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## Overview

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

## Entry requirements

### International students

### English language entry requirements

### Need help with English?

## Form

## Course structure

### Modules

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### Teaching and assessment

### Programme aims

### Learning outcomes

#### Knowledge and understanding

#### Intellectual skills

#### Subject-specific skills

#### Transferable skills

## Fees

## Additional costs

### General additional costs

## Funding

## The Complete University Guide

## Independent rankings

## Research areas

### Nonlinear differential equations

### Painlevé equations

### Mathematical biology

### Quantum integrable systems

### Topological solitons

### Algebra and representation theory

### Algebraic topology

### Invariant theory

### Financial mathematics

## Staff research interests

### Professor Peter A Clarkson : Professor of Mathematics

### Dr Clare Dunning : Reader in Mathematics

### Dr Steffen Krusch : Senior Lecturer in Mathematics

### Professor Stephane Launois : Professor of Mathematics

### Dr Bas Lemmens : Reader in Mathematics

### Dr Ana F. Loureiro : Senior Lecturer in Mathematics

### Dr Jaideep S Oberoi : Lecturer in Finance

### Dr Rowena E Paget : Senior Lecturer in Mathematics

### Dr Constanze Roitzheim : Senior Lecturer in Mathematics

### Dr R James Shank : Reader in Mathematics

### Dr Huamao Wang : Lecturer in Finance

### Professor Jing Ping Wang : Professor of Mathematics

### Dr Ian Wood : Senior Lecturer in Mathematics

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

## Careers

## Study support

### Postgraduate resources

### Support

### Dynamic publishing culture

### Global Skills Award

## Apply now

### Apply for entry to:

## Contact us

### United Kingdom/EU enquiries

#### MSc at Canterbury

### International student enquiries

#### School website

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.

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.

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, professional qualifications and relevant experience may also be taken into account when considering applications.

Please see our International website for entry requirements by country and other relevant information. Due to visa restrictions, international fee-paying students cannot study **part-time** unless undertaking a distance or blended-learning programme with no on-campus provision.

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.

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.

**Duration:** 1 year full-time

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 a total of 180 credits, including the two compulsory modules listed below.

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.

Compulsory modules currently include

MA7503 - Communicating Mathematics (15 credits)

There is no specific mathematical syllabus for this module; students will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their coursework assessments (different topics for levels 6 and 7). The coursework is supported by a series of workshops covering various forms of written and oral communication. These may include critically evaluating the following: a research article in mathematics, statistics or finance; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

Optional modules may include

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.

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.

MA995 - Graphs and Combinatorics (15 credits)

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 for level 7 students, the module will cover an advanced topic in combinatorics such as: problems in extremal set theory; enumerative problems; Principle of Inclusion and Exclusion; Ramsey theory; computational complexity; the P versus NP problem.

MA7524 - Metric and Normed Spaces (15 credits)

Metric spaces: Examples of metrics and norms, topology in metric spaces, sequences and convergence, uniform convergence, continuous maps, compactness, completeness and completions, contraction mapping theorem and applications.

Normed spaces: Examples, including function spaces, Banach spaces and completeness, finite and infinite dimensional normed spaces, continuity of linear operators and spaces of bounded linear operators, compactness in normed spaces, Arzela-Ascoli theorem, Weierstrass approximation theorem.

Additional topics, especially for level 7, may include:

• Tietze extension theorem and Urysohn's lemma

• Baire category theorem and applications

• Cantor sets, attractors and chaos

MA7532 - Topology (15 credits)

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

At level 7, topics will be studied and assessed to greater depth.

MA7544 - Nonlinear Systems and Applications (15 credits)

• 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.

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.

MA871 - Asymptotics and Perturbation Methods (15 credits)

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 the Mathematical techniques available and to illustrate some different applications which are amenable to such analysis.

The indicative syllabus is:

• 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.

• 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).

Level 7 Students will study selected topics in greater depth than level 6 students.

MA574 - Polynomials in Several Variables (15 credits)

This module provides a rigorous foundation for the solution of systems of polynomial equations in many variables. In the 1890s, David Hilbert proved four ground-breaking theorems that prepared the way for Emmy Nöther's famous foundational work in the 1920s on ring theory and ideals in abstract algebra. This module will echo that historical progress, developing Hilbert's theorems and the essential canon of ring theory in the context of polynomial rings. It will take a modern perspective on the subject, using the Gröbner bases developed in the 1960s together with ideas of computer algebra pioneered in the 1980s. The syllabus will include

• Multivariate polynomials, monomial orders, division algorithm, Gröbner bases;

• Hilbert's Nullstellensatz and its meaning and consequences for solving polynomials in several variables;

• Elimination theory and applications;

• Linear equations over systems of polynomials, syzygies.

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.

MA6518 - Games and Strategy (15 credits)

In this module we study the fundamental concepts and results in game theory. We start by analysing combinatorial games, and discuss game trees, winning strategies, and the classification of positions in so called impartial combinatorial games. We then move on to discuss two-player zero-sum games and introduce security levels, pure and mixed strategies, and prove the famous von Neumann Minimax Theorem. We will see how to solve zero-sum two player games using domination and discuss a general method based on linear programming. Subsequently we analyse arbitrary sum two-player games and discuss utility, best responses, Nash equilibria, and the Nash Equilibrium Theorem. The final part of the module is devoted to multi-player games and cooperation; we analyse coalitions, the core of the game, and the Shapley value.

Compulsory modules currently include

MA960 - Dissertation (60 credits)

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.

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.

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.

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.

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.

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).

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.

The 2020/21 annual tuition fees for this programme are:

- Home/EU full-time £11000
- International full-time £16200

For details of when and how to pay fees and charges, please see our Student Finance Guide.

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

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

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.

In *The* *Complete University Guide 2020*, the University of Kent was ranked in the top 10 for research intensity. This is a measure of the proportion of staff involved in high-quality research in the university.

Please see the University League Tables 2020 for more information.

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.

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.

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.

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.

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 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.

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.

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 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.

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

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

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

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

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

View ProfileNonlinear (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 ProfileOrthogonal polynomials; special functions and integral transforms; some aspects of combinatorics and approximation theory.

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

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

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

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

View ProfileAnalysis 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 ProfileP-adic analogues of classical functions; commutative algebra; algebraic geometry; modular invariant theory.

View ProfileA 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.

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.

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.

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.

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.

Learn more about the applications process or begin your application by clicking on a link below.

Once started, you can save and return to your application at any time.

T: +44 (0)1227 823254

E: internationalstudent@kent.ac.uk