing the module, students should be able to
 tackle project work, including writing up of reports detailing their solutions to problems;
 use computers to create documents containing formulae, tables, plots and references;
 use MAPLE to manipulate mathematical expressions and to solve simple problems,
 better understand the mathematical topics covered, through direct experimentation with the computer.
After completing the module, students should be able to
 list skills required by recruiters of graduates in mathematical sciences;
 recognise what constitutes evidence for those skills;
 identify their own skills gaps and plan to develop their skills.
Introduction to Programming (COMP101)
Level 1 Credit level 15 Semester First Semester Exam:Coursework weighting 0:100 Aims  To introduce the concepts and principles of problem solving using computational thinking.
 To identify and employ algorithms in the solution of identified problems.
 To develop sound principles in designing programming solutions to identified problems using appropriate data structures.
 To introduce the concepts of implementing solutions in a high level programming language.
Learning Outcomes Identify the principles and practice of using highlevel programming constructs to solve a problem
Use relevant data structures to solve problems
Produce documentation in support of a programmed solution
Use a suitable Integrated Development Environment to carry out Implementation, interpretation/compilation, testing and execution.
Identify appropriate design approaches to formulate a solution to a program
Design and apply effective test cases
Develop debugging skills to correct a program
Specific learning outcomes are listed above.
General learning outcomes:
 An understanding of the principles and practice of analysis and design in the construction of robust, maintainable programs which satisfy their requirements;
 A competence to design, write, compile, test and execute straightforward programs using a highlevel language;
 An appreciate of the principles of procedural programming;
 An awareness of the need for a professional approach to design and the importance of good documentation to the finished programs.
Newtonian Mechanics (MATH122)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.
Learning Outcomes:
After completing the module students should be able to analyse real world problems
involving:
 the motions of bodies under simple force systems
 conservation laws for momentum and energy
 rigid body dynamics using centre of mass,
angular momentum and moments of inertiaLearning Outcomes
After completing the module students should be able to analyse
realworld problems involving:the motions of bodies under simple force systems
conservation laws for momentum and energy
rigid body dynamics using centre of mass, angular momentum and moments
oscillation, vibration, resonance
Numbers, Groups and Codes (MATH142)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims · To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.
· To give an appreciation of the utility and power of group theory as the study of symmetries.
· To introduce publickey cryptosystems as used in the transmission of confidential data, and also errorcorrecting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.
Learning Outcomes Use the division algorithm to construct the greatest common divisor of a pair of positive integers;
Solve linear congruences and find the inverse of an integer modulo a given integer;
Code and decode messages using the publickey method
Manipulate permutations with confidence
Decide when a given set is a group under a specified operation and give formal axiomatic proofs;
Understand the concept of a subgroup and use Lagrange''s theorem;
Understand the concept of a group action, an orbit and a stabiliser subgroup
Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;
Introduction to Statistics (MATH162)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims To introduce topics in Statistics and to describe and discuss basic statistical methods.
To describe the scope of the application of these methods.
Learning Outcomes to describe statistical data;
to use the Binomial, Poisson, Exponential and Normal distributions;
to perform simple goodnessoffit tests
to use the package Minitab to present data, and to make statistical analysis
Programme Year Two
Choose at least one module from:
 Metric Spaces and Calculus (MATH241)
 Linear Algebra and Geometry (MATH244)
 Numbers, Groups and Codes (MATH142)
 Commutative Algebra (MATH247)
 Geometry of Curves (MATH248)
Choose at least 2 Modules, including MATH201 or MATH224, or both from:
 Ordinary Differential Equations (MATH201)
 Introduction to the Methods of Applied Mathematics (MATH224)
 Ordinary Differential Equations (MATH201)
 Vector Calculus With Applications in Fluid Mechanics (MATH225)
 Mathematical Models: Microeconomics and Population Dynamics (MATH227)
 Introduction to the Methods of Applied Mathematics (MATH224)
 Group Project Module (MATH206)
 Classical Mechanics (MATH228)
 Numerical Methods (MATH266)
 Newtonian Mechanics (MATH122)
Choose 8 modules from:
 Introduction to Programming (COMP101)
 Foundations of Computer Science (COMP109)
 Software Engineering I (COMP201)
 Database Development (COMP207)
 Internet Principles (COMP211)
 Advanced Objectoriented Programming (COMP213)
 Introduction to Methods of Operational Research (MATH261)
 Operational Research: Probabilistic Models (MATH268)
 Humancentric Computing (COMP106)
 Data Structures and Algorithms (COMP108)
 Complexity of Algorithms (COMP202)
 Group Software Project (COMP208)
 Distributed Systems (COMP212)
 Introduction to Statistics (MATH162)
 Financial Mathematics II (MATH262)
 Statistical Theory and Methods I (MATH263)
 Statistical Theory and Methods II (MATH264)
plus
30 Credits of Language Modules (French, German or Spanish) throughout the year
Year Two Compulsory Modules
Complex Functions (MATH243)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims To introduce the student to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.
