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.

## Overview

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. It offers the flexibility to choose a wide range of optional modules, allowing those who wish to specialise in a particular area the opportunity to do so.

The programme provides opportunities 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 doctoral level in mathematics and other closely related subjects. A year of Master's-level study in Stage 4 gives you the opportunity to explore more advanced topics, which draw on the highly rated research expertise of the School of Mathematics, Statistics and Actuarial Science.

#### Independent rankings

Mathematics at Kent was ranked 19th for course satisfaction in *The Guardian University Guide 2017*. For graduate prospects, Mathematics was ranked 19th in *The Complete University Guide 2017.*

## 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.

On most programmes, you study a combination of compulsory and optional modules. You may also be able to take ‘wild’ modules from other programmes so you can customise your programme and explore other subjects that interest you.

### Stage 1

Modules may include | Credits |
---|---|

MA306 - Statistics
Increasingly data are collected to inform future decisions, varying from which websites people access on a regular basis to how patients respond to new drugs, to how the stock market responds to global events, or to how animals move around their local habitat. Therefore, most professionals will need to extract useful information from data and to manage and present data in their working lives. This module explores some of the basic concepts of statistics, from data summarisation to the main methods of statistical inference. The techniques that are discussed can be used in their own right for simple statistical analyses, but serve as an important foundation for later, more advanced, modules. The statistical computing package R is used throughout the module for data analysis. The syllabus includes: an introduction to R and investigating data sets, sampling and sampling distributions, point and interval estimation, hypothesis testing, association between variables. Read more |
15 |

MA343 - Algebraic Methods
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. Read more |
15 |

MA344 - Application of Mathematics
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. Read more |
15 |

MA346 - Linear Algebra
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). Read more |
15 |

MA348 - Mathematical Methods 1
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 Read more |
15 |

MA349 - Mathematical Methods 2
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) Read more |
15 |

MA351 - Probability
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. Read more |
15 |

MA352 - Real Analysis 1
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. Read more |
15 |

### Stage 2

Modules may include | Credits |
---|---|

MA5505 - Linear Partial Differential Equations
In this module we will study linear partial differential equations, we will explore their properties and discuss the physical interpretation of certain equations and their solutions. We will learn how to solve first order equations using the method of characteristics and second order equations using the method of separation of variables. Read more |
15 |

MA5503 - Groups and Symmetries
The concept of symmetry is one of the most fruitful ideas through which mankind has tried to understand order and beauty in nature and art. This module first develops the concept of symmetry in geometry. It subsequently discusses links with the fundamental notion of a group in algebra. Outline syllabus includes: Groups from geometry; Permutations; Basic group theory; Action of groups and applications to (i) isometries of regular polyhedra; (ii) counting colouring problems; Matrix groups. Read more |
15 |

MA5513 - Real Analysis 2
This module builds on the Stage 1 Real Analysis 1 module. You will extend your knowledge of functions of one real variable, look at series, and study functions of several real variables and their derivatives. Outline syllabus includes: Continuity and uniform continuity of functions of one variable; Sequences of functions; Series; The Riemann integral; Functions of several variables; Differentiation of functions of several variables; Extrema; Inverse function and Implicit function theorems. Read more |
15 |

MA5514 - Rings and Fields
Can we square a circle? Can we trisect an angle? These two questions were studied by the Ancient Greeks and were only solved in the 19th century using algebraic structures such as rings, fields and polynomials. In this module, we introduce these ideas and concepts and show how they generalise well-known objects such as integers, rational numbers, prime numbers, etc. The theory is then applied to solve problems in Geometry and Number Theory. This part of algebra has many applications in electronic communication, in particular in coding theory and cryptography. Read more |
15 |

MA5504 - Lagrangian and Hamiltonian Dynamics
This module will present a new perspective on Newton's familiar laws of motion. First we introduce variational calculus with applications such as finding the paths of shortest distance. This will lead us to the principle of least action from which we can derive Newton's law for conservative forces. We will also learn how symmetries lead to constants of motion. We then derive Hamilton's equations and discuss their underlying structures. The formalisms we introduce in this module form the basis for all of fundamental modern physics, from electromagnetism and general relativity, to the standard model of particle physics and string theory. Read more |
15 |

MA566 - Number Theory
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. Read more |
15 |

MA5507 - Mathematical Statistics
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 MA348 and MA349. 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, the method of maximum likelihood, is 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; Transformations of random variables; Sampling distributions; Point and interval estimation; Properties of estimators; Maximum likelihood; Hypothesis testing; Neyman-Pearson lemma; Maximum likelihood ratio test. Read more |
15 |

