This has been an incredibly busy and hectic year, but overall an extraordinarily rewarding one. I will catalogue some of what I did here in case anyone is interested, and point out some of the things that will hopefully happen early next year. I do not know how often I plan to blog going forward, but I think having at the very least one reminiscence a year about things that I have been involved in is useful.
Almost on a whim, I applied for a postdoctoral position early in 2017, which meant that I frantically finished my DPhil (PhD). I submitted my thesis in July, and had my viva (defense) in September. Due to some delays I have not yet been awarded the degree officially, but I have been pushing on ahead as if it was all behind me. While I did really enjoy the field that I studied (bioactive porous media, with an eye on tissue engineering applications), I anticipate that once some of the work in the thesis has been published, I will primarily work in other areas, at least for some time. Two arXiv preprints have been posted related to these, although due to reviewer comments some details in these versions will likely change. A modelling paper can be found here, and a dynamical systems paper can be found here. I have now become very preoccupied with several other areas of mathematical biology, and a few areas of physics.
In the summer of 2016, I co-supervised two summer students with Robert Van Gorder. These students considered ecological models, both of which incorporated dispersal of populations throughout a continuous spatial domain. One student studied interactions between predators, prey, and a subsidy, and found some interesting clustering behaviours induced by heterogeneous subsidy input in the domain. This heterogeneity mimics many biological scenarios where a predator subsists off of prey and subsidy in different geographical regions. We have continued to explore these kinds of models over the past year in terms of incorporating additional temporal and spatial effects, so I may discuss these in a later post once they have been published.
Another student considered a generalized Lotka-Volterra type model in a two dimensional domain with both random dispersal (e.g. diffusion) and directed motion of populations. This directed motion arise from environmental effects as well as inter- and intra-species interactions. This second kind of movement was modelled as an advection toward favorable regions of the domain, or away from unfavorable regions. This model then consisted of coupled reaction-advection-diffusion equations, with additional elliptic equations determining the direction and magnitude of the advection. While this model was mathematically more complicated than classical reaction-diffusion equations modelling population dispersal, we showed that it did not give rise to spatial clustering of the populations in a spatially homogeneous domain (subject to some technical restrictions). So we investigated explicit spatial heterogeneity, corresponding to sources of food, or hazards within the domain, and found some interesting interactions between spatial heterogeneity and the advection of these interacting populations. We are also pursuing several other directions from this project, which mathematically correspond to understanding the role of advection in reaction-diffusion equations (as well as coupled elliptic-parabolic systems), and biologically correspond to understanding the directed motion of populations. Some of these extensions have resulted in projects involving waves in biological media, investigation of Turing instabilities in this more general setting, and emergence of patterns on a variety of manifolds.
In addition to these extensions, I have also been involved in a Saturday study group that has explored a variety of applied mathematics projects. We started by considering models for chaotic rotation of rigid bodies orbiting more massive objects, motivated by chaotic rotation observed in some of Pluto’s recently discovered moons (this is a good visualization of the rotation of Nix). Using a Melnikov function approach, we could analytically demonstrate that chaos will occur in a particular idealization of this problem, and we confirmed this numerically via Poincaré sections (see the paper here if you’re interested). While Robert originally suggested this project, a large amount of the modelling and the analysis was developed by a fellow graduate student, James Kwiecinski. This was intended to be a one-off kind of project, but has now spawned at least one interesting extension which I may discuss early next year.
We also studied models involving coupled Complex Ginzburgh-Laundau Equations. These arise in a huge variety of physical contexts, from superconductivity, to quantum field theory and nonlinear optics, and also give rise to a plethora of spatiotemporal behaviours. Despite being well studied, there are still many interesting phenomena that have not been thoroughly explored, and so we analytically and numerically investigated a particular generalization of the model (which can be thought of as a generalization of the cubic nonlinear Schrödinger equation). Our study was posted to the arXiv here, and along with another paper, was submitted to a journal last year.
There are a few other things that I was involved in last year, including student supervision of projects involving stochastic epidemiological models, as well as projects involving robots, but overall the above gives a flavour of the sorts of things that I have been up to. My postdoctoral work fits into a similar class of problems as those described above, and involves spatial and temporal heterogeneity in reaction-diffusion systems. It is especially nice as there are several theoretical projects, as well as an excellent collaboration with an experimental group in Edinburgh which is already producing some surprising connections (see here for some neat videos of size and wavelength modulation of spots in a growing domain). Once some of these projects have concluded, I hope to give a brief nontechnical summary of the results later next year.
I am incredibly indebted to all of the people who have helped me get to where I am now – my PhD and postdoc supervisors, my fellow students and colleagues, and my incredible wife and family have all made this an extraordinarily successful year. I plan to continue with the momentum from this year, and hopefully be able to give back to the world something useful for everything that I have been given. See you all in 2018!