# Mathematics and Music Technology BSc (Hons)

- Course length: 3 years
- UCAS code: G1W3
- Year of entry: 2021
- A-level requirements: ABB

## 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**50:50 **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**(LO1) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

(LO2) Apply the methods of calculus and real analysis to solve previously unseen problems (of a similar style to those covered in the course).

(LO3) Understand in interpret proofs in the context of real analysis and apply the theorems developed in the course to straightforward examples.

(LO4) Independently construct proofs of previously unseen mathematical results in real analysis (of a similar style to those demonstrated in the course).

(LO5) Differentiate and integrate a wide range of functions;

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

(S1) Numeracy

##### 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**(LO1) Use Taylor series to obtain local approximations to functions

(LO2) Obtain partial derivatives and use them in several applications such as, error analysis, stationary points change of variables.

(LO3) Evaluate double integrals using Cartesian and Polar Co-ordinates.

(LO5)

##### Introduction to Linear Algebra (MATH103)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **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**(LO1) Manipulate complex numbers and solve simple equations involving them, solve arbitrary systems of linear equations.

(LO2) Understand and use matrix arithmetic, including the computation of matrix inverses.

(LO3) Compute and use determinants.

(LO4) Understand and use vector methods in the geometry of 2 and 3 dimensions.

(LO5) Calculate eigenvalues and eigenvectors.

(S1) Numeracy

##### 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 inertia

**Learning Outcomes**(LO1) the motions of bodies under simple force systems

(LO2) conservation laws for momentum and energy

(LO3) rigid body dynamics using centre of mass, angular momentum and moments

(LO4) oscillation, vibration, resonance

(LO5) oscillation, vibration, resonance

(S1) Representing physical problems in a mathematical way

(S2) Problem Solving Skills

##### 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 public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

**Learning Outcomes**(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.

(LO3) Be able to perform a range of calculations and manipulations with permutations.

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.

(LO7) Engage in group project work to investigate applications of the theoretical material covered in the module.

##### 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**(LO1) To know how to describe statistical data.

(LO2) To be able to use the Binomial, Poisson, Exponential and Normal distributions.

(LO3) To be able to perform simple goodness-of-fit tests.

(LO4) To be able to use an appropriate statistical software package to present data and to make statistical analysis.

(S1) Numeracy

(S2) Problem solving skills

(S3) IT skills

(S4) Communication skills

##### Introduction to Sound Recording and Production (MUSI108)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To introduce students to a variety of recording and production techniques in a professional studio.

To enable students to use a professional level digital recording studio effectively.

To develop the students’ individual and group work skills within the studio environment.

To develop students’ knowledge of recording and production technologies.**Learning Outcomes**(LO1) An ability to record a song or instrumental in Pro Tools HD using the appropriate recording and production techniques demonstrated in class.

(LO2) An ability to edit and mix a song in Pro Tools HD using appropriate effects and plugins to a standard suitable for public broadcast.

(LO3) An ability to explain, justify and reflect on recording and production techniques used in the recording studio.

(LO4) An ability to explain, justify and reflect on the mixing techniques and effects used in the recording studio.

(LO5) An ability to successfully work, cooperate and present within a group.

(S1) Working in groups and teams - Group action planning

(S2) Organisational skills

(S3) Communication (oral, written and visual) - Following instructions/protocols/procedures

##### Introduction to Logic (MUSI109)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**To introduce students to MIDI sequencing and to consolidate their technical knowledge and skills, bringing them to an intermediate level, through the use of Logic Pro software. To enable students to compose music to digital video clips. To enable students to compose electronic music in a given style. To introduce students to software synthesis parameters and signal processing using a digital audio workstation. To introduce students to optimal mixing and routing procedures in a DAW.

**Learning Outcomes**(LO1) Proficiency to an intermediate level in the use of Logic Pro software for composing and editing MIDI sequenced music.

(LO2) The ability to transfer a sequenced composition to an audio CD or MP3 file.

(LO3) The ability to successfully compose music for digital video and render it to a Quicktime movie file.

(LO4) For students to be able be able to analyse a piece of popular electronic music in terms of structure, texture, instrumentation and technical resources.

(LO5) The ability to correctly use the mixing facilities in Logic Studio.

(LO6) Gain an enhanced understanding of software synthesis parameters.

(LO7) To gain proficiency in the Apple Mac software environment.

(S1) Gain proficiency in the Apple Mac software environment.

##### Introduction to Sound and Technology (MUSI171)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To provide students with an overview and understanding of key terminologies, technical concepts and sound theories that are important for music technology studies.

To provide students with an understanding of acoustics and sound measurement techniques and units.

To provide students with an understanding of analogue and digital audio theories and applications.

To enable students to apply their knowledge of digital audio and MIDI in an appropriate music programme.

