## Robert A. Van Gorder

### Bio-sketch:

I obtained BS (2009) and PhD (2014) degrees in mathematics from the University of Central Florida (UCF), 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, New Zealand, as a Senior Lecturer.

### Contact Details:

Physical address: Robert A. Van Gorder, Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand

email: rvangorder at maths dot otago dot ac dot nz

### Employment:

Senior Lecturer, Department of Mathematics and Statistics, University of Otago, New Zealand (2019 - present)

Violette and Samuel Glasstone Research Fellow in Science, Mathematical Institute, University of Oxford, United Kingdom (2015 - 2018)

Research Fellow in Nonlinear Dynamics, Mathematical Institute, University of Oxford, United Kingdom (2014 - 2015)

### Education:

PhD Mathematics (2014); University of Central Florida (UCF), Orlando, Florida, United States

National Science Foundation (NSF) Graduate Research Fellowship (2011-2014)

UCF Trustees Doctoral Fellowship (2009-2011)

B.S. Mathematics (2009); University of Central Florida (UCF), Orlando, Florida, United States

Minor areas of concentration: Economics, Physics, Statistics

### Research and Publications:

I am interested in understanding how physical phenomena can be explained, predicted, and even modified using mathematical methods. I specialize in the trifecta of mathematical modelling (writing down relations between physical quantities within the language of mathematics... *this is where all those equations come from*!), analytical and asymptotic solution methods (deriving an exact or approximate solution to a mathematical problem using *pen-and-paper*), and numerical simulations (using a computer program to solve a mathematical problem). Most of my current research interests lie in the following three areas:

**I. Fluid mechanics, heat and mass transfer**My interests in fluid mechanics include better understanding fundamental dynamics governing the likes of vortex filaments, bubbles, and boundary layer flows. I am also interested in problems involving heat and mass transfer, including problems involving chemical reactions within a flow and phase-change problems. More applied work in this area has been in collaboration with industry partners concerned with such topics as green manufacture of silicon and the roasting of coffee beans.

Selected Recent Publications:

B. M. Sloman, C. P. Please, and R. A. Van Gorder, *Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube*, Journal of Fluid Mechanics 905 (2020) A16.

R. A. Van Gorder, H. Kim, and A. L. Krause, *Diffusive instabilities and spatial patterning from the coupling of reaction-diffusion processes with Stokes flow in complex domains*, Journal of Fluid Mechanics 877 (2019) 759-823.

J. A. Kwiecinski and R. A. Van Gorder, *Dynamics of nearly parallel interacting vortex filaments*, Journal of Fluid Mechanics 835 (2018) 575-623.

R. A. Van Gorder, *Dynamics of the Rayleigh-Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid*, Journal of Fluid Mechanics 807 (2016) 478-508.

R. A. Van Gorder, *Self-similar vortex filament motion under the non-local Biot-Savart model*, Journal of Fluid Mechanics 802 (2016) 760-774.

R. A. Van Gorder, *Helical vortex filament motion under the non-local Biot-Savart model*, Journal of Fluid Mechanics 762 (2015) 141-155.

**II. Instabilities and pattern formation**Pioneering work in the middle of the 20th century lead to the Turing and Benjamin-Feir instabilities, which give routes for spatially homogeneous solutions to transition into spatially heterogeneous patterns or spatiotemporal structures, and have been suggested as mechanisms for the formation of patterns in many systems. My work has involved understanding how these instability mechanisms are properly extended to more generic non-autonomous systems (systems which have an explicit time dependence) and spatially heterogeneous systems (systems which have an explicit dependence on space variables). These kinds of systems arise from studies of reaction-diffusion processes in the presence of thermal forcing, fluid flows, or spatial domains which evolve in time. Biologically motivated applications have included biochemistry, epidemiology, and ecology, and I've ended up supervising a number of students on projects related to these areas. These kinds of systems arise on both continuum (e.g., PDEs) and discrete (networked dynamical systems) contexts.

Selected Recent Publications:

R. A. Van Gorder, *Pattern formation from spatially heterogeneous reaction-diffusion systems*, Philosophical Transactions of the Royal Society A 379 (2021) 20210001.

R. A. Van Gorder, *A theory of pattern formation for reaction-diffusion systems on temporal networks*, Proceedings of the Royal Society A 477 (2021) 20200753.

R. A. Van Gorder, V. Klika, and A. L. Krause, *Turing conditions for pattern forming systems on evolving manifolds*, Journal of Mathematical Biology 82 (2021) 4.

R. A. Van Gorder, *Influence of temperature on Turing pattern formation*, Proceedings of the Royal Society A 476 (2020) 20200356.

R. A. Van Gorder, *Turing and Benjamin-Feir instability mechanisms in non-autonomous systems*, Proceedings of the Royal Society A 476 (2020) 20200003.

**III. Quantum fluids, theoretical and mathematical physics, nonlinear waves**My interests in mathematical physics include quantum mechanics and quantum field theory, with particular application to low-temperature physics and condensed matter physics. Specific applications of interest include Bose-Einstein condensates and dynamics of quantized vortex filaments in superfluid helium. Nonlinear waves feature frequently in mathematical physics, and current interests on this topic involve the control of nonlinear waves through non-autonomous terms, waves which are dissipative, and waves on bounded space domains.