MA5509 - Numerical Methods
This module is an introduction to the methods, tools and ideas of numerical computation. In mathematics, one often encounters standard problems for which there are no easily obtainable explicit solutions, given by a closed formula. Examples might be the task of determining the value of a particular integral, finding the roots of a certain non-linear equation or approximating the solution of a given differential equation. Different methods are presented for solving such problems on a modern computer, together with their applicability and error analysis. A significant part of the module is devoted to programming these methods and running them in MATLAB. Read more |
15 |

MA5512 - Ordinary Differential Equations
This module introduces the basic ideas to solve certain ordinary differential equations, like first order scalar equations, second order linear equations and systems of linear equations. It mainly considers their qualitative and analytical aspects. Outline syllabus includes: First-order scalar ODEs; Second-order scalar linear ODEs; Existence and Uniqueness of Solutions; Autonomous systems of two linear first-order ODEs. Read more |
15 |

MA5501 - Applied Statistical Modelling 1
Constructing suitable models for data is a key part of statistics. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. Even if the temperature and pressure are fixed, there will be variation in the yield which motivates the use of a statistical model which includes a random component. In this module, we study how suitable models can be constructed, how to fit them to data and how suitable conclusions can be drawn. Both theoretical and practical aspects are covered, including the use of R. Read more |
15 |

MA5502 - Curves and Surfaces
The main aim of this module is to give an introduction to the basics of differential geometry, keeping in mind the recent applications in mathematical physics and the analysis of pattern recognition. Outline syllabus includes: Curves and parameterization; Curvature of curves; Surfaces in Euclidean space; The first fundamental form; Curvature of surfaces; Geodesics. Read more |
15 |

### Stage 3

Modules may include | Credits |
---|---|

MA569 - Starting Research in the Mathematical Sciences
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. Read more |
15 |

MA549 - Discrete Mathematics
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. Read more |
15 |

MA587 - Numerical Solution of Differential Equations
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. Read more |
15 |

MA595 - Graphs and Combinatorics
Combinatorics is a field in mathematics that studies discrete, usually finite, structures, such as graphs. It not only plays an important role in numerous parts of mathematics, but also has real world applications. In particular, it underpins a variety of computational processes used in digital technologies and the design of computing hardware. Among other things, this module provides an introduction to graph theory. Graphs are discrete objects consisting of vertices that are connected by edges. We will discuss a variety of concepts and results in graph theory, and some fundamental graph algorithms. Topics may include, but are not restricted to: trees, shortest paths problems, walks on graphs, graph colourings and embeddings, flows and matchings, and matrices and graphs. In addition to graphs, the module may cover other topics in combinatorics such as: problems in extremal set theory, enumerative problems, Principle of Inclusion and Exclusion, and, for M-level students, Ramsey theory, computational complexity and the P versus NP problem. Read more |
15 |

MA603 - Introduction to Lie Groups and Lie Algebras
Lie groups and their associated Lie algebras are studied by both pure and applied mathematicians and by physicists; this is a topic renowned for both its mathematical beauty and its immense utility. Lie groups include translation, rotation and scaling groups as well as unitary, symplectic and special linear matrix groups. We will study in detail the lower dimensional groups that arise in many applications, and more general theory such as the structure of their associated Lie algebras. Special topics include a look at the lowest dimensional exceptional Lie group G2, and Lie group actions and their invariants. Read more |
15 |

MA605 - Symmetries, Groups and Invariants
In this module we will study certain configurations with symmetries as they arise in real world applications. Examples include knots described by "admissible diagrams" or chemical structures described by colouring patterns. Different diagrams and patterns can describe essentially the same structure, so the problem of classification up to equivalence arises. This will be solved by attaching invariants which are then put in normal form to distinguish them. The syllabus will be as follows: (a) Review of basic methods from linear algebra, group theory and discrete mathematics; (b) Permutation groups, transitivity, primitivity, Burnside formula; (c) Finitely generated Abelian groups; (d) Applications to knot theory, Reidemeister moves, the Abelian knot group; (e) Examples, observations, generalizations and proofs. Read more |
15 |

MA617 - Asymptotics and Perturbation Methods
The lectures will introduce students to asymptotic and perturbation methods for the approximate evaluation of integrals and to obtain approximations for solutions of ordinary differential equations. These methods are widely used in the study of physically significant differential equations which arise in Applied Mathematics, Physics and Engineering. The material is chosen so as to demonstrate a range of mathematical techniques available and to illustrate some different applications which are amenable to such analysis. Read more |
15 |

MA636 - Stochastic Processes
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. Read more |
15 |

MA639 - Time Series Modelling and Simulation
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. Read more |
15 |

MA6524 - Metric and Normed Spaces
Many fundamental concepts and results in mathematical analysis in real and complex spaces rely on the notion of `being close'. It turns out that one can do mathematical analysis in a much wider context as long as there is a distance (or metric) that provides a way to measure closeness. Such spaces are called metric spaces and include the important class of normed spaces. In this module you will be introduced to theory and applications of metric and normed spaces. Much of the theory was developed in the previous century and has been a driving force in modern analysis. Read more |
15 |