Learning Outcomes After completing this module students should:
 appreciate the central role of complex numbers in mathematics;
 be familiar with all the classical holomorphic functions;
 be able to compute Taylor and Laurent series of such functions;
 understand the content and relevance of the various Cauchy formulae and theorems;
 be familiar with the reduction of real definite integrals to contour integrals;
 be competent at computing contour integrals.
Year Two Optional Modules
Metric Spaces and Calculus (MATH241)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims To introduce the basic elements of the theories of metric spaces and calculus of several variables.
Learning Outcomes After completing the module students should:
Be familiar with a range of examples of metric spaces.
Have developed their understanding of the notions of convergence and continuity.
Understand the contraction mapping theorem and appreciate some of its applications.
Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.
Understand the inverse function and implicit function theorems and appreciate their importance.
Have developed their appreciation of the role of proof and rigour in mathematics.
Linear Algebra and Geometry (MATH244)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics.
Learning Outcomes After completing the module students should be able to:
• appreciate the geometric meaning of linear algebraic ideas,
• appreciate the concept of an abstract vector space and how it is used in different mathematical
situations,
• apply a change of coordinates to simplify a linear map,
• manipulate matrix groups (in particular Gln, On and Son),
• understand bilinear forms from a geometric point of view.
Numbers, Groups and Codes (MATH142)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims · To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.
· To give an appreciation of the utility and power of group theory as the study of symmetries.
· To introduce publickey cryptosystems as used in the transmission of confidential data, and also errorcorrecting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.
Learning Outcomes Use the division algorithm to construct the greatest common divisor of a pair of positive integers;
Solve linear congruences and find the inverse of an integer modulo a given integer;
Code and decode messages using the publickey method
Manipulate permutations with confidence
Decide when a given set is a group under a specified operation and give formal axiomatic proofs;
Understand the concept of a subgroup and use Lagrange''s theorem;
Understand the concept of a group action, an orbit and a stabiliser subgroup
Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;
Commutative Algebra (MATH247)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory.
Learning Outcomes After completing the module students should be able to:
• Work confidently with the basic tools of algebra (sets, maps, binary operations and equivalence relations).
• Recognise abelian groups, different kinds of rings (integral, Euclidean, principal ideal and unique factorisation domains) and fields.
• Find greatest common divisors using the Euclidean algorithm in Euclidean domains.
• Apply commutative algebra to solve simple numbertheoretic problems.
Geometry of Curves (MATH248)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To introduce geometric ideas and develop the basic skills in handling them.
To study the line, circle, ellipse, hyperbola, parabola, cubics and many other curves.
To study theoretical aspects of parametric, algebraic and projective curves.
To study and sketch curves using an appropriate computer package.
Learning Outcomes After completing this module students should be able to:
 use a computer package to study curves and their evolution in both parametric and algebraic forms.
determine and work with tangents, inflexions, curvature, cusps, nodes, length and other features.
calculate envelopes and evolutes.
 solve the position and shape of some algebraic curves including conics.
The first learning outcome is assessed by coursework, the others by both coursework and examination.
Ordinary Differential Equations (MATH201)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.
To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.
Learning Outcomes After completing the module students should be:
 familiar with elementary techniques for the solution of ODE''s, and the idea of reducing a complex ODE to a simpler one;
 familiar with basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions;
 well versed in the solution of linear ODE systems (homogeneous and nonhomogeneous) with constant coefficients matrix;
 aware of a range of applications of ODE.
Introduction to the Methods of Applied Mathematics (MATH224)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics.
To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics.
Learning Outcomes After completing the module students should:
 be fluent in the solution of basic ordinary differential equations, including systems of first order equations;
 be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations;
 be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations;
 be able to solve simple first order partial differential equations;
 be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.
Ordinary Differential Equations (MATH201)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.
To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.
Learning Outcomes After completing the module students should be:
 familiar with elementary techniques for the solution of ODE''s, and the idea of reducing a complex ODE to a simpler one;
 familiar with basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions;
 well versed in the solution of linear ODE systems (homogeneous and nonhomogeneous) with constant coefficients matrix;
 aware of a range of applications of ODE.
Vector Calculus With Applications in Fluid Mechanics (MATH225)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 85:15 Aims To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them.
To give an appreciation of the many applications of vector calculus to physical situations.
To provide an introduction to the subjects of fluid mechanics and electromagnetism.
Learning Outcomes After completing the module students should be able to:
 Work confidently with different coordinate systems.
 Evaluate line, surface and volume integrals.
 Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes.
 Recognise the many physical situations that involve the use of vector calculus.
 Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow.
All learning outcomes are assessed by both examination and course work.
Mathematical Models: Microeconomics and Population Dynamics (MATH227)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims 1. To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models.
2. To do this in the context of two nonphysical applications, namely microeconomics and population dynamics.
3. To use and develop mathematical skills introduced in Year 1  particularly the calculus of functions of several variables and elementary differential equations.
Learning Outcomes After completing the module students should be able to:
 Use techniques from several variable calculus in tackling problems in microeconomics.
 Use techniques from elementary differential equations in tackling problems in population dynamics.
 Apply mathematical modelling methodology in these subject areas.
All learning outcomes are assessed by both examination and course work.
Introduction to the Methods of Applied Mathematics (MATH224)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics.