**Learning Outcomes**(LO1) To demonstrate knowledge and understanding of important sound and audio theories.

(LO2) To demonstrate knowledge and understanding sound and recording systems.

(LO3) To demonstrate knowledge and understanding of analogue and digital sound.

(LO4) To demonstrate knowledge and understanding of basic audio system interconnects and theories

(LO5) To demonstrate knowledge and understanding of sound system design, studio design and installations.

(S1) Problem solving skills.

(S2) Numeracy.

(S3) Communication skills.

(S4) Organisational skills.

(S5) IT skills.

##### The History of Electronic Music (MUSI172)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**To provide the student with a good understanding of the development of electronic music and the technology used to create it from the early 20th Century to the present day.

Introduce the student to the key composers and works associated with the electronic music repertoire.Provide the student with the ability to recognise the stylistic qualities and compositional techniques employed in a number of electronic music genres and how these techniques also cross the boundaries into other disciplines such as sound design for Film, TV and Gaming.

Develop the students' critical listening skills in relation to the electronic music repertoire.

Develop the students' ability to articulate themselves clearly in a written and aural manner.

**Learning Outcomes**(LO1) Demonstrate a knowledge of the history and development of electronic music and technology.

(LO2) Respond critically to a variety of musical styles from the electronic music repertoire referencing specific characteristics of the music and technology.

(LO3) The ability to conduct individual research into the history and development of electronic music using a variety of resources.

(S1) Communication, listening and questioning respecting others, contributing to discussions, communicating in a foreign language, influencing, presentations.

(S2) Research management developing a research strategy, project planning and delivery, risk management, formulating questions, selecting literature, using primary/secondary/diverse sources, collecting & using data, applying research methods, applying ethics.

(S3) Self-management readiness to accept responsibility (i.e. leadership), flexibility, resilience, self-starting, initiative, integrity, willingness to take risks, appropriate assertiveness, time management, readiness to improve own performance based on feedback/reflective learning.

(S4) Literacy application of literacy, ability to produce clear, structured written work and oral literacy - including listening and questioning.

(S5) Media literacy online critically reading and creatively producing academic and professional communications in a range of media.

#### Year Two Optional Modules

##### Math201 - Ordinary Differential Equations (MATH201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **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**(LO1) To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.

(LO2) To know the elementary techniques for the solution of ODEs.

(LO3) To understand the idea of reducing a complex ODE to a simpler one.

(LO4) To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.

(LO5) To understand a range of applications of ODE.

(S1) Problem solving skills

(S2) Numeracy

##### 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**(LO1) 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.

##### Vector Calculus With Applications in Fluid Mechanics (MATH225)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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**(LO1) 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**50:50 **Aims**•To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models.

•To do this in the context of two non-physical applications, namely microeconomics and population dynamics.

•To use and develop mathematical skills introduced in Year 1 - particularly the calculus of functions of several variables and elementary differential equations.**Learning Outcomes**(LO1) 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.

##### Metric Spaces and Calculus (MATH241)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To introduce the basic elements of the theory of metric spaces and calculus of several variables.

**Learning Outcomes**(LO1) 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.

(S1) Problem solving skills

##### Complex Functions (MATH243)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **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**(LO1) To understand the central role of complex numbers in mathematics;.

(LO2) To develop the knowledge and understanding of all the classical holomorphic functions.

(LO3) To be able to compute Taylor and Laurent series of standard holomorphic functions.

(LO4) To understand various Cauchy formulae and theorems and their applications.

(LO5) To be able to reduce a real definite integral to a contour integral.

(LO6) To be competent at computing contour integrals.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### 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**(LO1) 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 number-theoretic problems.

##### Introduction to Methods of Operational Research (MATH261)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**The aims of the module are to develop an understanding of how mathematical modelling and operational research techniques are applied to real-world problems and to gain an understanding of linear and convex programming, multi-objective problems, inventory control and sensitivity analysis.

**Learning Outcomes**(LO1) To understand the operational research approach.

(LO2) To be able to apply standard methods of operational research to a wide range of real-world problems as well as to problems in other areas of mathematics.

(LO3) To understand the advantages and disadvantages of particular operational research methods.

(LO4) To be able to derive methods and modify them to model real-world problems.

(LO5) To understand and be able to derive and apply the methods of sensitivity analysis.

(LO6) To understand the importance of sensitivity analysis.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Self-management readiness to accept responsibility (i.e. leadership), flexibility, resilience, self-starting, initiative, integrity, willingness to take risks, appropriate assertiveness, time management, readiness to improve own performance based on feedback/reflective learning

##### Statistical Theory and Methods I (MATH263)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**75:25 **Aims**To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation.

To introduce students to an appropriate statistical software package.