MA6544 - Nonlinear Systems and Applications
This module will give an introduction to nonlinear ordinary differential equations and difference equations. Such ordinary differential equations and difference equations have a variety of applications such as Mathematical Biology and Ecology. The emphasis will be on developing an understanding of ordinary differential equations and difference equations and using analytical and computational techniques to analyse them. Topics include: phase plane, equilibria and stability analysis; periodic solutions and limit cycles; Poincare-Bendixson theorem; dynamics of difference equations: cobwebs, equilibria, stability and periodic solutions; the discrete logistic model and chaos. The material is chosen so as to demonstrate the range of modern analytical and computational techniques available for solving nonlinear ordinary differential equations and difference equations and to illustrate the many different applications which are modelled by such equations. A range of Mathematical tools are drawn together to study the nonlinear equations, including computation through the use of MAPLE. Read more |
15 |

MA567 - Topology
This module is an introduction to point-set topology, a topic that is relevant to many other areas of mathematics. In it, we will be looking at the concept of topological spaces and related constructions. In an Euclidean space, an "open set" is defined as a (possibly infinite) union of open "epsilon-balls". A topological space generalises the notion of "open set" axiomatically, leading to some interesting and sometimes surprising geometric consequences. For example, we will encounter spaces where every sequence of points converges to every point in the space, see why for topologists a doughnut is the same as a coffee cup, and have a look at famous objects such as the Moebius strip or the Klein bottle. Read more |
15 |

MA568 - Orthogonal Polynomials and Special Functions
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. Read more |
15 |

MA572 - Complex Analysis
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. Read more |
15 |

MA574 - Polynomials in Several Variables
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. Read more |
15 |

### Stage 4

Modules may include | Credits |
---|---|

MA578 - Dissertation for MMath Mathematics
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. Read more |
30 |

MA561 - Introduction to Lie Groups and Algebras
Lie groups and their associated Lie algebras are studied by both pure and applied mathematicians and by physicists; this is a topic renowned for both its mathematical beauty and its immense utility. Lie groups include translation, rotation and scaling groups as well as unitary, symplectic and special linear matrix groups. We will study in detail the lower dimensional groups that arise in many applications, and more general theory such as the structure of their associated Lie algebras. Special topics include a look at the lowest dimensional exceptional Lie group G2, and Lie group actions and their invariants. Read more |
15 |

MA7524 - Metric and Normed Spaces
Many fundamental concepts and results in mathematical analysis in real and complex spaces rely on the notion of `being close'. It turns out that one can do mathematical analysis in a much wider context as long as there is a distance (or metric) that provides a way to measure closeness. Such spaces are called metric spaces and include the important class of normed spaces. In this module you will be introduced to theory and applications of metric and normed spaces. Much of the theory was developed in the previous century and has been a driving force in modern analysis. As more advanced topics we might discuss the extension of continuous functions from subsets to the whole space, famous fixed point theorems and the astonishing existence of continuous functions which are nowhere differentiable. Read more |
15 |

MA7526 - Orthogonal Polynomials and Special Functions
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 and a discussion of elliptic and theta functions. Read more |
15 |

MA871 - Asymptotics and Perturbation Methods
The lectures will introduce students to asymptotic and perturbation methods for the approximate evaluation of integrals and to obtaining approximations for solutions of ordinary differential equations. These methods are widely used in the study of physically significant differential equations which arise in Applied Mathematics, Physics and Engineering. The material is chosen so as to demonstrate a range of mathematical techniques available and to illustrate some different applications which are amenable to such analysis. Asymptotics. Ordering symbols. Asymptotic sequences, expansions and series. Differentiation and integration of asymptotic expansions. Dominant balance. Solution of algebraic and transcendental equations. Asymptotic evaluation of integrals. Integration by parts. Laplace's method and Watson's lemma. Method of stationary phase. Approximate solution of linear differential equations. Classification of singular points. Local behaviour at irregular singular points. Asymptotic expansions in the complex plane. Stokes phenomena: Stokes and anti-Stokes lines, dominance and sub-dominance. Connections between sectors of validity. Airy functions. Matched asymptotic expansions. Regular and singular perturbation problems. Asymptotic matching. Boundary layer theory: inner, outer and intermediate expansions and limits. Uniform approximation. WKB method. Schrödinger equation and Sturm-Liouville problems. Turning points. Multiple scales analysis and related methods. Secular terms. Multiple scales method. Method of strained coordinates (Lindstedt-Poincaré method). Read more |
15 |

