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Undergraduate Courses 2017
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Mathematics - MMath

Canterbury

Overview

Mathematics is important to the modern world. All quantitative science, including both physical and social sciences, is based on it. It provides the theoretical framework for physical science, statistics and data analysis as well as computer science. Our programmes reflect this diversity and the excitement generated by new discoveries within mathematics that affect not only the technicalities of science but also our general understanding of the world in which we live.

The programmes share a common core of Mathematics at Stage 1, and then move on to cover abstract, analytical and computational techniques that give you the opportunity to specialise in areas such as non-linear differential equations, computational algebra and geometry, financial mathematics, forecasting, design and analysis of experiments, inference and stochastic processes.

The MMath course is aimed at students who have a strong interest in pursuing a deeper study of mathematics with the flexibility to choose a wide range of optional modules to allow those who wish to specialise in a particular area the opportunity to do so.

The programme provides opportunities for students to develop and demonstrate key mathematical knowledge and understanding and will prepare successful students with the depth of mathematical knowledge to enter postgraduate studies at the doctorate level in mathematics and other closely related subjects. A year of Masters level study in Stage 4 gives students the opportunity to explore more advanced topics, which draws on the School's highly rated research expertise.

Independent rankings

In the National Student Survey 2015, 93% of Mathematics students were satisfied with the overall quality of their course.

Course structure

The following modules are indicative of those offered on this programme. This listing is based on the current curriculum and may change year to year in response to new curriculum developments and innovation.  Most programmes will require you to study a combination of compulsory and optional modules. You may also have the option to take ‘wild’ modules from other programmes offered by the University in order that you may customise your programme and explore other subject areas of interest to you or that may further enhance your employability.

Stage 1

Possible modules may include:

MA306 - Statistics (15 credits)

This module will introduce the student to the basic concepts of statistics. The material will be related to real data at every stage and MINITAB will be used to provide statistical computing facilities for all the material studied. Data description and data summary will be studied, followed by an introduction to the main methods of inference. Most material will be based on the Normal, t, and F distributions, but some simple non-parametric procedures will also be covered. The following is a brief summary of the topics to be covered in the module: graphical representation of data; numerical summaries of data; sampling distributions; point estimation; interval estimation; hypothesis tests; association between variables; introduction to nonparametric procedures.

Credits: 15 credits (7.5 ECTS credits).

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MA343 - Algebraic Methods (15 credits)

This module serves as an introduction to algebraic methods. These methods are central in modern mathematics and have found applications in many other sciences, but also in our everyday life. In this module, students will also gain an appreciation of the concept of proof in mathematics.

Credits: 15 credits (7.5 ECTS credits).

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MA344 - Application of Mathematics (15 credits)

This module introduces mathematical modelling and Newtonian mechanics. Tutorials and Maple worksheets will be used to support taught material.



The modelling cycle: General description with examples; Newton's law of cooling; population growth (Malthusian and logistic models); simple reaction kinetics (unimolecular and bimolecular reactions); dimensional consistency



Motion of a body: frames of reference; a particle's position vector and its time derivatives (velocity and acceleration) in Cartesian coordinates; mass, momentum and centre of mass; Newton's laws of motion; linear springs; gravitational acceleration and the pendulum; projectile motion



Orbital motion: Newton's law of gravitation; position, velocity and acceleration in plane polar coordinates; planetary motion and Kepler's laws.

Credits: 15 credits (7.5 ECTS credits).

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MA346 - Linear Algebra (15 credits)

This module is a sequel to Algebraic Methods. It considers the abstract theory of linear spaces together with applications to matrix algebra and other areas of Mathematics (and its applications). Since linear spaces are of fundamental importance in almost every area of mathematics, the ideas and techniques discussed in this module lie at the heart of mathematics.



Topics covered will include:

1 Vector Spaces: definition, examples, linearly independent and spanning sets, bases, dimension, subspaces.

2 Linear transformations: definition, examples, matrix of a linear transformation, change of basis, similar matrices.

3 Determinant of a linear transformation.

4 Eigenvalues/eigenvectors and diagonalisation: characteristic polynomial, invariant subspaces and upper triangular forms. Cayley-Hamilton Theorem.

5 Bilinear forms: inner products, norms, Cauchy-Schwarz inequality.

6 Orthonormal systems, the Gram-Schmidt process.

7 Symmetric Matrices. Every real symmetric matrix is diagonalisable.

8 Quadratic forms: Sylvester's Law of Inertia; signature of a quadratic form; application to conics (and quadrics if time permits).

Credits: 15 credits (7.5 ECTS credits).

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MA348 - Mathematical Methods 1 (15 credits)

This module introduces widely-used mathematical methods for functions of a single variable. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Complex numbers: Complex arithmetic, the complex conjugate, the Argand diagram, de Moivre's Theorem, modulus-argument form; elementary functions

Polynomials: Fundamental Theorem of Algebra (statement only), roots, factorization, rational functions, partial fractions

Single variable calculus: Differentiation, including product and chain rules; Fundamental Theorem of Calculus (statement only), elementary integrals, change of variables, integration by parts, differentiation of integrals with variable limits

Scalar ordinary differential equations (ODEs): definition; methods for first-order ODEs; principle of superposition for linear ODEs; particular integrals; second-order linear ODEs with constant coefficients; initial-value problems

Curve sketching: graphs of elementary functions, maxima, minima and points of inflection, asymptotes

Credits: 15 credits (7.5 ECTS credits).