To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics.
Learning Outcomes After completing the module students should:
 be fluent in the solution of basic ordinary differential equations, including systems of first order equations;
 be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations;
 be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations;
 be able to solve simple first order partial differential equations;
 be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.
Group Project Module (MATH206)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 0:100 Aims · To give students experience of working effectively in small groups.
· To train students to write about mathematics.
· To give students practice in delivering presentations.
· To develop students’ ability to study independently.
· To prepare students for later individual project work.
· To enhance students’ appreciation of the connections between different areas of mathematics.
· To encourage students to discuss mathematics with each other.
Learning Outcomes Work effectively in groups, and delegate common tasks.
Write substantial mathematical documents in an accessible form.Give coherent verbal presentations of more advanced mathematical topics.
Appreciate how mathematical techniques can be applied in a variety of different contextsClassical Mechanics (MATH228)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.
Learning Outcomes the motions of bodies under simple force systems, including calculations of the orbits of satellites, comets and planetary motions
motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravitry over the Earth''s surface
rigid body dynamics using centre of mass, angular momentum and moments of inertia
Numerical Methods (MATH266)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To provide an introduction to the main topics in Numerical Analysis and their relation to other branches of Mathematics
Learning Outcomes After completing the module students should be able to:
• write simple mathematical computer programs in Maple,
• understand the consequences of using fixedprecision arithmetic,
• analyse the efficiency and convergence rate of simple numerical methods,
• develop and implement algorithms for solving nonlinear equations,
• develop quadrature methods for numerical integration,
• apply numerical methods to solve systems of linear equations and to calculate eigenvalues and eigenvectors,
• solve boundary and initial value problems using finite difference methods.
Newtonian Mechanics (MATH122)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems.
Learning Outcomes:
After completing the module students should be able to analyse real world problems
involving:
 the motions of bodies under simple force systems
 conservation laws for momentum and energy
 rigid body dynamics using centre of mass,
angular momentum and moments of inertiaLearning Outcomes
After completing the module students should be able to analyse
realworld problems involving:the motions of bodies under simple force systems
conservation laws for momentum and energy
rigid body dynamics using centre of mass, angular momentum and moments
oscillation, vibration, resonance
Introduction to Programming (COMP101)
Level 1 Credit level 15 Semester First Semester Exam:Coursework weighting 0:100 Aims  To introduce the concepts and principles of problem solving using computational thinking.
 To identify and employ algorithms in the solution of identified problems.
 To develop sound principles in designing programming solutions to identified problems using appropriate data structures.
 To introduce the concepts of implementing solutions in a high level programming language.
Learning Outcomes Identify the principles and practice of using highlevel programming constructs to solve a problem
Use relevant data structures to solve problems
Produce documentation in support of a programmed solution
Use a suitable Integrated Development Environment to carry out Implementation, interpretation/compilation, testing and execution.
Identify appropriate design approaches to formulate a solution to a program
Design and apply effective test cases
Develop debugging skills to correct a program
Specific learning outcomes are listed above.
General learning outcomes:
 An understanding of the principles and practice of analysis and design in the construction of robust, maintainable programs which satisfy their requirements;
 A competence to design, write, compile, test and execute straightforward programs using a highlevel language;
 An appreciate of the principles of procedural programming;
 An awareness of the need for a professional approach to design and the importance of good documentation to the finished programs.
Foundations of Computer Science (COMP109)
Level 1 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims  To introduce the notation, terminology, and techniques underpinning the discipline of Theoretical Computer Science.
 To provide the mathematical foundation necessary for understanding datatypes as they arise in Computer Science and for understanding computation.
 To introduce the basic proof techniques which are used for reasoning about data and computation.
 To introduce the basic mathematical tools needed for specifying requirements and programs
Learning Outcomes Understand how a computer represents simple numeric data types; reason about simple data types using basic proof techniques; Interpret set theory notation, perform operations on sets, and reason about sets;
Understand, manipulate and reason about unary relations, binary relations, and functions;
Apply logic to represent mathematical statement and digital circuit, and to recognise, understand, and reason about formulas in propositional and predicate logic;
Apply basic counting and enumeration methods as these arise in analysing permutations and combinations.
Software Engineering I (COMP201)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims The module is intended to develop an understanding of the problems associated with the development of significant computing systems (that is, systems that are too large to be designed and developed by a single person,and are designed to be used by many users) and to appreciate the techniques and tools necessary to develop such systems efficiently, in a costeffective manner.
Learning Outcomes At the end of the module, the student is expected to
 realise the problems in designing and building significant computer systems;
 understand the need to design systems that fully meet the requirements of the intended users including functional and non functional elements
 appreciate the need to ensure that the implementation of a design is adequately tested to ensure that the completed system meets the specifications
 be fully aware of the principles and practice of an OO approach to the design and development of computer systems
 be able to apply these principles in practice.
