Associate Professor in Mathematics
University of Leeds
I have been a member of the geometry group in the School of Mathematics at the University of Leeds since 2013. Before that I was a lecturer at Loughborough, and had spells as post-doc in Durham and Hannover. I completed my PhD in Durham in 2008 under the supervision of Richard Ward.
I am broadly interested in geometry and mathematical physics.
A lot of my research focuses on topological solitons. These can roughly be understood as solutions in classical field theories or geometric variational problems which behave like particles. The number of particles is typically equal to an integer-valued topological invariant of the underlying fields.
Monopoles are quintessential examples of topological solitons. Some of my recent work has analysed the structures that arise when collections of monopoles coalesce, and the formation of so-called magnetic bags. I have also been thinking about the distribution of magnetic charge in monopoles and how this is encoded in the integrable systems machinery that was developed to construct monopoles. In ongoing work I am analysing harmonic forms on monopole moduli spaces and how this relates to their asymptotic geometry. I am also looking at ways of constructing chains of monopoles with a high degree of symmetry using relations with Hitchin's equations.
The Skyrme model is an effective model of quantum chromo-dynamics in which atomic nuclei emerge as solitons. I have been thinking about how to tune the model to better reflect data from experimental nuclear physics, and to this end have been developing a lightly bound Skyrme model. I have also been trying to extract electromagnetic properties of nucleons by properly coupling the model to a gauge field.
Instantons are gauge-theoretic solitons that exist in dimension four and higher. I have been developing a theory of instantons on manifolds that admit real Killing spinors, including nearly Kaehler six-manifolds (which are closely linked with G2-geometry). Important problems include the construction and counting of such instantons; one of my recent results is that the only known instanton on the six-sphere is rigid (so new instantons cannot be obtained by deforming it).
I am grateful to all of the following people for fruitful research collaborations:
Richard Ward, Seckin Kurkcuoglu, Olaf Lechtenfeld, Alexander Popov, Tatiana Ivanova, Christoph Nölle, Paul Sutcliffe, Martin Speight, David Foster, Christian Saemann, Sam Palmer, Mark Everitt, Tim Spiller, Yasha Shnir, Juha Jaykka, Mike Gillard, Stefano Bolognesi, Benoit Charbonneau, Daniel Nogradi, Eugenie Hunsicker.
I am looking to recruit PhD students; if you would like to study for a PhD here at Leeds please contact me.
I currently supervise one PhD student, Joseph Driscoll.
My student Josh Cork graduated in 2017.
Material for MATH2051 Geometry of Curves and Surfaces
Schedule for the Pure Mathematics Colloquium
The Yorkshire Durham Geometry Day, which I coorganise.
Photos from the Leeds maths walks.
The LEEDS conference that I organised in 2018.
The LMS Summer School that I organised in 2019.