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MA349 - Mathematical Methods 2 (15 credits)

This module introduces widely-used mathematical methods for vectors and functions of two or more variables. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Vectors: Cartesian coordinates; vector algebra; scalar, vector and triple products (and geometric interpretation); straight lines and planes expressed as vector equations; parametrized curves; differentiation of vector-valued functions of a scalar variable; tangent vectors; vector fields (with everyday examples)

Partial differentiation: Functions of two variables; partial differentiation (including the chain rule and change of variables); maxima, minima and saddle points; Lagrange multipliers

Integration in two dimensions: Double integrals in Cartesian coordinates; plane polar coordinates; change of variables for double integrals; line integrals; Green's theorem (statement – justification on rectangular domains only)

Credits: 15 credits (7.5 ECTS credits).

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MA351 - Probability (15 credits)

Introduction to Probability. Concepts of events and sample space. Set theoretic description of probability, axioms of probability, interpretations of probability (objective and subjective probability).

Theory for unstructured sample spaces. Addition law for mutually exclusive events. Conditional probability. Independence. Law of total probability. Bayes' theorem. Permutations and combinations. Inclusion-Exclusion formula.

Discrete random variables. Concept of random variable (r.v.) and their distribution. Discrete r.v.: Probability function (p.f.). (Cumulative) distribution function (c.d.f.). Mean and variance of a discrete r.v. Examples: Binomial, Poisson, Geometric.

Continuous random variables. Probability density function; mean and variance; exponential, uniform and normal distributions; normal approximations: standardisation of the normal and use of tables. Transformation of a single r.v.

Joint distributions. Discrete r.v.'s; independent random variables; expectation and its application.

Generating functions. Idea of generating functions. Probability generating functions (pgfs) and moment generating functions (mgfs). Finding moments from pgfs and mgfs. Sums of independent random variables.

Laws of Large Numbers. Weak law of large numbers. Central Limit Theorem.

Credits: 15 credits (7.5 ECTS credits).

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MA352 - Real Analysis 1 (15 credits)

Topics covered will include:

Real Numbers: Rational and real numbers, absolute value and metric structure on the real numbers, induction, countability and uncountability, infimum and supremum.

Limits of Sequences: Sequences, definition of convergence, epsilon terminology, uniqueness, algebra of limits, comparison principles, standard limits, subsequences and non-existence of limits, convergence to infinity.

Completeness Properties: Cantor's Intersection Theorem, limit points, Bolzano-Weierstrass theorem, Cauchy sequences.

Continuity of Functions: Functions and basic definitions, limits of functions, continuity and epsilon terminology, sequential continuity, Intermediate Value Theorem.

Differentiation: Definition of the derivative, product rule, quotient rule and chain rule, derivatives and local properties, Mean Value Theorem, L'Hospital's Rule.

Taylor Approximation: Taylor's Theorem, remainder term, Taylor series, standard examples, O and o notation, limits using Taylor series.

Credits: 15 credits (7.5 ECTS credits).

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Stage 2

Possible modules may include:

MA552 - Analysis (15 credits)

This module will consider many concepts you know from Calculus and put them on a more rigorous basis. The concept of a limit is basic to Calculus and, unless this concept is defined precisely, uncertainties and paradoxes will creep into the subject. Based on the foundation of the real number system, this module develops the theory of convergence of sequences and series and the study of continuity and differentiability of functions. The notion of Riemann integration is also explored. The syllabus includes the following: Sequences and their convergence. The convergence of bounded increasing sequences. Series and their convergence: the comparison test, the ratio test, absolute and conditional convergence, the alternating series test. Continuous functions: the boundedness theorem, the Intermediate Value Theorem. Differentiable functions: The Mean Value Theorem with applications, power series, Taylor expansions. Construction and properties of the Riemann integral.

Credits: 15 credits (7.5 ECTS credits).

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MA553 - Linear Algebra (15 credits)

Systems of linear equations appear in numerous applications of mathematics. Studying solution sets to such systems leads to the abstract notions of a vector space and a linear transformation. Matrices can be used to represent linear transformations and to do concrete calculations. This module is about the properties of vector spaces, linear transformations and matrices. The syllabus includes: vector spaces, linearly independent and spanning sets, bases, dimension, subspaces, linear transformations, the matrix of a linear transformation, similar matrices, the determinant, diagonalisation, bilinear forms, norms, and the Gram-Schmidt process.

Credits: 15 credits (7.5 ECTS credits).

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MA629 - Probability and Inference (15 credits)

This module is a pre-requisite for many of the other statistics modules at Stages 2, 3 and 4, but it can equally well be studied as a module in its own right, extending the ideas of probability and statistics met at Stage 1 and providing practice with the mathematical skills learned in MA321. It starts by revising the idea of a probability distribution for one or more random variables and looks at different methods to derive the distribution of a function of random variables. These techniques are then used to prove some of the results underpinning the hypothesis test and confidence interval calculations met at Stage 1, such as for the t-test or the F-test. With these tools to hand, the module moves on to look at how to fit models (probability distributions) to sets of data. A standard technique, known as the method of maximum likelihood, is introduced, which is then used to fit the model to the data to obtain point estimates of the model parameters and to construct hypothesis tests and confidence intervals for these parameters. Outline Syllabus includes: Joint, marginal and conditional distributions of discrete and continuous random variables; Generating functions; Transformations of random variables; Sampling distributions; Point and interval estimation; Properties of estimators; Maximum likelihood; Hypothesis testing; Neyman-Pearson lemma; Maximum likelihood ratio test.

Credits: 15 credits (7.5 ECTS credits).

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MA564 - Functions of Several Variables (15 credits)

Functions of several variables occur in many important applications. In this module we introduce the derivative for functions of several variables and derive an important consequence, namely the chain rule. We use this to calculate maxima and minima and Taylor series for functions of several variables. We also discuss the important problem of finding maxima and minima of functions subject to a constraint using the method of Lagrange multipliers. Furthermore, we define different ways to integrate functions of several variables such as arclength integrals, line integrals, surface integrals and volume integrals. Outline Syllabus includes: Continuity and Differentiation; tangent plane; swapping order of partial derivatives; implicit function theorem; inverse function theorem; paths independence of line integrals; use of polar, cylindrical and spherical polar coordinates; integral theorems such as Green's theorem.

