The topology of chaos of partial differential equations

The mathematical ideas of topology provide a powerful framework for understanding the qualitative behaviour of dynamical systems, offering new ways to describe and predict large-scale changes in complex systems. This project will build on previous work by the supervisors to extend these methods to more intricate chaotic partial differential equation (PDE) systems relevant to physics, such as the Navier–Stokes equations.

A key motivation for this research is the pressing need to develop new mathematical tools for studying systems that exhibit sudden and dramatic changes, such as "tipping points" in climate dynamics. By applying topological techniques to these problems, this project aims to uncover novel insights into the underlying structures governing turbulence, transport, and long-term system evolution.

The work will involve both theoretical and computational aspects, combining mathematical advances in applied topology with the development of numerical and data-driven methods for real-world and artificial systems. Depending on the student’s interests, there will be opportunities to explore topics such as topological invariants of fluid flows, computational homology in high-dimensional data, and the role of symmetry in chaotic dynamics.

The ideal candidate will have a strong mathematical background, particularly in analysis and differential equations, as well as an interest in physical applications. Experience with numerical methods, programming, or data analysis will be highly beneficial, but prior knowledge of topology is not required—enthusiasm for learning new mathematical ideas is more important.

This project will provide excellent training in both theoretical and computational mathematics, offering opportunities for interdisciplinary collaboration and real-world impact.

Diversity statement

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.