Database Development (COMP207)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims  To introduce students to the problems arising from concurrency in databases, information security considerations and how they are solved
 To introduce students to the problems arising from the integration of heterogeneous sources of information and the use of semistructured data;
 To introduce students to nonrelational databases and the economic factors involved in their selection
 To introduce students to techniques for analyzing large amounts of data, the security issues and commercial factors involved with them
Learning Outcomes At the end of this module the student will be able to identify and apply the principles underpinning transaction management within DBMS and the main security issues involved in securing transaction;
Demonstrate an understanding of advanced SQL topics;
Illustrate the issues related to Web technologies as a semistructured data representation formalism;
Identify the principles underlying object relational models and the economic factors in their uptake and development;
Interpret the main concepts and security aspects in data warehousing, and the concepts of data mining and commercial considerations involved in adopting the paradigm.
Internet Principles (COMP211)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 80:20 Aims  To introduce networked computer systems in general, and the Internet in particular.
 To introduce the basic principles that govern their operation.
 To introduce the design and organisation principles of successful computer networks.
 To introduce the key protocols and technologies that are used in the Internet.
Learning Outcomes By the end of this module, students should understand the basic theoretical principles of computer communications networks (eg the notion of band width, Shannon’s law etc);
Understand how the notion of layering and abstraction apply to the design of computer communication networks;
Understand the organisation of the Internet, and how this organisation relates to the OSI seven layer model;
Understand the structure and function of the OSI seven layer model of computer networks;
Understand the principles of the key protocols that govern the Internet.
Advanced Objectoriented Programming (COMP213)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 50:50 Aims To introduce data structures and advanced programming language features within the context of a highlevel programming language (Java). To demonstrate principles, provide indicative examples, develop problemsolving abilities and provide students with experience and confidence in the use of advanced features to implement algorithms in a contemporary software setting.
Learning Outcomes By the end of this module, students should be familiar with data structures and advanced programming concepts within Java; should be able to carry out the construction of software artefacts utilising these concepts; and should be capable of carrying out the development of complex elements, such as user interfaces, multiprocessing, and faulttolerant components.
Introduction to Methods of Operational Research (MATH261)
Level 2 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims  Appreciate the operational research approach.
 Be able to apply standard methods to a wide range of realworld problems as well as applications in other areas of mathematics.
 Appreciate the advantages and disadvantages of particular methods.
 Be able to derive methods and modify them to model realworld problems.
 Understand and be able to derive and apply the methods of sensitivity analysis.
Learning Outcomes Appreciate the operational research approach.Be able to apply standard methods to a wide range of realworld problems as well asapplications in other areas of mathematics.
Appreciate the advantages and disadvantages of particular methods.
Be able to derive methods and modify them to model realworld problems.Understand and be able to derive and apply the methods of sensitivity analysis. Appreciate the importance of sensitivity analysis.
Operational Research: Probabilistic Models (MATH268)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance.
Learning Outcomes The ability to understand and describe mathematically reallife optimization problems.
Understanding the basic methods of dynamical decision making.
Understanding the basics of forecasting and simulation.
The ability to analyse elementary queueing systems.
Humancentric Computing (COMP106)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims  To provide guidelines, concepts and models for designing and evaluating interactive systems.
To provide an introduction to designing and implementing graphical user interfaces.Learning Outcomes identify or describe the tasks and issues, such as establishing requirements, developing designs, and implementing designs, which are involved in the process of developing efficient, effective, and safe interactive products for people, and the tools and techniques used to perform these tasks write Java programs that demonstrate simple examples of graphical user interfaces
 To provide guidelines, concepts and models for designing and evaluating interactive systems.
Data Structures and Algorithms (COMP108)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 60:40 Aims 1. To introduce the notation, terminology, and techniques underpinning the study of algorithms.
2. To introduce basic data structures and associated algorithms.
3. To introduce standard algorithmic design paradigms and efficient use of data structures employed in the development of efficient algorithmic solutions.Learning Outcomes Be able to describe the principles of and apply a variety of data structures and their associated algorithms; Be able to describe standard algorithms, apply a given pseudo code algorithm in order to solve a given problem, and carry out simple asymptotic analyses of algorithms;
Be able to describe and apply different algorithm design principles and distinguish the differences between these principles;
Be able to choose and justify the use of appropriate data structures to enable efficient implementation of algorithms;
Complexity of Algorithms (COMP202)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims  To demonstrate how the study of algorithmics has been applied in a number of different domains.
 To introduce formal concepts of measures of complexity and algorithms analysis.
 To introduce fundamental methods in data structures and algorithms design.
 To make students aware of computationally hard problems and possible ways of coping with them.
Learning Outcomes At the conclusion of the module students should have an appreciation of the diversity of computational fields to which algorithmics has made significant contributions.
At the conclusion of the module students should have fluency in using basic data structures (queues, stacks, trees, graphs, etc) in conjunction with classical algorithmic problems (searching, sorting, graph algorithms, security issues) and be aware of basic number theory applications, etc.
At the conclusion of the module students should be familiar with formal theories providing evidence that many important computational problems are inherently intractable, e.g., NPcompleteness.
Group Software Project (COMP208)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 0:100 Aims Students will work in small groups to produce a working software system.
The deliverables and working methods will be prescribed. The aims of the module are:
1. to provide experience of group working;
2. to provide experience of all aspects of the development of a moderately sized software system;
3. to prepare students for their individual projects in the third year;
4. to consolidate material from the first semester of the second year, in particular COMP201 and COMP207.Learning Outcomes Show an awareness of the issues involved in working as part of a team.