Credits: 15 credits (7.5 ECTS credits).

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MA565 - Groups and Rings (15 credits)

Groups are sets with a single binary operation. They arise as symmetry groups in contexts from puzzles like Rubik's cube to chemistry, where they help list molecules with a given number of atoms involved. In contrast, rings have two binary operations, generalising the arithmetic of integer numbers. This part of algebra has many applications in electronic communication, in particular in coding theory and cryptography. Outline Syllabus includes: permutations and cycle decomposition, subgroups, cosets, Lagrange's theorem, normal subgroups, symmetry groups, group actions, homomorphisms of groups and rings, ideals, factorization in rings, polynomial rings, domains, fields, quotient fields, finite fields.

Credits: 15 credits (7.5 ECTS credits).

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MA588 - Mathematical Techniques and Differential Equations (15 credits)

We will study ordinary differential equations analytically, going beyond the exact techniques studied in MA321. We will also learn how to solve partial differential equations and apply the techniques to phenomena such as the vibration of a guitar string or a drum skin. Outline syllabus includes: Series Solutions of Linear Ordinary Differential Equations, Orthogonal polynomials and Special functions, Fourier Series and Transforms and Partial Differential Equations.

Credits: 15 credits (7.5 ECTS credits).

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MA584 - Computational Mathematics (15 credits)

The aim of the module is to provide an introduction to the methods, tools and ideas of numerical computation. In applications mathematics frequently generates specific instances of standard problems for which there are no easily obtainable analytic solutions. Examples might be the task of determining the value of a particular integral, or of finding the roots of a certain non- linear equation. Methods are presented for solving such problems on a modern computer. Besides a description of the basic numerical procedure, each method is analysed in terms of when it best works, how it compares with alternative approaches, and the way it may be implemented on a computer. Numerical computations are almost invariably contaminated by errors, and an important concern throughout the module is to understand the source, propagation and magnitude of these errors.The syllabus will cover: Introduction to numerics; solutions of equations in one variable; interpolation and polynomial approximation; numerical differentiation; numerical integration; direct methods for solving linear systems; iterative techniques for solving linear systems.

Credits: 15 credits (7.5 ECTS credits).

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MA590 - Mathematical Modelling (15 credits)

This module introduces mathematical modelling in a variety of contexts including using Newton's laws of motion, Newton’s law of gravitation, population models, exponential growth, density dependent growth, and predator-prey models. Outline syllabus may include topics from (i) deriving differential equations from data; dimensional analysis; (ii) discrete models and difference equations: steady states and their stability; (iii) continuous models and ordinary differential equations: steady states and their stability; the slope fields and phase lines; (iv) applications of Linear Algebra (in lower dimensions): systems of linear ordinary differential equations; linear phase plane analysis and stability; (v) electrical networks; (vi) vector algebra, vector geometry, vector equations, coordinate systems and vector differentiation; (vii) application in mechanics: Newton's laws for a single particle in 3-D; conserved quantities; angular velocity, angular momentum, moment of a force; harmonic motion.

Credits: 15 credits (7.5 ECTS credits).

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MA566 - Number Theory (15 credits)

The security of our phone calls, bank transfers, etc. all rely on one area of Mathematics: Number Theory. This module is an elementary introduction to this wide area and focuses on solving Diophantine equations. In particular, we discuss (without proof) Fermat's Last Theorem, arguably one of the most spectacular mathematical achievements of the twentieth century. Outline syllabus includes: Modular Arithmetic; Prime Numbers; Introduction to Cryptography; Quadratic Residues; Diophantine Equations.

Credits: 15 credits (7.5 ECTS credits).

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CB668 - Linear Programming and its Application (15 credits)

The broad areas will be as defined as shown below:



Modelling LP applications (management, finance, business, marketing)

The use of graphical method for small problems and the development Simplex Method (optimality and feasibility criteria) including the two-phase method.

The use of a computer software such as Excel to solve LP instances and discussion of results (through a couple of Labs).

Degeneracy issues in LP (brief)

Duality theory (dual problems, duality theorem, and complementary slackness conditions), and application of duality to other problems (brief)

Dual Simplex Method

Sensitivity analysis and brief pot-optimality analysis

Extension of LP to Integer Programming or Ratio Programming (DEA)

Credits: 15 credits (7.5 ECTS credits).

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MA632 - Regression Models (15 credits)

Regression is a fundamental technique of statistical modelling, in which we aim to model a response variable using one or more explanatory variables. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. The need for statistical modelling arises because even when temperature and pressure are fixed, there will typically be variation in the resulting yield, so the model must include a random component. In this module we study the broad class of linear regression models, which are widely used in practice. We learn how to formulate such models and fit them to data, how to make predictions with associated measures of uncertainty, and how to select appropriate explanatory variables. Both theory and practical aspects are covered, including the use of computer software for regression. Outline of the syllabus: simple linear regression; the method of least squares; sums of squares; the ANOVA table; residuals and diagnostics; matrix formulation of the general linear model; prediction; variable selection; one-way analysis of variance; practical regression analysis using software; logistic regression.

Credits: 15 credits (7.5 ECTS credits).