Selected Recent Publications:

R. A. Van Gorder, *Time-varying Bose-Einstein condensates*, Proceedings of the Royal Society A 477 (2021) 20210443.

R. A. Van Gorder, *Perturbation theory for Bose-Einstein condensates on bounded space domains*, Proceedings of the Royal Society A 476 (2020) 20200674.

R. A. Van Gorder, *The Biot-Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid*, Proceedings of the Royal Society A 471 (2015) 20150149.

Past interests have involved the application of specific analytical methods (perturbation theory, homotopy analysis, spectral methods) toward the solution of nonlinear differential equations. Additional work concerned nonlinear dynamics and chaos. Past work in econophysics was concerned with problems in economics (including nonlinear dynamics and optimal control theory), sociology, epidemiology of metapopulations, and game theory. Past interests in fluids involved topics such as boundary layer flows in classical fluids, as well as various applications involving a variety of flow configurations and fluid properties, along with multi-phase flows, fluid-particle systems, and chemical flows, and some of these were in collaboration with industrial partners. I am also occasionally interested in problems in elementary analytic number theory and related mathematics.

These days, my papers appear on Google Scholar quickly after they are published, and you can find a publication list sorted in reverse chronological order here:

Feel free to get in touch if you'd like a copy of a published paper emailed to you.

### Research student supervision:

I've supervised the following doctoral students:

E. K. Luckins, Mathematical modelling of electrical, thermal, and chemical processes occurring within a silicon furnace, 2018 - 2021, DPhil, Oxford InFoMM CDT; co-supervised with J. M. Oliver, C. P. Please; industry sponsor: Elkem. (Placement: Post-Doc at Oxford)

R. González Fariña, Microsilica Particle Formation and Growth, 2017 - 2020, DPhil, Oxford InFoMM CDT; co-supervised with A. Muench, J. M. Oliver; industry sponsor: Elkem. (Placement: Post-Doc at Cardiff)

J. Lee, Modelling and Computation for Petroleum Systems in Basins with Complex Structural Geometries, 2015-2019, DPhil, Oxford InFoMM CDT; co-supervised with K. Gillow, J. Whiteley; industry sponsor: Petrotechnical Data Systems

F. Brosa Planella, Modelling Solidification of Binary Alloys, 2015 - 2018, DPhil, Oxford InFoMM CDT; co-supervised with C. P. Please; industry sponsor: Elkem. (Placement: Post-Doc at Warwick, then Assistant Professor at Warwick)

N. T. Fadai, Multiphase Modelling of Coffee Bean Roasting, Oxford, 2015 - 2018, DPhil, Oxford InFoMM CDT; co-supervised with C. P. Please; industry sponsor: Jacobs Douwe Egberts. (Placement: Post-Doc at QUT, then Assistant Professor at Nottingham)

B. M. Sloman, Mathematical Modelling of Silicon Furnaces, 2015 - 2018, DPhil, Oxford InFoMM CDT; co-supervised with C. P. Please; industry sponsor: Elkem. (Placement: Research Scientist at Elkem ASA)

A. L. Krause, Network Modelling of Bioactive Porous Media, 2015 - 2017, DPhil, Oxford; co-supervised with S. L. Waters and D. Beliaev. (Placement: Post-Doc at Oxford, then Departmental Lecturer at Oxford, then Assistant Professor at Durham)

I've also supervised a number of MSc, BSc, and other short-term research projects.

### Teaching

University of Otago

Lectures: MATH140 Fundamentals of Modern Mathematics 2 - Calculus and ODEs half (S2 2022), MATH170 Mathematics 2 - Calculus and ODEs half (S1 2019, S1 & S2 2020, S1 & S2 2021), MATH302 Complex Analysis (S2 2022), MATH4AA Asymptotic Analysis (S2 2019, S2 2020), MATH4A1 Techniques in Applied Mathematics 1 (S1 2021, S2 2022), MATH4A2 Techniques in Applied Mathematics 2 (S2 2021, S2 2022)

University of Oxford

Intercollegiate Classes (Mathematical Institute): Applied Complex Variables (2015,2017), Perturbation Methods (2014), Viscous Flow (2015), Applied Partial Differential Equations (2015), Techniques of Applied Mathematics (2014), Modelling Case Study (2016, 2017, 2018)

College Tutorials (including Christ Church College, Oriel College, New College): Geometry (2015,2016), Dynamics (2016), Fluids and Waves (2016), Fourier Series and PDEs (2015), Integration (Measure Theory) (2016), Numerical Analysis (2015, 2016), Probability II (2015), Quantum Theory (2015, 2016), Special Relativity (2015), Statistics and Data Analysis (2016), Topology (2016)

University of Central Florida

Lectures: Calculus 1 (2010), Calculus 2 (2011), Calculus 3 (2010, 2013)

### Miscellaneous:

Erdős Number = 3