Here I have tried to list the main topics I’ve worked on, am working on, or would like to work on. You will see that things get a bit out of hand if I try to write everything I find interesting, but hopefully this will give you a good idea of what I do! In general, I am interested in nonlinearity and chaos in nature, particularly biology, as well as really interdisciplinary research. Anything that involves bringing together different branches of research knowledge and ways of thinking is exciting, especially when there is a practical application at the end of it.
If you are looking for potential projects, see if any of the following catch your eye. There are possibilities in many directions, so please send me an email to see if there might be something that could work for you. Even if you’re just curious about the possibilities, feel free to ask!
Active Turbulence
Currently my most, aha, active area of research. Active turbulence is the phenomenon of chaotic flows emerging in fluids at very low Reynolds number, i.e. very far from the typical onset of inertial turbulence at high Reynolds number (high velocity or low viscosity). Observed primarily in biological systems such as swarming bacteria or kinesin-microtubule suspensions, this is a fascinating example of chaos emerging in strange places with possible biological relevance. I am interested in understanding the transition to active turbulence; the properties of active turbulent flow, in particular intermittency and how similar or different active turbulence is to classical inertial turbulence; what can dynamical systems approaches tell us about active fluids; how might biological systems such as cells or embryos tame or harness active turbulence? I am also curious about the various “kinds” of turbulence that have been discovered — inertial, elastic, active, odd… Understanding how these fit together, what is common to all these chaotic flows, and how they differ, is a daunting but exciting prospect. There are a whole host of other amazing things to explore in this direction, so I won’t try to list them all!
High Performance Scientific Computing
This one requires perhaps the least explanation. In all my research, I use some form of High Performance Computing (HPC) and data analysis. From making niftier Python scripts to large-scale GPU parallelised PDE solvers, I thoroughly enjoy the process of thinking about complex code and how best to use modern technology to solve difficult problems. Counting floating point operations, trying to jam as many active threads as possible into a CUDA kernel, hacking around to find a “close enough” to optimal arrangement of scripting. I’ve worked in Fortran (mostly modern, but for my sins even Fortran77), C/C++, Matlab (eugh), Python, Julia, CUDA (C and Fortran), Mathematica, and do a bunch of bash scripting to make the magic happen. On the numerical simulation side, I’ve typically focused on simulating 1- or 2D lattices and fluids, using various integration techniques (e.g. symplectic integrators, finite difference and spectral methods, implicit methods). Some of this has been parallelised (CPU or GPU), and some were small enough to stay serial. In terms of data analysis it’s the usual mixed bag of science — lots and lots of statistics and curve fitting, managing terabytes of simulation and experimental data, some image analysis, solving all sorts of bizarrely specific problems, and data visualisation. The advantage is that a lot of the same techniques can be successfully applied to data from chemical reaction simulations, images from developing embryos, and atomic simulations of graphene. This is an exciting time to be a computational scientist, and I am keen to push forward new techniques and higher standards of scientific coding.
DNA Dynamics
The stuff I cut my teeth on as an Honours (fourth-year bachelors) student and on to my PhD. In the past I have studied a Hamiltonian model of DNA (the Peyrard-Bishop-Dauxois [PBD] model), exploring the impact of chaos on various arrangements of base pairs. The biophysics aspect came out more in the study of bubbles, thermally-induced openings in the DNA double strand, posited to have a relationship to transcription. The basic idea is that if DNA is more prone to spontaneously unzip at a certain point, that point should logically be a preferential binding site for the mRNA polymerase when it comes past to initiate transcription. Through many careful simulations, we probed the internal properties of DNA to find typical lengths and lifetimes of these bubbles, and then compared the typical results with those from actual promoter sequences with known transcription start sites. Essentially, the idea holds, with mutations that inhibit transcription resulting in fewer, shorter-lived bubbles upstream, and transcription-enhancing mutations resulting in more, longer-lived bubbles. However, there is a lot more to be explored here, especially in collaboration with experiments and with more complex models, to see how this theory can be effectively used in a bioinformatics context.
Chaos in 2- and 3D Lattices and Materials
For a mathematician or a physicist, probing 2- and 3D lattices rapidly becomes complicated, especially when nonlinearities and/or disorder are involved. In these structures, chaos is the norm, and nice linear behaviour hard to find. In the past I have worked on the chaotic properties of graphene, the supermaterial of the future (TM), and am generally interested in what we can learn about physical properties of these materials from the chaotic dynamics. This extends in various directions, to energy spreading in solid state condensates, to charge transport through lattices, and to active solids where the lattice is maintained out of equilibrium. Using HPC to tackle large-scale simulation limits is one way forward, and finding new links between chaotic dynamics and solid mechanics is another intriguing prospect.
Phase Space Transport and Chemical Reactions
The mathematics of phase space transport provides an incredibly elegant framework for exploring a vast array of physical systems. Given any dynamical process that has several possible initial states (think of reactants in a chemical reaction) and several possible final states (think products), the notions of phase space transport can provide concrete ways of understanding which initial states will result in which final states, given a certain set of conditions. Concretely, given some initial reaction coordinates for a chemical reaction, and the relevant conditions, it may be possible to predict with some certainty what the outcome of the reaction will be. And more importantly, it may be possible to vary certain relevant parameters and predict when these behaviours will change. However, this is only possible in limited cases or where we treat real chemical reactions with simplified models. Combining mathematical tools from phase space transport such as dividing surfaces and invariant manifolds with computational chemistry and free energy surfaces would be a very exciting way of potentially exploring transition states and properties of chemical reactions. What’s more, this provides a wonderful way of seeing fractals resulting in chaos! Fun for the whole family.
Neuroscience
This is a new one for me, but something I’m fascinated to work more on in the future. Neuroscience is a beautiful convergence of practical biology, pure mathematics, biophysics, and dynamical systems. The brain itself is incredibly complex, and scientists use so many different types of models to explain all the various kinds of behaviours that we observe. A long-term goal of mine is to work with people from different corners of the neuro world to find new insights, providing my flavour of dynamical systems and biophysics. How does one correctly model the metabolic aspects of epilepsy across scales, locally at a neuron level up to the entire brain? What would we learn from thinking about ion channels through both a dynamics and a physics lens? Why does my brain work, anyway? These are some of the questions I ask myself.
Maths and Physics in Biology
This is the most general one, because it is hard to put down everything. So this is the catch-all “if it has to do with mathematical or physical biology, especially with dynamics, then I’m interested in it”. From relatively classic things like analysing biological data or constructing models for biophysical processes to ecology and how to think about metabolism mathematically, there are plenty of things that one can do in the pursuit of understanding the incredible world we live in. If you’re reading this and want to discuss more ideas, just send me an email!