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Stage 3

Possible modules may include:

MA569 - Starting Research in the Mathematical Sciences (15 credits)

This module has two components: (i) A series of workshops on key skills: The interactive workshops will cover general research methods possibly including library & information systems, mathematical reading, referencing conventions, coherent mathematical writing, typesetting using LaTeX and presentation skills. (ii) Independent project work: A list of possible topics will be offered. The description of each project will include a reference to a single journal article. The student will carry out independent study on one topic based on the journal article listed.

Credits: 15 credits (7.5 ECTS credits).

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MA568 - Orthogonal Polynomials and Special Functions (15 credits)

This module provides an introduction to the study of orthogonal polynomials and special functions. They are essentially useful mathematical functions with remarkable properties and applications in mathematical physics and other branches of mathematics. Closely related to many branches of analysis, orthogonal polynomials and special functions are related to important problems in approximation theory of functions, the theory of differential, difference and integral equations, whilst having important applications to recent problems in quantum mechanics, mathematical statistics, combinatorics and number theory. The emphasis will be on developing an understanding of the structural, analytical and geometrical properties of orthogonal polynomials and special functions. The module will utilise physical, combinatorial and number theory problems to illustrate the theory and give an insight into a plank of applications, whilst including some recent developments in this field. The development will bring aspects of mathematics as well as computation through the use of MAPLE. The topics covered will include: The hypergeometric functions, the parabolic cylinder functions, the confluent hypergeometric functions (Kummer and Whittaker) explored from their series expansions, analytical and geometrical properties, functional and differential equations; sequences of orthogonal polynomials and their weight functions; study of the classical polynomials and their applications as well as other hypergeometric type polynomials.

Credits: 15 credits (7.5 ECTS credits).

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MA572 - Complex Analysis (15 credits)

This module is concerned with complex functions, that is functions which are both defined for and assume complex values. Their theory follows a quite different development from that of real functions, is remarkable in its directness and elegance, and leads to many useful applications.Topics covered will include: Complex numbers. Domains and simple connectivity. Cauchy-Riemann equations. Integration and Cauchy's theorem. Singularities and residues. Applications.

Credits: 15 credits (7.5 ECTS credits).

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MA574 - Polynomials in Several Variables (15 credits)

Systems of polynomial equations arise naturally in many applications of mathematics. This module focuses on methods for solving such systems and understanding the solutions sets. The key abstract concept is an ideal in a commutative ring and the fundamental computational concept is Buchberger's algorithm for computing a Groebner basis for an ideal in a polynomial ring. The syllabus includes: multivariate polynomials, Hilbert's Basis Theorem, monomial orders, division algorithms, Groebner bases, Hilbert's Nullstellensatz, elimination theory, linear equations over systems of polynomials, and syzygies.

Credits: 15 credits (7.5 ECTS credits).

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MA576 - Groups and Representations (15 credits)

Groups arise naturally in many areas of mathematics as well as in chemistry and physics. A concrete way to approach groups is by representing them as a group of matrices, in which explicit computations are easy. This approach has been very fruitful in developing our understanding of groups over the last century. It also helps students to understand aspects of their mathematical education in a broader context, in particular concepts from earlier modules (From Geometry to Algebra/Groups and Symmetries and Linear Algebra) have been amalgamated into more general and powerful tools.



This module will provide a rigorous introduction to the main ideas and notions of groups and representations. It will also have a strong computational strand: a large part of the module will be devoted to explicit computations of representations and character tables (a table of complex numbers associated to any finite group).

Credits: 15 credits (7.5 ECTS credits).

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MA636 - Stochastic Processes (15 credits)

A stochastic process is a process developing in time according to probability rules, for example, models for reserves in insurance companies, queue formation, the behaviour of a population of bacteria, and the persistence (or otherwise) of an unusual surname through successive generations.The syllabus will include coverage of a wide variety of stochastic processes and their applications: Markov chains; processes in continuous-time such as the Poisson process, the birth and death process and queues.



Marks on this module can count towards exemption from the professional examination CT4 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

Credits: 15 credits (7.5 ECTS credits).

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MA639 - Time Series Modelling and Simulation (15 credits)

A time series is a collection of observations made sequentially in time. Examples occur in a variety of fields, ranging from economics to engineering, and methods of analysing time series constitute an important area of statistics. This module focuses initially on various time series models, including some recent developments, and provides modern statistical tools for their analysis. The second part of the module covers extensively simulation methods. These methods are becoming increasingly important tools as simulation models can be easily designed and run on modern PCs. Various practical examples are considered to help students tackle the analysis of real data.The syllabus includes: Difference equations, Stationary Time Series: ARMA process. Nonstationary Processes: ARIMA Model Building and Testing: Estimation, Box Jenkins, Criteria for choosing between models, Diagnostic tests.Forecasting: Box-Jenkins, Prediction bounds. Testing for Trends and Unit Roots: Dickey-Fuller, ADF, Structural change, Trend-stationarity vs difference stationarity. Seasonality and Volatility: ARCH, GARCH, ML estimation. Multiequation Time Series Models: Spectral Analysis. Generation of pseudo – random numbers, simulation methods: inverse transform and acceptance-rejection, design issues and sensitivity analysis.



Marks on this module can count towards exemption from the professional examination CT6 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

Credits: 15 credits (7.5 ECTS credits).

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MA690 - Symmetry Methods for Differential Equations (15 credits)

Over a century ago, the Norwegian mathematician Sophus Lie made a simple but profound observation: each well-known method for solving a class of ordinary differential equations (ODEs) uses a change of variables that exploits symmetries of the class. Lie went on to develop this idea into a systematic method for attacking the problem of solving unknown differential equations. Essentially, one can use mathematical tools to force a given differential equation to reveal whether or not it has certain symmetries – provided it has, they can be used to simplify or solve the equation. This module is designed to enable students to understand the mathematics behind Lie's methods and to become proficient in using these powerful tools.

The following topics are covered.