Demonstrate improved personal, interpersonal and communication skills.
Demonstrate a more in depth understanding of the software development process.
Specify the requirements of a software system.
Demonstrate some experience in the design of a software system.
Demonstrate practical experience in the implementation and testing of a moderately sized software system.Show an awareness of the typical project management issues.
Understand the process and role of software documentation.Demonstrate some experience in the writing of a sizeable report on a software project.
Distributed Systems (COMP212)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims This module is intended to provide an understanding of the technical issues involved in the design of modern distributed systems. Besides conveying the central principles involved in designing distributed systems, this module also aims to present some of the major current paradigms (see learning outcomes below).
Learning Outcomes An appreciation of the main principles underlying distributed systems: processes, communication, naming, synchronisation, consistency, fault tolerance, and security.
Familiarity with some of the main paradigms in distributed systems: objectbased systems, file systems, and coordinationbased systems.
Knowledge and understanding of the essential facts, concepts, principles and theories relating to Computer Science in general, and Distributed Computing in particular.
A sound knowledge of the criteria and mechanisms whereby traditional and distributed systems can be critically evaluated and analysed to determine the extent to which they meet the criteria defined for their current and future development.
An in depth understanding of the appropriate theory, practices, languages and tools that may be deployed for the specification, design, implementation and evaluation of both traditional and Internet related distributed computer systems.
Introduction to Statistics (MATH162)
Level 1 Credit level 15 Semester Second Semester Exam:Coursework weighting 80:20 Aims To introduce topics in Statistics and to describe and discuss basic statistical methods.
To describe the scope of the application of these methods.
Learning Outcomes to describe statistical data;
to use the Binomial, Poisson, Exponential and Normal distributions;
to perform simple goodnessoffit tests
to use the package Minitab to present data, and to make statistical analysis
Financial Mathematics II (MATH262)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims to provide an understanding of the basic financial mathematics theory used in the study process of actuarial/financial interest,
to provide an introduction to financial methods and derivative pricing financial instruments ,
to understand some financial models with applications to financial/insurance industry,
to prepare the students adequately and to develop their skills in order to be ready to sit for the CT1 & CT8 subject of the Institute of Actuaries (covers 20% of CT1 and material of CT8).
Learning Outcomes After completing the module students should:
(a) Understand the assumptions of CAMP, explain the no riskless lending or borrowing and other lending and borrowing assumptions, be able to use the formulas of CAMP, be able to derive the capital market line and security market line,
(b) Describe the Arbitrage Theory Model (APT) and explain its assumptions, perform estimating and testing in APT,
(c) Be able to explain the terms long/short position, spot/delivery/forward price, understand the use of future contracts, describe what a call/put option (European/American) is and be able to makes graphs and explain their payouts, describe the hedging for reducing the exposure to risk, be able to explain arbitrage, understand the mechanism of short sales,
(d) Explain/describe what arbitrage is, and also the risk neutral probability measure, explain/describe and be able to use (perform calculation) the binomial tree for European and American style options,
(e) Understand the probabilistic interpretation and the basic concept of the random walk of asset pricing,
(f) Understand the concepts of replication, hedging, and delta hedging in continuous time,
(g) Be able to use Ito''s formula, derive/use the the Black‐Scholes formula, price contingent claims (in particular European/American style options and forward contracts), be able to explain the properties of the Black‐Scholes formula, be able to use the Normal distribution function in numerical examples of pricing,
(h) Understand the role of Greeks , describe intuitively what Delta, Theta, Gamma is, and be able to calculate them in numerical examples.
Statistical Theory and Methods I (MATH263)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 85:15 Aims To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation. The emphasis is on applications.
Learning Outcomes After completing the module students should have a conceptual and practical understanding of a range of commonly applied statistical procedures. They should have also developed some familiarity with the statistical package MINITAB.
Statistical Theory and Methods II (MATH264)
Level 2 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory.
Learning Outcomes After completing the module students should understand basic probability calculus. They should be familiar with a range of techniques for solving real life problems of the probabilistic nature.
Programme Year Three
Year abroad
Year Three Optional Modules
Year Abroad Erasmus Semester 1 (MODL215)
Level 2 Credit level 60 Semester First Semester Exam:Coursework weighting 0:100 Aims By allowing students to study a semester abroad at university level, this module aims to provide them with the opportunity to significantly improve their skills in the target language.
To provide students with the opportunity to follow content modules at university level in the target language that enhance their knowledge in areas of study relevant to their degree programme.
To provide students with the opportunity to widen and consolidate their intercultural competence.
Learning Outcomes When finishing MODL215, students will have successfully fulfilled the learning outcomes of the language and the content modules they have completed at their host university.
By successfully adapting to the system of study at their host university, students will have demonstrated that they have acquired the academic skills and competence to study abroad.
By successfully adapting to everyday life outside the UK, students will have demonstrated that they have acquired the intercultural competence to live abroad.
