## Robert 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 Department of Mathematics and Statistics at the University of Otago, New Zealand as a Senior Lecturer, where I was later "confirmed" (the closest approximation to "tenured" in Otago/NZ parlance) in 2023 and promoted to Associate Professor in 2024. (To see what these ranks signify outside of the four-tiered New Zealand system, click here).

I enjoy traveling and carrying out my work in myriad locales, preferring longer research visits to brief, hectic conferences. I spent the first half of 2023 on leave while a Visiting Scholar at Merton College and the Mathematical Institute, University of Oxford, and a CNRS-funded guest scientist in the Institut de Physique de Nice, France.

### Contact Details:

Mailing address:

Robert A. Van Gorder,

Department of Mathematics and Statistics

University of Otago

P.O. Box 56

Dunedin 9054 New Zealand

email:

robert dot vangorder at otago dot ac dot nz

### Employment:

Associate Professor, Department of Mathematics and Statistics, University of Otago, New Zealand (2024 -)

Senior Lecturer, Department of Mathematics and Statistics, University of Otago, New Zealand (2019 - 2023)

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 Interests and Selected Publications:

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 falling into one of the following areas:

I. Fluids

If you’re alive, you’re interacting with fluids: The air that you breathe and the water that 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 fluid mechanics include better understanding the dynamics governing fundamental structures –such as vortices, bubbles, waves, and boundary layers – that can then be used as the 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.

Selected Publications:

R. A. Van Gorder and P. A. Fisher, Spatial eigenvalue problems for stars in hydrostatic equilibrium: Generalized Lane–Emden equations as boundary value problems, Monthly Notices of the Royal Astronomical Society 523 (2023) 2059-2073

R. A. Van Gorder, Finite time blowup of incompressible flows surrounding compressible bubbles evolving under soft equations of state, Proceedings of the Royal Society A 478 (2022) 20220172

E. K. Luckins, J. M. Oliver, C. P. Please, B. M. Sloman, R, A. Van Gorder, Modelling and analysis of an endothermic reacting counter-current flow, Journal of Fluid Mechanics 949 (2022) A21

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

F. Brosa Planella, C. P. Please, and R. A. Van Gorder, Extended Stefan problem for solidification of binary alloys in a finite planar domain, SIAM Journal on Applied Mathematics 79 (2019) 876-913

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

D. T. Kennedy and R. A. Van Gorder, Motion of open vortex-current filaments under the Biot–Savart model, Journal of Fluid Mechanics 836 (2018) 532-559.

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

Cross-sections of the velocity field (top) and scalar pressure (bottom) for an incompressible fluid surrounding a compressible "soft" bubble undergoing collapse and subsequent blowup. For more details, see: R. A. Van Gorder, Proceedings of the Royal Society A 478 (2022) 20220172.

II. Patterns

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?

Selected Publications:

Y. Önder and R. A. Van Gorder, Modification of Turing patterns through the use of time-varying anisotropic diffusion, Proceedings of the Royal Society A 479 (2023) 20230487

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

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

Control of spatial patterns within a disc using a sharp diffusion and reaction rate differential. For more details, see R. A. Van Gorder, Philosophical Transactions of the Royal Society A 379 (2021) 20210001.

III. 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 interesting scientific challenges. The study of waves is pertinent to 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 imposed heterogeneity.

Selected Publications:

R. A. Van Gorder, Quantum mechanics on time-varying space domains, Proceedings of the Royal Society A 479 (2023) 20220759

R. A. Van Gorder, Compressed hydrogen atoms confined within generic boxes, Proceedings of the Royal Society A 478 (2022) 20220467

R. A. Van Gorder, Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged, Physics Letters A 446 (2022) 128284

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

L. W. S. Baines and R. A. Van Gorder, Soliton wave-speed management: slowing, stopping, or reversing a solitary wave, Physical Review A 97 (2018) 063814

R. A. Van Gorder, Breathers and nonlinear waves on open vortex filaments in the nonrelativistic Abelian Higgs model, Physical Review D 95 (2017) 096007

R. A. Van Gorder, Solitons and nonlinear waves along quantum vortex filaments under the low-temperature two-dimensional local induction approximation, Physical Review E 93 (2016) 052208

R. A. Van Gorder, Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation, Physical Review E 91 (2015) 053201

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

R. A. Van Gorder, Decay of helical Kelvin waves on a quantum vortex filament, Physics of Fluids 26 (2014) 075101

Density plots for the ground state of an attractive Bose-Einstein condensate on space domains of different shapes. For more details, see R. A. Van Gorder, Proceedings of the Royal Society A 476 (2020) 20200674.

Other research interests have included:

applications of specific analytical methods (perturbation theory, homotopy analysis, spectral methods) toward the solution of nonlinear differential equations;

nonlinear dynamics and chaos;

solitons and solitary waves;

econophysics (including nonlinear dynamics and optimal control theory), sociology, epidemiology of metapopulations, and game theory;

mathematical biology and ecology;

fluid dynamics problems involving 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 (some of these were in collaboration with industrial partners);

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. I am happy to supervise mathematics or physics PhD and honours projects in any of the three areas mentioned above.

### Research student supervision:

I've supervised the following doctoral students:

Shabina Khalid, Mathematical modelling of antibubbles, 2023 - present, PhD, Otago.

Ellen 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, then Zeeman Lecturer at Warwick)

Raquel 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)

Jane 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

Ferran 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)

Nabil 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)

Benjamin 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)

Andrew 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

Current Year Lectures:

MATH140 Fundamentals of Modern Mathematics 2 (full course, S2 2024)

MATH302 Complex Analysis (S2 2024)

MATH424 Techniques in Applied Mathematics 1 (S2 2024)

MATH426 Techniques in Applied Mathematics 2 (S2 2024)Previous Lectures:

MATH140 Fundamentals of Modern Mathematics 2 (Calculus half in S2 2022, full course in S2 2023)

MATH170 Mathematics 2 - Calculus half (S1 2019, S1 & S2 2020, S1 & S2 2021)

MATH302 Complex Analysis (S2 2022, S2 2023)

MATH4AA Asymptotic Analysis (S2 2019, S2 2020)

MATH4A1 Techniques in Applied Mathematics 1 (S1 2021, S2 2022, S2 2023)

MATH4A2 Techniques in Applied Mathematics 2 (S2 2021, S2 2022, S2 2023)

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