Introduction: Symmetries of geometrical objects, symmetries of some first-order ODEs, solution via symmetries.

Lie symmetries of first-order ODEs: The infinitesimal generator, canonical coordinates, invariant points, Lie symmetries and standard solution methods.

How to find Lie symmetries: The linearized symmetry condition, solution of overdetermined systems, the Lie algebra of point symmetry generators.

Solution of higher-order ODEs: Solvability, differential invariants, reduction of order, invariant solutions.

Credits: 15 credits (7.5 ECTS credits).

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MA691 - Linear and Nonlinear Waves (15 credits)

Linear PDEs. Dispersion relations. Review of d'Alembert's solutions of the wave equation.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).

Credits: 15 credits (7.5 ECTS credits).

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MA692 - Operators and Matrices (15 credits)

Matrix theory: Hermitian and symmetric matrices, spaces of these matrices and the associated inner product, diagonalization, orthonormal basis of eigenvectors, spectral properties, positive definite matrices and their roots

Hilbert space theory: inner product spaces and Hilbert spaces, L^2 and l^2 spaces, orthogonality, bases, Gram-Schmidt procedure, dual space, Riesz representation theorem

Linear operators: the space of bounded linear operators with the operator norm, inverse and adjoint operators, Hermitian operators, infinite matrices, spectrum, compact operators, Hilbert-Schmidt operators, the spectral theorem for compact Hermitian operators.

Credits: 15 credits (7.5 ECTS credits).

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MA538 - Applied Bayesian Modelling (15 credits)

The origins of Bayesian inference lie in Bayes' Theorem for density functions; the likelihood function and the prior distribution combine to provide a posterior distribution which reflects beliefs about an unknown parameter based on the data and prior beliefs. Statistical inference is determined solely by the posterior distribution. So, for example, an estimate of the parameter could be the mean value of the posterior distribution. This module will provide a full description of Bayesian analysis and cover popular models, such as the normal distribution. Initially, the flavour will be one of describing the Bayesian counterparts to well known classical procedures such as hypothesis testing and confidence intervals. Outline Syllabus includes: Bayes Theorem for density functions; Exchangeability; Choice of priors; Conjugate models; Predictive distribution; Bayes estimates; Sampling density functions; Gibbs samplers; OpenBUGS; Bayesian hierarchical models; Applications of hierarchical models; Bayesian model choice.

Credits: 15 credits (7.5 ECTS credits).

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MA549 - Discrete Mathematics (15 credits)

Recently some quite novel applications have been found for "Discrete Mathematics", as opposed to the “Continuous Mathematics” based on the Differential and Integral Calculus. Thus methods for the encoding of information in order to safeguard against eavesdropping or distortion by noise, for example in online banking and digital television, have involved using some basic results from abstract algebra. This module will provide a self-contained introduction to this general area and will cover most of the following topics: (a) Modular arithmetic, polynomials and finite fields: Applications to orthogonal Latin squares, cryptography, “coin-tossing over a telephone”, linear feedback shift registers and m-sequences. (b) Error correcting codes: Binary block, linear and cyclic codes including repetition, parity-check, Hamming, simplex, Reed-Muller, BCH, Golay codes; channel capacity; Maximum likelihood, nearest neighbour, syndrome and algebraic decoding.

Credits: 15 credits (7.5 ECTS credits).

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MA607 - Quantum Mechanics (15 credits)

Quantum mechanics provides an accurate description of nature on a subatomic scale, where the standard rules of classical mechanics fail. It is an essential component of modern technology and has a wide range of fascinating applications. This module introduces some of the key concepts of quantum mechanics from a mathematical point of view.

The joint level 6/level 7 curriculum will consist of the following:

• The necessity for quantum mechanics. The wavefunction and Born's probabilistic interpretation.

• Solutions of the time-dependent and time-independent Schrödinger equation for a selection of simple potentials in one dimension.

• Reflection and transmission of particles incident onto a potential barrier. Probability flux. Tunnelling of particles.

• Wavefunctions and states, Hermitian operators, outcomes and collapse of the wavefunction.

• Heisenberg's uncertainty principle.

Additional topics may include applications of quantum theory to physical systems, quantum computing or recent developments in the quantum world.

Credits: 15 credits (7.5 ECTS credits).

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MA609 - Applied Differential Geometry (15 credits)

The main aim is to give an introduction to the basics of differential geometry, keeping in mind the recent application in mathematical physics and the analysis of pattern recognition.

The synopsis may include:

(a) Theory of curves: Regular plane and space curves. Tangent vectors. Arclength parameterisation. Curvature and Euclidean invariants. The Frenet formula.



(b) Geometry of surfaces: Regular parameterised surface. The tangent plane. Curvature of a curve on a surface. First and second fundament forms. Shape operator. Gaussian curvature and mean curvature.

(c) Geodesics and Minimal surfaces: The Christoffel symbols. Geodesics. The Euler-Lagrange equations. The Gauss-Bonnet Theorem. Minimal surfaces.



Possible other topics may include: Evolution of curves and surfaces as integrable systems: Invariant curve evolution. The mean curvature flow. Riemannian metrics, connections, curvatures and geodesics.

In addition, for M-level students, the connection with integrable systems; curves evolution in Riemannian manifolds with constant curvature and Moving frames.

Credits: 15 credits (7.5 ECTS credits).