Year Abroad Erasmus Semester 2 (MODL216)
Level 2 Credit level 60 Semester Second Semester Exam:Coursework weighting 0:100 Aims  By allowing students to study a semester abroad at university level, this module aims to provide them with the opportunity to significantly improve their skills in the target language. .
 To provide students with the opportunity to follow content modules at university level in the target language that enhance their knowledge in areas of study relevant to their degree programme.
 To provide students with the opportunity to widen and consolidate their intercultural competence.
Learning Outcomes When finishing MODL216, students will have successfully fulfilled the learning outcomes of the language and the content modules they have completed at their host university.
By successfully adapting to the system of study at their host university, students will have demonstrated that they have acquired the academic skills and competence to study abroad.
By successfully adapting to everyday life outside the UK, students will have demonstrated that they have acquired the intercultural competence to live abroad.
All Year Placements Essay (MODL219)
Level 2 Credit level 120 Semester Whole Session Exam:Coursework weighting 0:100 Aims  The aim of the YAPE is to give students the opportunity to carry out independent academic research into a cultural, literary, linguistic or area studies topic of interest to them.
Learning Outcomes In completing their YAPE, students will have demonstrated the ability to utilise a variety of bibliographical tools to locate a range of primary and secondary sources on which to base a research project.
Construct, focus and structure an independent project, in discussion with a personal supervisor working in that subject area.
Produce a written text, following scholarly conventions of referencing and bibliography.
Analyse source materials, and develop coherent and original arguments on the basis of their research.
Engage critically with relevant cultural, literary, linguistic, area studies and / or theoretical debates on the topic.
Organise the material gathered in the course of their research.
Manage their time effectively and efficiently and plan a process of research, reading and writing over a period of about 7 months.
Conduct independent research.
Present a coherent argument in clear written prose.
All Year Placements Portfolio (MODL224)
Level 2 Credit level 120 Semester Whole Session Exam:Coursework weighting 0:100 Aims  The aim of the Portfolio is to give students, in the course of the year abroad, the opportunity to reflect in 3 different 1,000word Milestones on their personal development as well as on linguistic or cultural or political issues they come across at their placement
Learning Outcomes To explain the topics/issues selected for and examined in the Milestones.
To explain the relevance of the materials used.
To explain the analytical methods employed.
To further expand on one of the linguistic, cultural or political topics/issues examined in the Milestones.
To provide a summary of the results achieved in the Milestones.
To provide selfcritical reflections on the Milestones produced.
To provide a summary of his or her personal development during the semester abroad.
To explain in which particular ways feedback received from their supervisor on the Milestones has helped him or her in writing the Final Report.
Programme Year Four
Choose six optional modules.
Year Four Optional Modules
Further Methods of Applied Mathematics (MATH323)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims To give an insight into some specific methods for solving important types of ordinary differential equations.
To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.
To build on the students'' existing knowledge of partial differential equations of first and second order.
Learning Outcomes After completing the module students should be able to:
 use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.
 solve simple integral extremal problems including cases with constraints;
 classify a system of simultaneous 1storder linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;
 classify 2ndorder linear partial differential equations and, in appropriate cases, find general or specific solutions. [This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]
Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.
Learning Outcomes After completing the module, students should be able to understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, and apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.
Quantum Mechanics (MATH325)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time. The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.
Learning Outcomes After completing the module students should be able to solve Schrodinger''s equation for simple systems, and have some intuitive understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.
Population Dynamics (MATH332)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims  To provide a theoretical basis for the understanding of population ecology
 To explore the classical models of population dynamics
 To learn basic techniques of qualitative analysis of mathematical models
Learning Outcomes The ability to relate the predictions of the mathematical models to experimental results obtained in the field.
The ability to recognise the limitations of mathematical modelling in understanding the mechanics of complex biological systems.
The ability to use analytical and graphical methods to investigate population growth and the stability of equilibrium states for continuoustime and discretetime models of ecological systems.
Group Theory (MATH343)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.
Learning Outcomes Understanding of abstract algebraic systems (groups) by concrete, explicit realisations (permutations, matrices, Mobius transformations).
The ability to understand and explain classification results to users of group theory.
The understanding of connections of the subject with other areas of Mathematics.
To have a general understanding of the origins and history of the subject.
Combinatorics (MATH344)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims To provide an introduction to the problems and methods of Combinatorics, particularly to those areas of the subject with the widest applications such as pairings problems, the inclusionexclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions.
Learning Outcomes After completing the module students should be able to:
 understand of the type of problem to which the methods of Combinatorics apply, and model these problems;
 solve counting and arrangement problems;
 solve general recurrence relations using the generating function method;
 appreciate the elementary theory of partitions and its application to the study of symmetric functions.
Applied Probability (MATH362)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for ‘‘dynamic" events occuring over time. To familiarise students with an important area of probability modelling.
Learning Outcomes 1. Knowledge and Understanding
After the module, students should have a basic understanding of:
(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes
(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain
(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.
2. Intellectual Abilities
After the module, students should be able to:
(a) formulate appropriate situations as probability models: random processes
(b) demonstrate knowledge of standard models
(c) demonstrate understanding of the theory underpinning simple dynamical systems
3. General Transferable Skills
(a) numeracy through manipulation and interpretation of datasets
(b) communication through presentation of written work and preparation of diagrams
(c) problem solving through tasks set in tutorials
(d) time management in the completion of practicals and the submission of assessed work
(e) choosing, applying and interpreting results of probability techniques for a range of different problems.
