**Studying Mathematics at postgraduate level gives you a chance to begin your own research, develop your own creativity and be part of a long tradition of people investigating analytic, geometric and algebraic ideas. **

## Overview

This programme allows you to further enhance your knowledge, creativity and computational skills in core mathematical subjects and their applications giving you a competitive advantage in a wide range of mathematically based careers. The modules, which are designed and taught by internationally known researchers, are accessible, relevant, interesting and challenging.

### About the School of Mathematics, Statistics and Actuarial Science (SMSAS)

The School has a strong reputation for world-class research as indicated by our results in the latest Research Excellence Framework (REF). Postgraduate students develop analytical, communication and research skills. Developing computational skills and applying them to mathematical problems forms a significant part of the postgraduate training in the School.

The Research Excellence Framework (REF) 2014 rated 72% of the School's research output as International Quality or above. The School is in the top 25 in the UK when considered on weighted GPA.

The Mathematics Group also has an excellent track record of winning research grants from the Engineering and Physical Sciences Research Council (EPSRC), the Royal Society, the EU, the London Mathematical Society and the Leverhulme Trust.

### National ratings

In the Research Excellence Framework (REF) 2014, research by the School of Mathematics, Statistics and Actuarial Science was ranked 25th in the UK for research power and 100% or our research was judged to be of international quality.

An impressive 92% of our research-active staff submitted to the REF and the School’s environment was judged to be conducive to supporting the development of world-leading research.

## Course structure

At least one modern application of mathematics is studied in-depth by each student. Mathematical computing and open-ended project work forms an integral part of the learning experience. There are opportunities for outreach and engagement with the public on mathematics.

You take eight modules in total: six from the list below; a short project module and a dissertation module. The modules concentrate on a specific topic from: analysis; applied mathematics; geometry; and algebra.

### Modules

The following modules are indicative of those offered on this programme. This list 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 modules from other programmes so that you may customise your programme and explore other subject areas that interest you.

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

MA7503 - Communicating Mathematics
The aim of this module is to equip students with the skills needed to communicate efficiently the findings of a recent piece of research. This module is supported by a series of workshops covering various forms of written and oral communication. Each student will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their three coursework assessments which include a scientific writing assessment and an oral presentation. 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 |

MA7532 - Topology
The module is intended to serve as an introduction to point-set topology, focusing on examples and applications. This will also enhance other modules by providing examples and concepts relevant to Functional Analysis, Algebra and Mathematical Physics. The syllabus will include but is not restricted to topics from the following list: Basic definitions and examples (Euclidean and discrete spaces and non-metrizable examples such as the finite complement topology) Continuity and convergence in general topological spaces (especially related to the examples above) Product topology, subspace topology, quotient topology (including real and complex projective spaces) Compactness, including comparing different characterisations of compactness Homotopy and paths Homeomorphisms and homotopy equivalence, contractibility Connectedness and path-connectedness Winding number Fixed point theorems In addition, for level 7 students: Advanced topic such as a topological proof of the Fundamental Theorem of Algebra; simply connected spaces. Read more |
15 |

MA7544 - Nonlinear Systems and Applications
Scalar autonomous nonlinear first-order ODEs. Review of steady states and their stability; the slope fields and phase lines. Autonomous systems of two nonlinear first-order ODEs. The phase plane; Equilibra and nullclines; Linearisation about equilibra; Stability analysis; Constructing phase portraits; Applications. Nondimensionalisation. Stability, instability and limit cycles. Liapunov functions and Liapunov's theorem; periodic solutions and limit cycles; Bendixson's Negative Criterion; The Dulac criterion; the Poincare-Bendixson theorem; Examples. Dynamics of first order difference equations. Linear first order difference equations; Simple models and cobwebbing: a graphical procedure of solution; Equilibrium points and their stability; Periodic solutions and cycles. The discrete logistic model and bifurcations. Level 7 Students only: Further applications of phase portraits and the Poincare-Bendixson theorem; Higher order difference equations. 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 |