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MA968 - Mathematics and Music (15 credits)

This module is divided into two - one part is about the mathematics of sound both acoustic and digital, and the other is about the structure of music as it affects musical composition. The mathematics of sound includes the study of the linear wave equation, in particular, the mathematics of drums and Chladni patterns. We then move on the mathematics of digital sound - the discrete Fourier transform, the short time Fourier transform and the Gabor transform. Here we can answer questions like, does Louis Armstrong play the trumpet the same way he sings? And, how to slow down music without losing pitch? The mathematics of rhythm and harmony are two very different fields of study. Many world music rhythms can be studied using the Euclidean algorithm, and we will also take a look at the theory of rhythmic tilings which underpin cannons and more modern compositions. Finally, the harmonic progression of a musical composition can be modelled as a path in chord space. In this part of the module, we will look at how simple geometric ideas are used to model voice leading and harmony. For this last part, familiarity with the keyboard would be helpful but is not a pre-requisite.

Credits: 15 credits (7.5 ECTS credits).

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MA587 - Numerical Solution of Differential Equations (15 credits)

Most differential equations which arise from physical systems cannot be solved explicitly in closed form, and thus numerical solutions are an invaluable way to obtain information about the underlying physical system. The first half of the module is concerned with ordinary differential equations. Several different numerical methods are introduced and error growth is studied. Both initial value and boundary value problems are investigated. The second half of the module deals with the numerical solution of partial differential equations. The syllabus includes: initial value problems for ordinary differential equations; Taylor methods; Runge-Kutta methods; multistep methods; error bounds and stability; boundary value problems for ordinary differential equations; finite difference schemes; difference schemes for partial differential equations; iterative methods; stability analysis.

Credits: 15 credits (7.5 ECTS credits).

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Stage 4

Possible modules may include:

MA578 - Dissertation for MMath Mathematics (30 credits)

The module offers students the opportunity to work independently, under limited supervision, on an area of mathematics of their choice. There is no specific mathematical syllabus for this module. The topic of the dissertation will depend on the mutual interests of the student and the student's chosen supervisor. The coursework will consist of writing a dissertation plan, an oral presentation of material from the dissertation to examiners and an interview of the student by the examiners. There will be four workshops on key skills relevant to dissertation planning and oral presentation.

Credits: 30 credits (15 ECTS credits).

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MA969 - Applied Differential Geometry (15 credits)

Differential geometry studies geometrical objects using analytical methods. It originates in classical mechanics. Modern differential geometry has made a huge impact in the development of nonlinear mathematical physics including integrable systems and string theory. Nowadays differential geometry is at the centre of the analysis of pattern recognition, image processing and computer graphics.



Indicative specific subtopics are:

• Theory of curves. Plane and space curves. Euclidean invariants of curves. Frenet frame.

• Theory of surfaces. Metrics on regular surface. Curvature of a curve on a surface. Gaussian curvature and mean curvature. Covariant derivative and geodesics. The Euler-Lagrange equations. Minimal surfaces.

• Evolution of curves and surfaces as integrable systems: Invariant curve evolution. The mean curvature flows. The connection with integrable systems. The modified Korteweg de-Vries equation.

• Curves in Riemannian manifolds: Riemannian metrics, connections, curvatures and geodesics. Curves evolution in Riemannian manifold with constant curvature.

• Modern applications.

i. 2D and 3D projective geometry and application to multiple view geometry in computer vision;

ii. Moving frames, invariant signatures in pattern recognition;

iii. Poisson manifold and Hamiltonian systems.

Credits: 15 credits (7.5 ECTS credits).

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MA972 - Algebraic Curves in Nature (15 credits)

In this module we will study plane algebraic curves and the way that they arise in applications to other parts of mathematics and physics. Examples include the use of elliptic functions to solve problems in mechanics (e.g. the pendulum, or Euler's equations for rigid body motion), spectral curves of separable Hamiltonian systems, and algebraic curves over finite fields that are used in cryptography. The geometrical properties of a curve are not altered by coordinate transformations, so it is important to identify quantities that are invariant under such transformations. For curves, the most basic invariant is the genus, which is most easily understood in terms of the topology of the associated Riemann surface: it counts the number of handles or “holes”. The case of genus zero (corresponding to the Riemann sphere) is well understood, but curves of genus one (also known as elliptic curves) lead to some of the most interesting and difficult problems in modern number theory.



• Review of basic results from complex analysis and topology;

• Riemann surfaces and plane curves in complex affine and projective space;

• The genus of a curve: degree-genus and Riemann-Hurwitz formulae;

• Genus one: elliptic curves and their group structure; elliptic functions and elliptic integrals, with applications;

• Higher genus: functions, divisors and differentials on algebraic curves; Riemann-Roch theorem; example: hyperelliptic curves.

Credits: 15 credits (7.5 ECTS credits).

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MA776 - Groups and Representations (15 credits)

Groups arise naturally in many areas of mathematics as well as in chemistry and physics. A concrete way to approach groups is by representing them as a group of matrices, in which explicit computations are easy. This approach has been very fruitful in developing our understanding of groups over the last century. It also helps students to understand aspects of their mathematical education in a broader context, in particular concepts from earlier modules (From Geometry to Algebra/Groups and Symmetries and Linear Algebra) have been amalgamated into more general and powerful tools.



This module will provide a rigorous introduction to the main ideas and notions of groups and representations. It will also have a strong computational strand: a large part of the module will be devoted to explicit computations of representations and character tables (a table of complex numbers associated to any finite group).



Syllabus:

1. Review of basic group theory (including matrix groups, the symmetric groups, permutation groups and symmetry groups, subgroups, conjugacy, normal subgroups and quotient groups, homomorphisms, group actions);

2. A concrete approach to groups via representations (including examples via group actions and the language modules);

3. Irreducible representations, Maschke's theorem, Schur's lemma;

4. Characters and their basic properties;

5. Character tables: theory and computations for small groups. Consequences.

In addition, for level 7 students:

6. Simple groups, composition series and the Jordan--Hölder theorem.

Credits: 15 credits (7.5 ECTS credits).

