## MSc, MPhil, PhD - Mathematics

University of Kent, School of Mathematics, Statistics & Actuarial Science

## MSc, MPhil, PhD - Mathematics

*At SMSAS you could be joining a vibrant research community of almost 100 postgraduate and postdoctoral researchers and academic staff.*

You would have the opportunity to engage with a very wide range of research topics within a well-established system of support and training, with a high level of contact between staff and research students.

A very active research seminar programme further enhances the Mathematics research experience.

## Course structure

The research interests of the Mathematics Group cover a wide range of topics following our strategy of cohesion with diversity. The areas outlined provide focal points for these varied interests.

## 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.

## Fees

As a guide only, the 2014/15 annual tuition fees for this programme are:

**Mathematics - PhD at Canterbury:**

- Full-time: £3996 UK/EU; £12450 Overseas
- Part-time: £1998 UK/EU; £6240 Overseas

**Mathematics - MPhil at Canterbury:**

- Full-time: £3996 UK/EU; £12450 Overseas
- Part-time: £1998 UK/EU; £6240 Overseas

**Mathematics - MSc at Canterbury:**

- Full-time: £3996 UK/EU; £12450 Overseas
- Part-time: £1998 UK/EU; £6240 Overseas

## Key facts

- Subject area: Mathematics
- Award: MSc, MPhil, PhD
- Course type: Research
- Location: Canterbury
- Mode of study: Full-time or part-time
- Attendance mode: Campus
- Duration: MSc one year full-time, two years part-time; MPhil two years full-time, three years part-time; PhD three to four years full-time, five to six years part-time
- Start: September

** This school offers programs in:**

- English