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 |

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 |

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 |

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 |

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 |

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 |

MA960 - Dissertation
The dissertation represents the culmination of the students work in the programme. It offers the students the opportunity to carry out a piece of extended independent scholarship, and to show their ability to organise and present their ideas in a coherent and convincing fashion. Students will be expected to discuss possible dissertation topics with academic staff members of the Mathematics group within SMSAS in the spring term. An initial supervision will be arranged in the Spring term during which the student and supervisor will discuss the approaches to the topic and draw up a timetable plan which will include some meetings to discuss progress and areas of difficulty. The supervisor will comment on a draft before submission. The topic of the dissertation will depend on the mutual interests of the student and the student's chosen supervisor. Read more |
60 |

### Teaching and Assessment

Assessment is by closed book examinations, take-home problem assignments and computer lab assignments (depending on the module). The project and dissertation modules are assessed mainly on the reports or work you produce, but also on workshop activities during the teaching term.

### Programme aims

This programme aims to:

- provide a Master’s level mathematical education of excellent quality, informed by research and scholarship
- provide an opportunity to enhance your mathematical creativity, problem-solving skills and advanced computational skills
- provide an opportunity for you to enhance your oral communication, project design and basic research skills
- provide an opportunity for you to experience and engage with a creative, research-active professional mathematical environment
- produce graduates of value to the region and nation by offering you opportunities to learn about mathematics in the context of its application.

### Learning outcomes

#### Knowledge and understanding

You will gain knowledge and understanding of:

- the applications of mathematical theories, methods and techniques
- the power of generalisation, abstraction and logical argument
- the processes and pitfalls of mathematical approximation
- nonlinear phenomena
- geometric thinking
- non-commutative phenomena
- algebraic thinking
- analytic thinking
- mathematical computation.

#### Intellectual skills

You develop intellectual skills in:

- problem solving: the ability to work with self-direction and originality in tackling and solving problems, as well as an ability to provide an analytic approach to mathematical problem-solving
- independent critical reading of technical material
- independent creative mathematical inquiry: to develop an understanding of how techniques of research and enquiry are used to create and interpret mathematical knowledge, to show initiative in the application of knowledge
- logical argument: the ability to formulate detailed rigorous arguments and to deal with complex issues both systematically and creatively.

#### Subject-specific skills

You gain subject-specific skills in:

- mathematical typesetting (LaTeX)
- the ability to speak with clarity to both a mathematical and a non-specialist audience
- symbolic computation (eg Maple)
- numerical computation (eg Matlab).

#### Transferable skills

You will gain the following transferable skills:

- oral and written communication: the ability to communicate technical material, ideas and results to specialist and non-specialist audiences
- project design: to independently plan, implement and complete a project to professional level
- basic research: to be able to select and critically evaluate appropriate material from a variety of sources, be able to use appropriate IT tools, be able to write a literature survey, to investigate a mathematical topic in-depth
- organisational, decision-making, self-management and time-management skills, including the ability to manage your own learning and self-development, and to plan and implement tasks autonomously.

## Careers

A postgraduate degree in Mathematics is a flexible and valuable qualification that gives you a competitive advantage in a wide range of mathematically oriented careers. Our programmes enable you to develop the skills and capabilities that employers are looking for including problem-solving, independent thought, report-writing, project management, leadership skills, teamworking and good communication.

Many of our graduates have gone on to work in international organisations, the financial sector, and business. Others have found postgraduate research places at Kent and other universities.

## Study support

### Postgraduate resources

The University’s Templeman Library houses a comprehensive collection of books and research periodicals. Online access to a wide variety of journals is available through services such as ScienceDirect and SpringerLink. The School has licences for major numerical and computer algebra software packages. Postgraduates are provided with computers in shared offices in the School. The School has two dedicated terminal rooms for taught postgraduate students to use for lectures and self-study.

### Support

The School has a well-established system of support and training, with a high level of contact between staff and research students. There are two weekly seminar series: The Mathematics Colloquium at Kent attracts international speakers discussing recent advances in their subject; the Friday seminar series features in-house speakers and visitors talking about their latest work. These are supplemented by weekly discussion groups. The School is a member of the EPSRC-funded London Taught Course Centre for PhD students in the mathematical sciences, and students can participate in the courses and workshops offered by the Centre. The School offers conference grants to enable research students to present their work at national and international conferences.