MA962 - Geometric Integration
The equations studied in this module will be ordinary differential systems, especially Hamiltonian systems. The aim of this subject area is to obtain and study numerical solutions of these systems that preserve specific qualitative and geometric properties. For certain differential equations, these geometric methods can be far superior to standard numerical methods. The syllabus includes: A review of basic numerical methods, variational methods and Hamiltonian mechanics; Properties that numerical methods can preserve (first integrals, symplecticity, time reversibility); Geometric numerical methods (modified Euler and Runge-Kutta methods, splitting methods); Use and misuse of the various notions of error. Read more |
15 |

MA964 - Applied Algebraic Topology
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. Read more |
15 |

MA965 - Symmetries, Groups and Invariants
In this module we will study certain configurations with symmetries as they arise in real world applications. Examples include knots described by "admissible diagrams" or chemical structures described by "colouring patterns". Different diagrams and patterns can describe essentially the same structure, so the problem of classification up to equivalence arises. This will be solved by attaching "invariants" which are then put in "normal form" to distinguish them. The syllabus will be as follows: (a) Review of basic methods from linear algebra, group theory and discrete mathematics; (b) Permutation groups, transitivity, primitivity, Burnside formula; (c) Finitely generated Abelian groups; (d) Applications to knot theory, Reidemeister moves, the Abelian knot group; (e) Examples, observations, generalizations and proofs; (f) General Polya-enumeration (as an extension of the Burnside formula). Read more |
15 |

MA995 - Graphs and Combinatorics
Combinatorics is a field in mathematics that studies discrete, usually finite, structures, such as graphs. It not only plays an important role in numerous parts of mathematics, but also has real world applications. In particular, it underpins a variety of computational processes used in digital technologies and the design of computing hardware. Among other things, this module provides an introduction to graph theory. Graphs are discrete objects consisting of vertices that are connected by edges. We will discuss a variety of concepts and results in graph theory, and some fundamental graph algorithms. Topics may include, but are not restricted to, trees, shortest paths problems, walks on graphs, graph colourings and embeddings, flows and matchings, and matrices and graphs. In addition to graphs the module may cover other topics in combinatorics such as Ramsey theory, problems in extremal set theory, enumerative problems, Principle of Inclusion and Exclusion, and the P versus NP problem. Read more |
15 |

## Teaching and assessment

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. Lectures are given by a wide variety of lecturers, all with different research backgrounds.

Assessment is carried out by means of: coursework involving problems; computer assignments; projects; tests; and written examinations. MMath students complete a dissertation as part of their Stage 4 studies.

### 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 solve 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 probability 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
- calculation and manipulation of the material within the programme
- the application of 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-specific 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 online computer searches
- information technology skills such as word-processing, spreadsheet use and internet communication
- personal and interpersonal skills needed to 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 open up a wide variety of careers. MMath Mathematics graduates typically find employment in areas involving applications of the subject or 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 three-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 Mathematics (not Use of Mathematics). Either General Studies or Critical Thinking (but not both) can be accepted against the requirements. |

Access to HE Diploma | The University will not necessarily make conditional offers to all Access candidates but will continue to assess them on an individual basis. If we make you an offer, you will need 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 for further advice on your individual circumstances. |

International Baccalaureate | 34 points overall or 17 points at HL including Mathematics 6 at HL |

### International students

The University welcomes applications from international students. Our international recruitment team can guide you on entry requirements. See our International Student website for further information about entry requirements for your country.

If you need to increase your level of qualification ready for undergraduate study, we offer a number of International Foundation Programmes.

#### Meet our staff in your country

For more advice about applying to Kent, you can meet our staff at a range of international events.

#### 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.

Please note that if you are required to meet an English language condition, we offer a number of pre-sessional courses in English for Academic Purposes through Kent International Pathways.

### General entry requirements

Please also see our general entry requirements.

## Fees

The 2018/19 regulated UK/EU tuition fees have not yet been set. The University intends to set fees at the maximum permitted level for new and returning UK/EU students. Please see further information below.

As a guide only the 2017/18 full-time UK/EU tuition fees for this programme are £9,250 unless otherwise stated:

UK/EU | Overseas | |
---|---|---|

Full-time |
TBC | £15200 |

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.*

### Your fee status

The University will assess your fee status as part of the application process. If you are uncertain about your fee status you may wish to seek advice from UKCISA before applying.

### General additional costs

Find out more about accommodation and living costs, plus general additional costs that you may pay when studying at Kent.

## Funding

#### University funding

Kent offers generous financial support schemes to assist eligible undergraduate students during their studies. See our funding page for more details.

#### Government funding

You may be eligible for government finance to help pay for the costs of studying. See the Government's student finance website.

### Scholarships

#### 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.

For 2018/19 entry, the scholarship will be awarded to any applicant who achieves a minimum of AAA over three A levels, or the equivalent qualifications (including BTEC and IB) as specified on our scholarships pages.

The scholarship is also extended to those who achieve AAB at A level (or specified equivalents) where one of the subjects is either Mathematics or a Modern Foreign Language. Please review the eligibility criteria.