Linear Statistical Models (MATH363)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims · to understand how regression methods for continuous data extend to include multiple continuous and categorical predictors, and categorical response variables.
· to provide an understanding of how this class of models forms the basis for the analysis of experimental and also observational studies.
· to understand generalized linear models.
· to develop familiarity with the computer package SPSS.
Learning Outcomes After completing the module students should be able to:
understand the rationale and assumptions of linear regression and analysis of variance.
· understand the rationale and assumptions of generalized linear models.
· recognise the correct analysis for a given experiment.
· carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.
· carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.
· perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.
Measure Theory and Probability (MATH365)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 90:10 Aims The main aim is to provide a sufficiently deepintroduction to measure theory and to the Lebesgue theory of integration. Inparticular, this module aims to provide a solid background for the modernprobability theory, which is essential for Financial Mathematics.
Learning Outcomes After completing the module students should be ableto:
master the basic results about measures and measurable functions;
master the basic results about Lebesgue integrals and their properties;to understand deeply the rigorous foundations ofprobability theory;
to know certain applications of measure theoryto probability, random processes, and financial mathematics.
Networks in Theory and Practice (MATH367)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims To develop an appreciation of network models for real world problems.
To describe optimisation methods to solve them.
To study a range of classical problems and techniques related to network models.
Learning Outcomes After completing the module students should
. be able to model problems in terms of networks.
· be able to apply effectively a range of exact and heuristic optimisation techniques.
Chaos and Dynamical Systems (MATH322)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims To develop expertise in dynamical systems in general and study particular systems in detail.
Learning Outcomes After completing the module students should be able to:
understand the possible behaviour of dynamical systems with particular attention to chaotic motion;
be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points;
understand how fractal sets arise and how to characterise them.
Relativity (MATH326)
Level 3 Credit level 15 Semester First Semester Exam:Coursework weighting 100:0 Aims To impart
(i) a firm grasp of the physical principles behind Special and General Relativity and their main consequences;
(ii) technical competence in the mathematical framework of the subjects  Lorentz transformation, coordinate transformations and geodesics in Riemann space;
(iii) knowledge of some of the classical tests of General Relativity  perihelion shift, gravitational deflection of light;
(iv) basic concepts of black holes and (if time) relativistic cosmology.
Learning Outcomes After completing this module students should
(i) understand why spacetime forms a nonEuclidean fourdimensional manifold;
(ii) be proficient at calculations involving Lorentz transformations, energymomentum conservation, and the Christoffel symbols.
(iii) understand the arguments leading to the Einstein''s field equations and how Newton''s law of gravity arises as a limiting case.
(iv) be able to calculate the trajectories of bodies in a Schwarzschild spacetime.
Mathematical Economics (MATH331)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims · To explore, from a gametheoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur.
· To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc..
· To treat fully a number of specific games including the famous examples of "The Prisoners'' Dilemma" and "The Battle of the Sexes".
· To treat in detail twoperson zerosum and nonzerosum games.
· To give a brief review of nperson games.
· In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved. To see how the Prisoner''s Dilemma arises in the context of public goods.
Learning Outcomes After completing the module students should:
· Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.
· Be able to formulate, in gametheoretic terms, situations of conflict and cooperation.
· Be able to solve mathematically a variety of standard problems in the theory of games.
· To understand the relevance of such solutions in real situations.
Riemann Surfaces (MATH340)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.
Learning Outcomes Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.
Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.
Number Theory (MATH342)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims To give an account of elementary number theory with use of certain algebraic methods and to apply the concepts to problem solving.
Learning Outcomes After completing this module students should be able to understand and solve a wide range of problems about the integers, and have a better understanding of the properties of prime numbers.
The Magic of Complex Numbers: Complex Dynamics, Chaos and the Mandelbrot Set (MATH345)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims 1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects;
2. To introduce students to some topics of current and recent research in the field;
3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting;
4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a realvariable context, and the implications of this in complex dynamics;
5. To explain how complexvariable methods have been instrumental in questions purely about realvalued onedimensional dynamical systems, such as the logistic family.
6. To deepen students'' appreciations for formal reasoning and proof.
After completing the module, students should be able to:
1. understand the compactification of the complex plane to the Riemann sphere, and use spherical distances and derivatives.2. use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.
3. state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.
4. determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.
5. apply advanced results from complex analysis in the setting of complex dynamics.
6. determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.Learning Outcomes will understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives
will be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems
will be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties
will be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set
will know how to apply advanced results from complex analysis in a dynamical setting
will be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not
Differential Geometry (MATH349)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 85:15 Aims This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3space. While forming a selfcontained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.
Learning Outcomes 1. Knowledge and understanding
After the module, students should have a basic understanding of
a) invariants used to describe the shape of explicitly given curves and surfaces,
b) special curves on surfaces,
c) the difference between extrinsically defined properties and those which depend only on the surface metric,
d) understanding the passage from local to global properties exemplified by the GaussBonnet Theorem.