**Learning Outcomes**(LO1) To have a conceptual and practical understanding of a range of commonly applied statistical procedures.

(LO2) To have developed some familiarity with an appropriate statistical software package.

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

(S4) Communication skills

##### Numerical Methods for Applied Mathematics (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**(LO1) To strengthen students’ knowledge of scientific programming, building on the ideas introduced in MATH111.

(LO2) To provide an introduction to the foundations of numerical analysis and its relation to other branches of Mathematics.

(LO3) To introduce students to theoretical concepts that underpin numerical methods, including fixed point iteration, interpolation, orthogonal polynomials and error estimates based on Taylor series.

(LO4) To demonstrate how analysis can be combined with sound programming techniques to produce accurate, efficient programs for solving practical mathematical problems.

(S1) Numeracy

(S2) Problem solving skills

##### 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**(LO1) The ability to understand and describe mathematically real-life optimization problems.

(LO2) Understanding the basic methods of dynamical decision making.

(LO3) Understanding the basics of forecasting and simulation.

(LO4) The ability to analyse elementary queueing systems.

(S1) Problem solving skills

(S2) Numeracy

#### Year Three Optional Modules

##### Chaos and Dynamical Systems (MATH322)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To develop expertise in dynamical systems in general and study particular systems in detail.

**Learning Outcomes**(LO1) After completing the module students will be able to understand the possible behaviour of dynamical systems with particular attention to chaotic motion;

(LO2) After completing the module students will be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points;

(LO3) After completing the module students will understand how fractal sets arise and how to characterise them.

(S1) Problem solving skills

(S2) Numeracy

##### Further Methods of Applied Mathematics (MATH323)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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**(LO1) 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 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order 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**50:50 **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**(LO1) To understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations.

(LO2) To apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### Quantum Mechanics (MATH325)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**The aim of the module is to lead the student to an understanding of the way that relatively simple mathematics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

**Learning Outcomes**(LO1) To be able to solve Schrodinger's equation for simple systems.

(LO2) To have an understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

(S1) Problem solving skills

(S2) Numeracy

##### Relativity (MATH326)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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**(LO1) After completing this module students should understand why space-time forms a non-Euclidean four-dimensional manifold.

(LO2) After completing this module students should be proficient at calculations involving Lorentz transformations, energy-momentum conservation, and the Christoffel symbols.

(LO3) After completing this module students should understand the arguments leading to the Einstein's field equations and how Newton's law of gravity arises as a limiting case.

(LO4) After completing this module students should be able to calculate the trajectories of bodies in a Schwarzschild space-time.

##### Game Theory (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To explore, from a game-theoretic 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 two-person zero-sum and non-zero-sum games. To give a brief review of n-person 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**(LO1) To extend the appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

(LO2) To be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

(LO3) To be able to solve mathematically a variety of standard problems in the theory of games and to understand the relevance of such solutions in real situations.

##### Population Dynamics (MATH332)

**Level**3 **Credit level**15 **Semester**Second 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**(LO1) The ability to relate the predictions of the mathematical models to experimental results obtained in the field.

(LO2) The ability to recognise the limitations of mathematical modelling in understanding the mechanics of complex biological systems.

(LO3) The ability to use analytical and graphical methods to investigate population growth and the stability of equilibrium states for continuous-time and discrete-time models of ecological systems.

(S1) Problem solving skills

(S2) Numeracy

##### Riemann Surfaces (MATH340)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:0 **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**(LO1) Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

(LO2) 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.

(LO3) Students should know some different techniques to calculate the genus and the dimensions of spaces of meromorphic functions, and they should have acquired some understanding of uniformisation.

(S1) Problem solving skills

##### 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**(LO1) To understand and solve a wide range of problems about integers numbers.

(LO2) To have a better understanding of the properties of prime numbers.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

##### Group Theory (MATH343)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.

**Learning Outcomes**(LO1) Understanding of abstract algebraic systems (groups) by concrete, explicit realisations (permutations, matrices, Mobius transformations).

(LO2) The ability to understand and explain classification results to users of group theory.

(LO3) The understanding of connections of the subject with other areas of Mathematics.

(LO4) To have a general understanding of the origins and history of the subject.

(S1) Problem solving skills

(S2) Logical reasoning

##### Combinatorics (MATH344)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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 inclusion-exclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions.

**Learning Outcomes**(LO1) 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.

##### 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 3-space. While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

**Learning Outcomes**(LO1) 1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces.

(LO2) 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

(LO3) 1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.

(LO4) 1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

(LO5) 2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.

(LO6) 2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.

(LO7) 3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.

(LO8) 3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.

(LO9) 4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,

(LO10) 4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.