### Dynamic publishing culture

Staff publish regularly and widely in journals, conference proceedings and books. Among others, they have recently contributed to: *Advances in Mathematics*; *Algebra and Representation Theory*; *Journal of Physics A*; *Journal of Symbolic Computations*; *Journal of Topology and Analysis*. Details of recently published books can be found within the staff research interests.

### Global Skills Award

All students registered for a taught Master's programme are eligible to apply for a place on our Global Skills Award Programme. The programme is designed to broaden your understanding of global issues and current affairs as well as to develop personal skills which will enhance your employability.

## Entry requirements

A first or second class honours degree in a subject with a significant mathematical content (or equivalent). Students not meeting the entry requirement may take the two year full-time International Master’s in Mathematics and its Applications.

All applicants are considered on an individual basis and additional qualifications, and professional qualifications and experience will also be taken into account when considering applications.

### International students

Please see our International Student website for entry requirements by country and other relevant information for your country.

### English language entry requirements

The University requires all non-native speakers of English to reach a minimum standard of proficiency in written and spoken English before beginning a postgraduate degree. Certain subjects require a higher level.

For detailed information see our English language requirements web pages.

### Need help with English?

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.

## Research areas

### Nonlinear differential equations

The research on nonlinear differential equations primarily studies algorithms for their classification, normal forms, symmetry reductions and exact solutions. Boundary value problems are studied from an analytical viewpoint, using functional analysis and spectral theory to investigate properties of solutions. We also study applications of symmetry methods to numerical schemes, in particular the applications of moving frames.

### Painlevé equations

Current research on the Painlevé equations involves the structure of hierarchies of rational, algebraic and special function families of exact solutions, Bäcklund transformations and connection formulae using the isomonodromic deformation method. The group is also studying analogous results for the discrete Painlevé equations, which are nonlinear difference equations.

### Mathematical biology

Artificial immune systems use nonlinear interactions between cell populations in the immune system as the inspiration for new computer algorithms. We are using techniques of nonlinear dynamical systems to analyse the properties of these systems.

### Quantum integrable systems

Current research on quantum integrable systems focuses on powerful exact analytical and numerical techniques, with applications in particle physics, quantum information theory and mathematical physics.

### Topological solitons

Topological solitons are stable, finite energy, particle-like solutions of nonlinear wave equations that arise due to the general topological properties of the nonlinear system concerned. Examples include monopoles, skyrmions and vortices. This research focuses on classical and quantum behaviour of solitons with applications in various areas of physics including particle, nuclear and condensed matter physics. The group employs a wide range of different techniques including numerical simulations, exact analytic solutions and geometrical methods.

### Algebra and representation theory

A representation of a group is the concrete realisation of the group as a group of transformations. Representation theory played an important role in the proof of the classification of finite simple groups, one of the outstanding achievements of 20th-century algebra. Representations of both groups and algebras are important in diverse areas of mathematics, such as statistical mechanics, knot theory and combinatorics.

### Algebraic topology

In topology, geometry is studied with algebraic tools. An example of an algebraic object assigned to a geometric phenomenon is the winding number: this is an integer assigned to a map of the n-dimensional sphere to itself. The methods used in algebraic topology link in with homotopy theory, homological algebra and modern category theory.

### Invariant theory

Invariant theory has its roots in the classical constructive algebra of the 19th century and motivated the development of modern algebra by Hilbert, Noether, Weyl and others. There are natural applications and interactions with algebraic geometry, algebraic topology and representation theory. The starting point is an action of a group on a commutative ring, often a ring of polynomials on several variables. The ring of invariants, the subring of fixed points, is the primary object of study. We use computational methods to construct generators for the ring of invariants, and theoretical methods to understand the relationship between the structure of the ring of invariants and the underlying representation.

### Financial mathematics

Research includes work on financial risk management, asset pricing and optimal asset allocation, along with models to improve corporate financial management.

## Staff research interests

Full details of staff research interests can be found on the School's website.