2. Intellectual abilities
After the module, students should be able to
a) use differential calculus to discover geometric properties of explicitly given curves and surface,
b) understand the role played by special curves on surfaces.
3. Subjectbased practical skills
Students should learn to
a) compute invariants of curves and surfaces,
b) interpret the invariants of curves and surfaces as indicators of their geometrical properties.
4. General transferable skills
Students will improve their ability to
a) think logically about abstract concepts,
b) combine theory with examples in a meaningful way.
Applied Stochastic Models (MATH360)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for ''dynamic'' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base.
Learning Outcomes After completing the module students should have a grounding in the theory of continuoustime Markov chains and diffusion processes. They should be able to solve corresponding problems arising in epidemiology, mathematical biology, financial mathematics, etc.
Theory of Statistical Inference (MATH361)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 90:10 Aims To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.
Learning Outcomes After completing the module students should have a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.
The students should also gain an appreciation of the blossoming area of Bayesian approach to inference
Medical Statistics (MATH364)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims The aims of this module are to:
 demonstrate the purpose of medical statistics and the role it plays in the control of disease and promotion of health
 explore different epidemiological concepts and study designs
 apply statistical methods learnt in other programmes, and some new concepts, to medical problems and practical epidemiological research
 enable further study of the theory of medical statistics by using this module as a base.
Learning Outcomes identify the types of problems encountered in medical statistics
demonstrate the advantages and disadvantages of different epidemiological study designs
apply appropriate statistical methods to problems arising in epidemiology and interpret results
explain and apply statistical techniques used in survival analysis
critically evaluate statistical issues in the design and analysis of clinical trials
discuss statistical issues related to systematic review and apply appropriate methods of metaanalysisapply Bayesian methods to simple medical problems.
Mathematical Risk Theory (MATH366)
Level 3 Credit level 15 Semester Second Semester Exam:Coursework weighting 100:0 Aims to provide an understanding of the mathematical risk theory used in the study process of actuarial interest,
to provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities),
to develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry,
to prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).
Learning Outcomes After completing the module students should be able to:
(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria''s, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.
(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables,
(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.
(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships ,
(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples,
(f) Understand and be able to use Panjer''s equation when the number of claims belongs to the
R(a, b, 0) class of distributions, use the Panjer''s recursion in order to derive/evaluate the probability function for the total aggregate claims,(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables,
(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),
(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process,
(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg''s inequality for exponential and mixtures of exponential claim severities,
(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,
(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions),
(m) Derive Lundberg''s equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes,
(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data,
(o) Explain the difference and adjust the chain ladder method, when inflation is considered,
(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table),
(q) Use loss ratios to estimate the eventual loss and hence outstanding claims,
(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method), use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).
Projects in Mathematics (MATH399)
Level 3 Credit level 15 Semester Whole Session Exam:Coursework weighting 0:100 Aims a) To study in depth an area of pure mathematics and report on it; or
b) To construct and study mathematical models of a chosen problem; or
c) To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.; or
d) To study in depth a particular problem in statistics, probability or operational research.
Learning Outcomes a) (Pure Maths option)  After completing the report with suitable guidance the student should have
· gained a greater understanding of the chosen mathematical topic
· gained experience in applying his/her mathematical skills
· had experience in consulting relevant literature
· learned how to construct a written project report
· had experience in making an oral presentation
b) (Applied Mathematics)  After completing the project with suitable guidance the students should have:
 learned strategies for simple model building
 gained experience in choosing and using appropriate mathematics
 understood the nature of approximations used
 made critical appraisal of results
 had experience in consulting related relevant literature
 learned how to construct a written project report
 had experience in making an oral presentation.
c) (Applied Maths/Theoretical Physics)  After researching and preparing the mathematical essay the student should have:
· gained a greater understanding of the chosen mathematical topic
· gained an appreciation of the historical context
· learned how to abstract mathematical concepts and explain them
· had experience in consulting related relevant literature
· learned how to construct a written project report
· had experience in making an oral presentation.
d) (Statistics, Probability and Operational Research) 
After completing the project the student should have:
· gained an indepth understanding of the chosen topic
· had experience in consulting relevant literature
· learned how to construct a written project report;
· had experience in making an oral presentation.
e) Mathematics in Society Projects. Only available to G1X3 students
Students interested in doing such a project should see Dr A Pratoussevitch and Dr T Eckl initially.
The programme detail and modules listed are illustrative only and subject to change.
Teaching and Learning
Your learning activities will consist of lectures, tutorials, practical classes, problem classes, private study and supervised project work. In Year One, lectures are supplemented by a thorough system of group tutorials and computing work is carried out in supervised practical classes. Key study skills, presentation skills and group work start in firstyear tutorials and are developed later in the programme. The emphasis in most modules is on the development of problem solving skills, which are regarded very highly by employers. Project supervision is on a onetoone basis, apart from group projects in Year Two.
Assessment
Most modules are assessed by a two and a half hour examination in January or May, but many have an element of coursework assessment. This might be through homework, class tests, miniproject work or key skills exercises.