Director of Studies for 100-level Mathematics
I obtained BS (2009) and PhD (2014) degrees in mathematics from the University of Central Florida (UCF), USA, where I held a Trustees Doctoral Fellowship (2009-2011) and then a National Science Foundation Graduate Research Fellowship (2011-2014). After completion of my PhD, I moved to the Mathematical Institute at the University of Oxford, UK, and took up the post of Research Fellow in Nonlinear Dynamics (2014-2015), followed by a Glasstone Research Fellowship in Science (2015-2018). In 2019, I joined the faculty of the Department of Mathematics and Statistics at the University of Otago as a Senior Lecturer.
Robert's teaching responsibilities include:
- MATH 140 Fundamentals of Modern Mathematics 2
- MATH 302 Complex Analysis
- MATH424 Techniques in Applied Mathematics 1
- MATH426 Techniques in Applied Mathematics 2
In my research I seek to better understand how physical phenomena can be described, predicted, and even modified using tools from applied mathematics. I specialize in the trifecta of mathematical modelling (writing down relations between physical quantities within the language of mathematics... this is where all of those equations come from!), analytical and asymptotic solution methods (deriving an exact or approximate solution to a mathematical problem using pen-and-paper approaches), and numerical simulations (using a computer program to solve a mathematical problem). I apply these methods to study physical phenomena, primarily those falling into one of the following areas:
If you're alive, you're interacting with fluids: The air you breathe and the water you drink are examples of fluids. Despite the ubiquitous nature of fluids, there are still many questions we have about their behavior. My interests in fluids include better understanding the dynamics governing fundamental structures – such as vortices, bubbles, waves, and boundary layers – that can then be used as building blocks for more complex fluid flows. What forms can these structures take? Do they persist, or break apart over time? What happens if we attempt to manipulate or control these structures? These are the kinds of questions I am interested in.
Spatial instabilities and pattern formation
Diffusive instabilities – such as the Turing and Benjamin-Feir instabilities – have been proposed as mechanisms for the formation of patterns in many real-world systems, ranging from spot formation on the coats of big cats to optical turbulence in lasers. I am interested in understanding how these instability mechanisms extended to more generic non-autonomous or spatially heterogeneous systems, where they result in messier yet perhaps more realistic patterning. These heterogeneous systems arise from models of reaction-diffusion processes such as chemical reactions in the presence of thermal forcing, fluid flows, or evolving space domains, and are found in applications ranging from physics and chemistry to biology and epidemiology. Under what conditions can spatial or spatiotemporal patterns emerge from such systems? How might these patterns be controlled?
Quantum theory, quantum fluids, and nonlinear waves
My interests in theoretical physics include quantum mechanics and quantum field theory, with a particular focus on modelling problems in low-temperature physics and condensed matter physics as quantum fluids. Some of my work in this area has involved understanding the dynamics Bose-Einstein condensates, quantized vortex filaments in superfluid helium, and confined quantum systems. While these topics are farther removed from our daily experiences, they exhibit exotic and interesting behaviors, providing a number of fascinating scientific challenges. Nonlinear waves arise in these and other applications, and I am interested in understanding how nonlinear waves evolve under various conditions relevant to realistic experimental configurations, such as confined space domains or external forcing.
I am happy to supervise PhD and honours projects in any of these areas.
Van Gorder, R. A. (2023). Quantum mechanics on time-varying space domains. Proceedings of the Royal Society A, 479, 20220759. doi: 10.1098/rspa.2022.0759
Van Gorder, R. A., & Fisher, P. A. (2023). Spatial eigenvalue problems for stars in hydrostatic equilibrium: Generalized Lane–Emden equations as boundary value problems. Monthly Notices of the Royal Astronomical Society, 523(2), 2059-2073. doi: 10.1093/mnras/stad1506
Van Gorder, R. A. (2022). Compressed hydrogen atoms confined within generic boxes. Proceedings of the Royal Society A, 478(2268), 20220467. doi: 10.1098/rspa.2022.0467
Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & Van Gorder, R. A. (2022). Modelling and analysis of an endothermic reacting counter-current flow. Journal of Fluid Mechanics, 949, A21. doi: 10.1017/jfm.2022.702
Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., Valderhaug, A. M., & Van Gorder, R. A. (2022). Modelling alternating current effects in a submerged arc furnace. IMA Journal of Applied Mathematics, 87, 492-520. doi: 10.1093/imamat/hxac012