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 of solitons, especially for nuclear skyrmions;
Identifying rigorous foundations for field theories using differential geometry and topology, and links between different field theories.
You can read about some of these aspects in more detail via the tabs below.
The majority of partial differential equations which arise in geometry and physics are highly complex and do not allow exact solutions. However, there are a handful of equations which are regarded as integrable, which more or less means they are solvable in analytic terms (as opposed to requiring approximation or numerical methods). A major example of such equations (which some have even speculated are the source of all integrable equations) are called the self-dual Yang-Mills equations. Solutions of these with finite energy are broadly known as instantons. The study of these equations originally came from physics: Yang-Mills theory was introduced as a way to generalise Maxwell's equations of electromagnetism in order to apply to other fundamental forces (specifically the strong nuclear force), and the instanton solutions represent tunneling trajectories between different vacuum states, with the transition in energy states being highly localised in time (hence the terminology instant-on).
Mathematically the self-dual Yang-Mills equations are a collection of first order partial differential equations in four dependent variables (i.e. on a four dimensional manifold). The equations have a deep geometric interpretation as equations for curvature, and the Yang-Mills energy is a measure of total curvature. In this way instantons could be interpreted as solutions which are the next best things to describing flat curvature. Very loosely speaking, the term curvature here refers to the idea of translating an object around a closed loop on some space, in such a way that at all times the object remains parallel to itself relative to the space, and curvature is a measure of the obstruction to the object ending up back in its exact original state. For example, if you are at the equator and face due north, take a (very long and difficult) walk forwards then at the north pole step sideways (don't turn) and walk "left" in a southerly direction until you reach the equator, then walk backwards along the equator, you will have traversed a closed loop while remaining parallel, yet you are now facing west, which is not how you started. On the contrary, if you performed the same trip on a flat triangle you would end up exactly how you started. The reason the situation is different on a sphere (Earth) is because of curvature. The Yang-Mills equations however are not directly measuring curvature of a physical object like a surface, but rather this process of parallel translating objects, which could be of more exotic geometric data, not just directions like our toy example on the sphere.
Instantons are classified by topological invariants. Broadly this is a measure of winding around the interior of the four-dimensional manifold on which they are defined. In the simplest case where the 4-dimensional space is a higher-dimensional sphere (called the 4-sphere), this can be superficially understood as the winding of a function defined on the "equator" of the 4-sphere, which is a 3-dimensional sphere, into itself. This classifies instantons into distinct boxes labelled by the integers. Now recall that the equations defining instantons are integrable. Somewhat remarkably the set of solutions of these equations (up to a suitable notion of equivalence) is itself an 8|k|-dimensional manifold, where k is this topological invariant. These moduli spaces are interesting in their own right, both in physical applications (e.g. as a way to model low-energy dynamics), and mathematically (e.g. as examples of special geometries called hyperkaehler manifolds, and also in defining geometric invariants for four manifolds).
Most remarkably, in several cases there exists a complete explicit construction of the solution space, which in the simplest case of euclidean space, reduces the self-dual Yang-Mills equations to a set of algebraic equations for k by k matrices (i.e. no differential equations at all!). This general construction is called the Nahm transform (or Atiyah-Drinfeld-Hitchin-Manin construction in the simplest case). Heuristically the Nahm transform relates self-dual solutions defined on euclidean space invariant under translations in some directions to solutions of a set of differential equations in these invariant directions. So when there are no invariant directions, the equations are defined over a point, thus are algebraic equations as described above. When there is one invariant direction, the associated equations are ordinary differential equations (called Nahm's equations) defined on a one-dimensional space: either a line or a circle. Self-dual solutions invariant under one direction are also known as monopoles, and loosening the invariance under this direction to instead ask for periodicity one obtains objects called calorons.
The Nahm transform allows for explicit parameterisation of solutions of the self-dual Yang-Mills equations which may not otherwise be obtainable via more direct methods (i.e. directly solving the PDEs). In my research I have had a lot of success in using this machinery to understand moduli spaces of instantons. For example, during my PhD I classified infinitely many calorons with special cyclic symmetries, and have also described large families of instantons with symmetry and families of instantons without symmetry in order to better understand the Skyrme model (see the next section). More recently I have been working on a new type of Nahm transform to describe instantons invariant under glide rotations, and my research PhD student has been working on describing highly symmetric solutions of Nahm's equations which give rise to instantons on curved manifolds called ALF (Asymptotically-locally-flat) spaces.
(coming soon)
(coming soon)
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".