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MA790 - Symmetry Methods for Differential Equations (15 credits)

Over a century ago, the Norwegian mathematician Sophus Lie made a simple but profound observation: each well-known method for solving a class of ordinary differential equations (ODEs) uses a change of variables that exploits symmetries of the class. Lie went on to develop this idea into a systematic method for attacking the problem of solving unknown differential equations. Essentially, one can use mathematical tools to force a given differential equation to reveal whether or not it has certain symmetries – provided it has, they can be used to simplify or solve the equation. This module is designed to enable students to understand the mathematics behind Lie's methods and to become proficient in using these powerful tools.

The following topics are covered.

Introduction: Symmetries of geometrical objects, symmetries of some first-order ODEs, solution via symmetries.

Lie symmetries of first-order ODEs: The infinitesimal generator, canonical coordinates, invariant points, Lie symmetries and standard solution methods.

How to find Lie symmetries: The linearized symmetry condition, solution of overdetermined systems, the Lie algebra of point symmetry generators.

Solution of higher-order ODEs: Solvability, differential invariants, reduction of order, invariant solutions.

In addition, for level 7 students:

Advanced topic: This will be selected from the following:

• Symmetry methods for PDEs.

• First integrals and dynamical symmetries.

• Discrete symmetries of ODEs

• Symmetries of difference equations.

Credits: 15 credits (7.5 ECTS credits).

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MA791 - Linear and Nonlinear Waves (15 credits)

Linear PDEs. Dispersion relations. Review of d'Alembert's solutions of the wave equation.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).

Level 7 Students only. Further applications of shock waves and nonlinear waves.

Credits: 15 credits (7.5 ECTS credits).

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MA792 - Operators and Matrices (15 credits)

Matrix theory: Hermitian and symmetric matrices, spaces of these matrices and the associated inner product, diagonalization, orthonormal basis of eigenvectors, spectral properties, positive definite matrices and their roots

Hilbert space theory: inner product spaces and Hilbert spaces, L^2 and l^2 spaces, orthogonality, bases, Gram-Schmidt procedure, dual space, Riesz representation theorem

Linear operators: the space of bounded linear operators with the operator norm, inverse and adjoint operators, Hermitian operators, infinite matrices, spectrum, compact operators, Hilbert-Schmidt operators, the spectral theorem for compact Hermitian operators.

Additional topics, especially for level 7 students may include:

- the Rayleigh quotient and variational characterisations of eigenvalues,

- the functional calculus,

- applications to Sturm-Liouville systems.

Credits: 15 credits (7.5 ECTS credits).

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MA964 - Applied Algebraic Topology (15 credits)

There is growing interest in applying the methods of algebraic topology to data analysis, sensor networks, robotics, etc. The module will develop the necessary elements of algebra and topology, and investigate how these techniques are used in various applications. The syllabus will include: an introduction to manifolds, CW complexes and simplicial complexes; an investigation of the elements of homotopy theory; an exploration of homological and computational methods; applications such as homological sensor networks and topological data analysis.

Credits: 15 credits (7.5 ECTS credits).

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MA967 - Quantum Physics (15 credits)

Quantum mechanics provides an accurate description of nature on a subatomic scale, where the standard rules of classical mechanics fail. It is an essential component of modern technology and has a wide range of fascinating applications. This module introduces some of the key concepts of quantum mechanics from a mathematical point of view. Outline syllabus: why classical mechanics fails; the Schrödinger equation and interpretation of the wavefunction; applications of quantum mechanics.

Credits: 15 credits (7.5 ECTS credits).

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Teaching & Assessment

Teaching

Lectures are given by a wide variety of lecturers all with different research backgrounds; small supervised example classes; computer laboratory classes, dissertation module. Teaching amounts to typically 16 hours of lectures and classes per week. Modules involving programming or working with computer software packages usually include practical sessions.

Assessment

Coursework involving problems; computer assignments; projects; tests; dissertation; written unseen examinations.

Programme aims

The programme aims to:

  • provide an excellent quality of mathematical education, informed by research and scholarship
  • equip students with a broad base of knowledge and skills to analyse and solve mathematically based problems showing a level of originality where necessary
  • ensure students are competent in communicating the knowledge, rationale and conclusions, both orally and by writing
  • ensure students are competent in the use of information technology and can use appropriate software to sovle problems
  • develop in students the ability to work independently with a minimum amount of supervision within agreed guidelines
  • prepare successful students with the depth of mathematical knowledge to enter postgraduate studies at the doctorate level in mathematics and other closely related subjects
  • produce graduates of value to the region and nationally, in possession of key mathematical knowledge and personal skills, with the capacity to learn

Learning outcomes

Knowledge and understanding

You gain knowledge and understanding of:

  • the fundamental concepts and techniques of calculus, algebra, analysis, geometry, differential equations, numerical mathematics and porbability and inference
  • nonlinear phenomena and related mathematical methods
  • applications of mathematical theories, methods and techniques to a range of associated problems
  • the role of logical mathematical argument and deductive reasoning including formal process of mathematical proof
  • more advanced material with mathematical ideas from more than one area
  • project work on an advanced topic based on substantial independent work

Intellectual skills

You develop your intellectual skills in the following areas:

  • the ability to demonstrate a reasonable understanding of mathematics
  • the calculation and manipulation of the material written within the programme
  • the ability to apply a range of concepts and principles in various contexts
  • the ability to construct and develop mathematical logical argument
  • the ability to solve mathematical problems by various appropriate methods
  • the relevant computer skills
  • the ability to work independently.

