Students preparing for their graduation ceremony at Canterbury Cathedral

Mathematics and its Applications - MSc


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.



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.


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

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.

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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.

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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.

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

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

• Review of basic results from complex analysis and topology;

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

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

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

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

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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.

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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

• Advanced topic such as a topological proof of the Fundamental Theorem of Algebra; simply connected spaces.

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

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


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

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

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

4. Characters and their basic properties;

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

In addition, for level 7 students:

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

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

The following topics are covered.

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

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

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

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

In addition, for level 7 students:

Advanced topic: This will be selected from the following:

• Symmetry methods for PDEs.

• First integrals and dynamical symmetries.

• Discrete symmetries of ODEs

• Symmetries of difference equations.

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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.

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

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

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

Additional topics, especially for level 7 students may include:

- the Rayleigh quotient and variational characterisations of eigenvalues,

- the functional calculus,

- applications to Sturm-Liouville systems.

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Discrete mathematics has found new applications in the encoding of information. Online banking requires the encoding of information to protect it from eavesdroppers. Digital television signals are subject to distortion by noise, so information must be encoded in a way that allows for the correction of this noise contamination. Different methods are used to encode information in these scenarios, but they are each based on results in abstract algebra. This module will provide a self-contained introduction to this general area of mathematics.

Syllabus: Modular arithmetic, polynomials and finite fields. Applications to

• orthogonal Latin squares,

• cryptography, including introduction to classical ciphers and public key ciphers such as RSA,

• "coin-tossing over a telephone",

• linear feedback shift registers and m-sequences,

• cyclic codes including Hamming,

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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.

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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.

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Combinatorial games, game trees, strategy, classification of positions. Two-player zero-sum games, security levels, pure and mixed strategies, von Neumann's minimax theorem. Solving zero-sum two player games using linear programming. Arbitrary sum games, utility, and matrix games. Nash equilibrium, Nash equilibrium theorem, applications, and cooperation. Multi-player games, coalitions, and the Shapley value.

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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.

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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.


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.


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, 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.  Please note that international fee-paying students cannot undertake a part-time programme due to visa restrictions.

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.

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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.

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Professor Peter Fleischmann: Professor of Pure Mathematics

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

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Dr Steffen Krusch: Lecturer in Applied Mathematics

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

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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.

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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.

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Dr Ana F. Loureiro: Lecturer in Mathematics

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

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Professor Elizabeth L Mansfield: Professor of Mathematics

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

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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.

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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.

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Dr Constanze Roitzheim: Lecturer in Mathematics

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

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Dr R James Shank: Reader in Mathematics

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

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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.

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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.

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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.

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Dr Chris F Woodcock: Senior Lecturer in Pure Mathematics

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

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The 2019/20 annual tuition fees for this programme are:

Mathematics and its Applications - MSc at Canterbury:
UK/EU Overseas
Full-time £10740 £15700

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

General additional costs

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


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