I am a University Academic Fellow in the School of Mathematics of the University of Leeds. Until recently I also held an EPSRC Early Career Fellowship, with the research project Bringing set theory and algebraic topology together.
I currently have 4 doctoral students, John Howe (co-supervised with John Truss), Richard Matthews (co-supervised with Michael Rathjen), Bea Adam-Day (co-supervised with Dugald Macpherson), and Alexandra Gouveia. My former student Stamatis Dimopoulos graduated in 2018.
The short, sweet, layman's summary of my research is that I study infinity. There's an excellent introductory video on YouTube made by Vsauce that starts very basic but goes pretty deeply into the concepts involved.
In more technical detail, I mainly work in set theory and its applications to other areas of mathematics, and am especially interested in large cardinal axioms. These posit the existence of cardinals (infinities) so large that they cannot be proven to exist from the standard axioms (assumptions) for mathematics. By assuming that such large cardinals do exist, we strengthen the theory, and so are able to draw more conclusions and do more mathematics than we could otherwise.
I have particularly worked on applications of set theory to category theory and algebraic topology; indeed, this was the topic of my EPSRC Early Career Fellowship. The same part of category theory that arises in this work - the theory of accessible categories, which is significantly affected by large cardinal assumptions - also turns out to be relevant in the context of abstract model theory.
Another part of set theory whose applications I am interested in is the study of Borel reducibility as a way to gauge complexity, for example in my work with Sheila Miller and Filippo Calderoni on quandles, which are algebraic invariants for knots.
In a point-set topology vein, I recently resolved the old question of precisely when the product of two CW complexes (endowed with the product topology) is itself a CW complex.
Probably the best source for my papers is my arXiv author page. They are reviewed on my author page at MathSciNet (subscription required), and there is a fairly up-to-date list of them with links to the published, journal versions at the bottom of my standard School of Mathematics webpage.