Subject-specific skills

You gain subject-skills in the following areas:

  • the ability to demonstrate knowledge of key mathematical concepts and topics, both explicitly and by applying them to the solution of problems
  • the ability to comprehend problems, abstract the essentials of problems and formulate them mathematically and in symbolic form so as to facilitate their analysis and solution
  • the use of computational and more general IT facilities as an aid to mathematical processes
  • the presentation of mathematical arguments and conclusions with clarity and accuracy.

Transferable skills

You gain transferable skills in the following areas:

  • problem-solving skills, relating to qualitative and quantitative information
  • communication skills
  • numeracy and computational skills
  • information-retrieval skills, in relation to primary and secondary information sources, including through on-line computer searches
  • information technology skills such as word-processing, spreadsheet use and internet communication
  • personal and interpersonal skills, work as a member of a team
  • time-management and organisational skills, as shown by the ability to plan and implement effective modes of working
  • study skills needed for continuing professional development.

Careers

Students studying this degree programme will develop a broad range of skills and mathematical understanding that are highly sought after by employers and which open up a wide variety of careers. MMath Mathematics graduates typically find employment in areas involving applications of the subject or they directly enter postgraduate studies at the doctoral level. Recent graduates of the School have gone into careers in medical statistics, the pharmaceutical industry, the aerospace industry, software development, teaching, Civil Service statistics, chartered accountancy, the oil industry and PhD training.

Entry requirements

Home/EU students

The University will consider applications from students offering a wide range of qualifications, typical requirements are listed below, students offering alternative qualifications should contact the Admissions Office for further advice. It is not possible to offer places to all students who meet this typical offer/minimum requirement.

Students can also enter the MMath programme by transfer from the standard 3-year degree  programmes at the end of Stage 2, provided they have passed the core modules and met the average mark threshold of Stage 2 of the MMath programme.

Qualification Typical offer/minimum requirement
A level

AAA including A in Mathematics (not Use of Mathematics). Only one General Studies and Critical Thinking can be accepted against the requirements

Access to HE Diploma

The University of Kent will not necessarily make conditional offers to all access candidates but will continue to assess them on an individual basis. If an offer is made candidates will be required to obtain/pass the overall Access to Higher Education Diploma and may also be required to obtain a proportion of the total level 3 credits and/or credits in particular subjects at merit grade or above.

BTEC Level 3 Extended Diploma (formerly BTEC National Diploma)

The University will consider applicants holding BTEC National Diploma and Extended National Diploma Qualifications (QCF; NQF;OCR) on a case by case basis please contact us via the enquiries tab for further advice on your individual circumstances.

International Baccalaureate

34 points overall or 17 at HL including Mathematics HL 6

International students

The University receives applications from over 140 different nationalities and consequently will consider applications from prospective students offering a wide range of international qualifications. Our International Development Office will be happy to advise prospective students on entry requirements. See our International Student website for further information about our country-specific requirements.

Please note that if you need to increase your level of qualification ready for undergraduate study, we offer a number of International Foundation Programmes through Kent International Pathways.

Qualification Typical offer/minimum requirement
English Language Requirements

International students will need to demonstrate their proficiency in English: Average 6.5 in IELTs test with minimum 6.0 in reading and writing or equivalent.

Please see our English language entry requirements web page. 

General entry requirements

Please also see our general entry requirements.

Funding

Kent offers generous financial support schemes to assist eligible undergraduate students during their studies. Our funding opportunities for 2017 entry have not been finalised. However, details of our proposed funding opportunities for 2016 entry can be found on our funding page.  

General scholarships

Scholarships are available for excellence in academic performance, sport and music and are awarded on merit. For further information on the range of awards available and to make an application see our scholarships website.

The Kent Scholarship for Academic Excellence

At Kent we recognise, encourage and reward excellence. We have created the Kent Scholarship for Academic Excellence. Details of the scholarship for 2017 entry have not yet been finalised. However, for 2016 entry, the scholarship will be awarded to any applicant who achieves a minimum of AAA over three A levels, or the equivalent qualifications as specified on our scholarships pages. Please review the eligibility criteria on that page. 

Enquire or order a prospectus

Resources

Read our student profiles

Contacts

Related schools

Enquiries

T: +44 (0)1227 827272

Fees

The 2017/18 tuition fees for this programme are:

UK/EU Overseas
Full-time £9250 £13810

For students continuing on this programme fees will increase year on year by no more than RPI + 3% in each academic year of study except where regulated.* If you are uncertain about your fee status please contact information@kent.ac.uk

The Government has announced changes to allow undergraduate tuition fees to rise in line with inflation from 2017/18.

The University of Kent intends to increase its regulated full-time tuition fees for all Home and EU undergraduates starting in September 2017 from £9,000 to £9,250. This is subject to us satisfying the Government's Teaching Excellence Framework and the access regulator's requirements. The equivalent part-time fees for these courses will also rise by 2.8%.

Key Information Sets


The Key Information Set (KIS) data is compiled by UNISTATS and draws from a variety of sources which includes the National Student Survey and the Higher Education Statistical Agency. The data for assessment and contact hours is compiled from the most populous modules (to the total of 120 credits for an academic session) for this particular degree programme. Depending on module selection, there may be some variation between the KIS data and an individual's experience. For further information on how the KIS data is compiled please see the UNISTATS website.

If you have any queries about a particular programme, please contact information@kent.ac.uk.

The University of Kent makes every effort to ensure that the information contained in its publicity materials is fair and accurate and to provide educational services as described. However, the courses, services and other matters may be subject to change. Full details of our terms and conditions can be found at: www.kent.ac.uk/termsandconditions.

*Where fees are regulated (such as by the Department of Business Innovation and Skills or Research Council UK) they will be increased up to the allowable level.

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The University of Kent, Canterbury, Kent, CT2 7NZ, T: +44 (0)1227 764000