### Professor Peter A Clarkson: Professor of Mathematics

Soliton theory, in particular the Painlevé equations, and Painlevé analysis. Asymptotics, Bäcklund transformations, connection formulae and exact solutions for nonlinear ordinary differential and difference equations, in particular the Painlevé equations and discrete Painlevé equations. Orthogonal polynomials and special functions, in particular nonlinear special functions such as the Painlevé equations. Symmetry reductions and exact solutions of nonlinear partial differential equations, in particular using nonclassical and generalized techniques.

View Profile### Dr Clare Dunning: Senior Lecturer in Applied Mathematics

Exactly solvable models in mathematical physics; integrable quantum field theory and spectral theory of ordinary differential equations.

View Profile### Professor Peter Fleischmann: Professor of Pure Mathematics

Representation theory and structure theory of finite groups; constructive invariant theory; applied algebra and discrete mathematics.

View Profile### Dr Steffen Krusch: Lecturer in Applied Mathematics

Topological solitons in mathematical physics, in particular the classical and quantum behaviour of Skyrmions.

View Profile### Dr Stephane Launois: Reader in Pure Mathematics

Non-commutative algebra and non-commutative geometry, in particular, quantum algebras and links with their (semi-)classical counterparts: enveloping algebras and Poisson algebras.

View Profile### Dr Bas Lemmens: Senior Lecturer in Mathematics

Nonlinear (functional) analysis, dynamical systems theory and metric geometry. In particular, the theory of monotone dynamical systems and its applications, and the geometry of Hilbert's metric spaces.

View Profile### Dr Ana F. Loureiro: Lecturer in Mathematics

Orthogonal polynomials; special functions and integral transforms; some aspects of combinatorics and approximation theory.

View Profile### Professor Elizabeth L Mansfield: Professor of Mathematics

Nonlinear differential and difference equations; variational methods; moving frames and geometric integration.

View Profile### Dr Jaideep S Oberoi: Lecturer in Finance

Identification and quantification of liquidity risk in financial markets and the implications of incomplete information for asset price co-variation.

View Profile### Dr Rowena E Paget: Lecturer in Pure Mathematics

Representation theory of groups and algebras, with emphasis on algebras possessing a quasihereditary or cellular structure, such as the group algebras of symmetric groups, Brauer algebras and other diagram algebras.

View Profile### Dr Constanze Roitzheim: Lecturer in Mathematics

Stable homotopy theory, in particular model categories and chromatic homotopy theory; homological algebra; A-infinity algebras.

View Profile### Dr R James Shank: Reader in Mathematics

The invariant theory of finite groups and related aspects of commutative algebra, algebraic topology and representation theory.

View Profile### Dr Huamao Wang: Lecturer in Finance

Developing mathematical models; numerical methods and practical application of portfolio optimisation; derivative pricing and hedging; risk management based on stochastic calculus, optimal control, filtering and simulation.

View Profile### Dr Jing Ping Wang: Reader in Applied Mathematics

Geometric and algebraic properties of nonlinear partial differential equations; test and classification of integral systems and asymptotic normal forms of partial differential equations.

View Profile### Dr Ian Wood: Lecturer in Mathematics

Analysis of PDEs and spectral theory, in particular, the study of spectral properties of non-self adjoint operators via boundary triples and M-functions (generalised Dirichlet-to-Neumann maps), regularity to solutions of PDEs in Lipschitz domains and waveguides in periodic structures.

View Profile### Dr Chris F Woodcock: Senior Lecturer in Pure Mathematics

P-adic analogues of classical functions; commutative algebra; algebraic geometry; modular invariant theory.

View Profile## Fees

The 2018/19 annual tuition fees for this programme are:

Mathematics and its Applications - MSc at Canterbury: | ||

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

Full-time |
£7300 | £15200 |

Part-time |
£3650 | £7600 |

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

### General additional costs

Find out more about general additional costs that you may pay when studying at Kent.

## Funding

Search our scholarships finder for possible funding opportunities. You may find it helpful to look at both:

- University and external funds
- Scholarships specific to the academic school delivering this programme