Pure Mathematics – (Ph.D.)

Contents

Areas of interest include nonlinear differential equations, the Painlevé equations, classification of integrable systems, geometric integration, quantum integrable systems and topological solitons, functional analysis, representation theory and invariant theory of finite groups, non-commutative geometry, and computational commutative algebra.

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.

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.