Activities
Representative research, teaching and outreach activities are described below.
Research 1
Bright students with a genuine interest in pursuing a PhD on one of the topics listed below are encouraged to email me. Please provide a brief description of which topic interests you and why, and how your particular set of skills will enable you to make a solid contribution. A range of scholarship opportunities are available.
Geometric Design of Numerical Algorithms for Signal Processing
The aim is to apply the mathematical tools of (algebraic and differential) topology and geometry to the perennial problem of designing better numerical algorithms for signal processing. These mathematical tools can be applied in two ways. One way is to seek generalisations of algorithms for problems in Euclidean space to problems on curved surfaces. This motivates our work on optimisation, parameter estimation and filtering on manifolds (with potential future extensions to algebraic varieties). The other way we apply these tools is to study the geometry of a family of problems as a whole, and use knowledge of the geometry of the family to find efficient algorithms for solving any particular problem in that family. This is particularly relevant for a wide variety of optimisation problems arising in signal processing which we term "real-time optimisation" problems. (A more detailed explanation can be found on the Research Goals webpage of the NSP Lab site.)
This area may appeal to students with a mathematics or engineering undergraduate degree wishing to work at the intersection of pure and applied mathematics and statistics. Students will have the opportunity, should they so desire, to learn advanced mathematical techniques from areas such as differential topology, differential geometry, information geometry, stochastic differential equations on manifolds, and algebraic topology. Furthermore, there are strong connections with the following research area.
Control and Optimisation of Large-Scale Distributed Systems
From power and water distribution networks to swarms of autonomous vehicles and disaster management, the control and optimisation of large-scale distributed systems is a current "hot topic" in control and signal processing. New paradigms will be required to deal with communication issues (limited bandwidth, delays in communication, outages), uncertainties (changes to the connectivity of the system, potential faults) and the lack of a central authority to monitor the system from above.
The broad nature of this area means there are challenges on both the fundamental and the applied sides of research, hence this area may likewise appeal to a broad range of students, from those wishing to build and experiment with distributed systems in the lab through to those wishing to study mathematically how, for example, the geometry of a distributed system can influence its properties such as stability and robustness to faults.
Architectures for Biological and Artificial Computation
The fundamental theories underpinning control and signal processing are applicable to any system. Therefore, systems in nature, such as neuronal networks, should be understood in terms of these theories. For this to happen though, these theories need to be developed further in a number of different directions. Multi-scale computational modelling of actual biological systems is an important subcomponent of this work, providing insight into how current theories should be extended. Additional information can be found on the Multi-scale Computational Modelling of Biological Systems on NSP Lab website.
This area may appeal to students with an interest in how nature computes and/or who wish to explore new paradigms for computation and control in engineering.
Teaching
I teach two informal graduate level courses. Both are designed to add to the breadth of our department by being outside of what is normally taught in engineering, and both are premised on the importance of rigorous mathematics in engineering. Neither course has an end; once one textbook is finished, another is found. The first course aims to strengthen the foundations and make it apparent that just because a particular topic is frequently taught at the undergraduate level does not mean all aspects of it are trivial. The second course aims to choose topics strategically, topics that a student or early career researcher may not have been exposed to before but which could prove exceedingly useful in control and signal processing. Examples of the books that have been used for these two courses are: Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra, A Topological Introduction to Nonlinear Analysis and Differential Topology.
For smaller topics, I have published some notes on my blog.
Outreach
Informing students at high school, and even possibly in primary school, why science (especially mathematics) and engineering are interesting and useful, will help students decide what they would like to do in life. I'm currently writing a non-traditional textbook on high school mathematics that endeavours to portray what mathematics is truly about. Other outreach activities are described on the Control and Signal Processing Lab webpage.