Topological solitons: geometry, dynamics, and applications

Topological solitons arise in field theories where the dynamical field is twisted into a stable, knot-like configuration that cannot be untied without cutting. They provide a mathematically natural description of particle-like structures within continuous physical systems, and appear across all scales of physics, from cosmic strings that stretch across the universe to the subatomic interactions of quarks. These configurations appear as solutions to nonlinear PDEs that generally do not admit closed-form solutions, making it challenging to understand their properties and how they interact with each other.

For example, the dynamics of these twisted fields reveal a profound geometric principle: interacting solitons can often be understood as motion on a moduli space — the geometric manifold of all static multi-soliton configurations. Through this perspective, highly nonlinear field equations become problems in differential geometry, exposing hidden structure and symmetry. The universality of topological solitons allows ideas developed in one context, such as nuclear and particle physics, to yield powerful insights in others, including superconductivity, magnetism and optics.

This project will explore the geometry of topological solitons and its intimate connection to their dynamics, in part as a conduit for understanding this universality. This is achieved by understanding key principles of classical field theories and PDEs. It offers exceptional flexibility: you might pursue a differential-geometric approach (studying moduli space geometry), a numerical approach (simulating vortex dynamics in superconductors or quantising the Skyrme model), or a more physically driven direction (modelling topological quantum computers based on vortices in superconductors, or spintronic devices that compute using magnetic skyrmions). Each direction contributes to a broader goal: understanding why the same topological principles reappear across such diverse physical systems.

The successful candidate will join a large, vibrant international collaboration spanning pure mathematics, theoretical physics, and experimental science, with partners across the UK, Sweden, Greece, Spain, and China. There are substantial opportunities for travel to conferences, workshops, and extended research visits throughout Europe through the EU-wide COST Action CA23134, Topological Textures in Condensed Matter, reflecting the international recognition and momentum of this rapidly developing field.

Topological ideas are reshaping modern physics and emerging technologies, with growing strategic and governmental recognition, including strong alignment with the UK’s Digital and Technologies Sector Plan through applications to topological quantum computing and novel magnetic computing devices. This project offers the opportunity to contribute to fundamental mathematics and physics while engaging directly with rapidly advancing technological frontiers.

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Our research community thrives on the diversity of students and staff which helps to make the University of Dundee a UK university of choice for postgraduate research. We welcome applications from all talented individuals and are committed to widening access to those who have the ability and potential to benefit from higher education.