Welcome to my research page! Click on the drop-down links below to read about my broad research interests and further detailed descriptions about my different focused research areas. The descriptions are aimed at a general (mathematically-interested) audience. You can also find details of my current and upcoming publications, and information on research talks (including some recordings), at the bottom of the page.
Mathematical Physics; Differential Geometry; Gauge theories; Sigma models; Topological Solitons; Yang-Mills theory; Skyrme model; Nahm transforms; Constructions of Instantons; Adiabatic approximations.
Many problems in geometry and physics (and other broader applications) involve minimisation of "energy". The term energy here is used as an umbrella term for many things, for instance purely geometric quantities such as lengths, areas, volumes, curvature, etc., more directly physical things such as total electromagnetic field strength, the potential and kinetic energy of a mechanical system, etc., and many others. In these contexts, the mathematical objects which are input into the energy are smooth functions from one space to another (e.g. maps of riemannian manifolds), and the condition for minimising the energy is that these functions satisfy specific partial differential equations (PDEs).
For most physically-relevant PDEs, solutions are unstable and eventually decay to the vacuum. However, remarkably there are theories whose solutions are stable; they do not collapse or decay even if the solution is deformed. Such solutions are very broadly known as solitons. Much of my research is interested in topological solitons. These are examples which owe their stability to properties inherited from the underlying spaces on which they are defined. As an intuitive example, imagine a choice of unit vector in the plane assigned to each point on the circle. Such vector fields are classified topologically by the winding number, that is, the number of times the vectors complete a full loop as you go around the circle. This notion of winding number generalises nicely to higher-dimensions by looking at unit vector-fields on spheres (in other words mappings of spheres of the same dimension). Modern physics is understood in the language of fields, and topological soliton models involve critical energy fields which have finite energy. This latter condition requires boundary conditions, and this gives rise to objects like these described above, which are classified topologically by a discrete set such as the integers (for example the winding number of a map of spheres). Physically these values are identified as numbers of particles or anti-particles in the theory.
The study of topological solitons is beautifully nonlinear, combining several sophisticated areas of mathematics including topology and geometry (both algebraic and differential), analysis of PDEs, to name a few, and often requires development of bespoke numerical methods. Their applications are widespread, from particle physics, condensed matter physics, and cosmology, to technological applications such as liquid crystal displays, large-scale storage devices, and quantum computing.
My research is generally focused on three main avenues, typically in the context of topological solitons:
Constructing explicit or approximate examples, and analysing and classifying moduli spaces;
Developing mathematical tools for probing physical and qualitative properties;
Identifying links between different field theories, and with aspects of differential geometry and topology.
You can read about some of these aspects in more detail via the tabs below.
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P1. "Twisted calorons" (with Derek Harland).
J9. "Locations of JNR skyrmions", Math. Phys. Anal. Geom. 28, 17 (2025), e-print: arxiv:2505.00075 (with Linden Disney-Hogg).
J8. "Quantization of skyrmions using instantons", Phys. Rev. D 110, 016027, e-print: arxiv:2403.17080 (with Chris Halcrow).
J7. "Finkelstein-Rubinstein constraints from ADHM data and rational maps", Phys. Lett. B. 850, 138542, 2024, e-print: arxiv:2401.16494 (with Derek Harland).
J6. "Geometry of gauged skyrmions", SIGMA 19 (2023), 071, e-print: arxiv:2303.02623 (with Derek Harland).
J5. "ADHM skyrmions", Nonlinearity 35 (2022) 3944, e-print: arxiv:2110.15190 (with Chris Halcrow).
J4. "A model for gauged skyrmions with low binding energies", 2022 J. Phys. A: Math. Theor. 55 015204, e-print: arxiv:2109.06886 (with Derek Harland, and Thomas Winyard).
J3. "A low-energy limit of Yang-Mills theory on de Sitter space", J. High Energ. Phys. 09 (2021) 89, e-print: arxiv:2104.02075 (with Emine Şeyma Kutluk, Olaf Lechtenfeld, and Alexander D Popov).
J2. "Skyrmions from calorons", J. High Energ. Phys. 11 (2018) 137, e-print: arxiv:1810.04143.
J1. "Symmetric calorons and the rotation map", J. Math. Phys. 59 (2018), no. 6, 062902, e-print: arxiv:1711.04599.
For more information about my published research, see Google Scholar, Inspire, ResearchGate, nlab, Web of Science, and ORCID.
March 2025: York Geometry, Analysis, and Mathematical Physics seminar, "On the instanton approximation of skyrmions".
August 2024: Geometric Models of Matter, Leeds, "Finkelstein-Rubinstein constraints from ADHM data and rational maps".
May 2024: Solitons at Work online seminar, "Quantizing the Skyrme model with instantons". Watch on YouTube.
June 2023: SIG XI conference, Krakow Poland, "Geometry of gauged skyrmions".
June 2023: Instantons, Holography, Strong Interactions and Nuclear Physics conference, Henan China (online), "ADHM skyrmions".
May 2023: Loughborough mathematical physics seminar, "Geometry and gauge theory of skyrmions".
March 2023: Mathematics seminar, Leicester, "On the Atiyah-Manton approximation of skyrmions".
November 2022: Leeds geometry seminar, "Near BPS and BPS gauged skyrmions".
September 2022: Geometric models of matter, Kent, "ADHM skyrmions". Watch on YouTube.
April 2022: DESY Hamburg (online) journal club, "On the instanton approximation of skyrmions".
July 2021: SIG IX (online), "Skyrmions from instantons" (joint with Chris Halcrow and Tom Winyard). Watch on YouTube.
January 2021: Waterloo, Canada, (online) geometry seminar, "Gauged skyrmions".
September 2020: Geometric models of matter (online), "A model for gauged skyrmions with low binding energies". Watch on YouTube.
February 2020: DAMTP Cambridge mathematical physics seminar, "Skyrmions from monopoles and calorons".
October 2019: Lancaster analysis seminar "Instantons, monopoles, and the Skyrme model".
November 2018: Bath geometry seminar, "Symmetric calorons and the rotation map".
September 2018: Dynamics Days Europe conference, Loughborough, "Symmetric calorons and the rotation map".
July 2018: M:iv mini workshop, Leicester, "Symmetric calorons and the rotation map".
July 2018: Low Energy Effective Dynamics of Skyrmions, Leeds, "Skyrmions and calorons".
March 2018: Yorkshire Durham Geometry Day, Durham, "Symmetric calorons and the rotation map".