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.

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

Describe the theoretical properties of methods based on machine learning to solve forward, inverse and operator design problems associated with partial differential equations.

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.

Symplectic vortice and Hamiltonian Gromov-Witten theory.

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.