(S1) Problem solving skills

(S2) Numeracy

##### Analysis and Number Theory (MATH351)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to uniform distribution modulo one as a meeting ground for other branches of pure mathematics such as topology, analysis and number theory.

To give an idea of how these subjects work together in this context to shape modern analysis by focusing on one particular rich context.

**Learning Outcomes**After completing the module students should be able to:

understand completions and irrationality;

understand diophantine approximation and its relation to uniform distribution;

appreciate that analysis has a complex unity and to have a feel for basic computations in analysis.

calculate rational approximations to real and p adic numbers and to put this to use in number theoretic situations.

calculate approximations to functions from families of simpler functions e.g. continuous functions by polynomials and step functions by trigonometric sums.

work with basic tools from analysis, like Fourier series and continuous functions to prove distributional properties of sequences of numbers.

##### 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**(LO1) To understans the theory of continuous-time Markov chains.

(LO2) To understans the theory of diffusion processes.

(LO3) To be able to solve problems arising in epidemiology, mathematical biology, financial mathematics, etc. using the theory of continuous-time Markov chains and diffusion processes.

(LO4) To acquire an undertanding of the standard concepts and methods of stochastic modelling.

(S1) Problem solving skills

(S2) Numeracy

##### 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**(LO1) To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

(LO2) To acquire an understanding of the blossoming area of Bayesian approach to inference.

(S1) Problem solving skills

(S2) Numeracy

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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 occurring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**(LO1) 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**40:60 **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 skills in using an appropriate statistical software package.

**Learning Outcomes**(LO1) Be able to understand the rationale and assumptions of linear regression and analysis of variance.

(LO2) Be able to understand the rationale and assumptions of generalized linear models.

(LO3) Be able to recognise the correct analysis for a given experiment.

(LO4) Be able to carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

(LO5) Be able to carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

(LO6) Be able to perform linear regression, analysis of variance and generalised linear model analysis using an appropriate statistical software package.

##### 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**(LO1) identify the types of problems encountered in medical statistics

(LO2) demonstrate the advantages and disadvantages of different epidemiological study designs

(LO3) apply appropriate statistical methods to problems arising in epidemiology and interpret results

(LO4) explain and apply statistical techniques used in survival analysis

(LO5) critically evaluate statistical issues in the design and analysis of clinical trials

(LO6) discuss statistical issues related to systematic review and apply appropriate methods of meta-analysis

(LO7) apply Bayesian methods to simple medical problems.

(S1) Problem solving skills

##### Measure Theory and Probability (MATH365)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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**(LO1) After completing the module students should be ableto:

(LO2) master the basic results about measures and measurable functions;

(LO3) master the basic results about Lebesgue integrals and their properties;

(LO4) to understand deeply the rigorous foundations ofprobability theory;

(LO5) to know certain applications of measure theoryto probability, random processes, and financial mathematics.

(S1) Problem solving skills

(S2) Logical reasoning

##### 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**(LO1) 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 theR(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).##### Networks in Theory and Practice (MATH367)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **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**(LO1) After completing the module students should be able to model problems in terms of networks and be able to apply effectively a range of exact and heuristic optimisation techniques.

##### 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;

orc) 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**(LO1) a) (Pure Maths)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 in-depth 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.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

(S4) Organisational skills

(S5) Communication skills

(S6) IT skills

##### School of the Arts Work Placements Module (SOTA300)

**Level**3 **Credit level**30 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**To develop materials and/or undertake tasks within a practical or vocational context. To apply within that practical or vocational context professional, pedagogical, theoretical and other knowledge relevant to the development and delivery of the placement materials and/or tasks. To apply academic and/or theoretical knowledge within a practical context, and reflect and report on the relationship between the two. To develop and identify a range of personal/ employability skills, and reflect and report on this development.

**Learning Outcomes**(LO1) To demonstrate an ability to develop materials and/or undertake tasks, according to a given specification and requirement, within a practical or vocational context.

(LO2) To reflect on and evaluate the efficacy of the materials developed and/or the tasks undertaken.

(LO3) To identify the connection between academic and/or theoretical knowledge and its practical or vocational application.

(LO4) To identify, reflect and report on a range of personal/employability skills.

(S1) Commercial awareness - Relevant understanding of organisations

(S2) Improving own learning/performance - Self-awareness/self-analysis

(S3) Improving own learning/performance - Personal action planning

(S4) Improving own learning/performance - Record-keeping

(S5) Communication (oral, written and visual) - Presentation skills – oral

(S6) Communication (oral, written and visual) - Academic writing (inc. referencing skills)

(S7) Communication (oral, written and visual) - Report writing

(S8) Critical thinking and problem solving - Critical analysis

(S9) Skills in using technology - Using common applications (work processing, databases, spreadsheets etc.)

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 first-year 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 one-to-one 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, mini-project work or key skills exercises.