Current research topics in representation theory and related areas explored via computer algebra systems.

Dirichlet's theorem on primes in an arithmetic progression, Riemann zeta function, distribution of prime numbers.

For people involved in the Mathematical Modelling and Computation program, research opportunities exist in formulation of fault-tolerant numerical schemes, implementation of fault-tolerant schemes on supercomputers, simulation of hardware failure events on ultrascale supercomputers, and application of these techniques to scientific computing.

Scattering theory is about properties of solutions of the eigenfunction equation, usually closely related to the Laplacian.

Sensitivity analysis is an important tool in evaluating model behaviour and assessing which parameters are significant, as well as the interactions between parameters.

Shaping value of information to real world conditions in water decision making

Spectral sequences in algebraic topology

I am interested in using a multi-parameter study of invariants from algebraic topology to do statistical shape analysis. The goal is to quantitatively compare geometric objects such as a set of bones, tumours, leaves, bird beaks, etc. I have both theory and application projects.

The symmetries of a manifold are described by its mapping class group, and the mapping class group (MCG) of a surface is a particularly important object.

The basic problem is to understand under what conditions it is possible to find a convex surface of prescribed Gauss curvature which also satisfies some boundary conditions.

This project aims to characterise the properties of network-packet captures (both batched and streaming), and use a combination of statistical techniques , Fourier and higher order spectral methods and correlation analysis techniques to develop candidate reduced “forward-models” for the network parameters.

Fusion plasmas can support a wide range of electromagnetic waves, ranging from pressure and current gradient driven modes to those driven unstable by fast particle-wave resonance. The diagnosis and control of fusion plasmas is contingent on the accurate modelling, prediction, and reliable measurement of such modes.

The Origin of Life on Earth

This project seeks to provide a fundamental understanding of the process of energetic particle redistribution from the perspective of thermodynamics and entropy.

Possible topics include Legendrian and transverse knot theory, contact geometry in three-manifolds, symplectic field theory, and connections with symplectic geometry.