Research and scholarship
Research and scholarship
My research interests are in algebra and number theory. Much of my work is connected with Hopf-Galois theory, a generalization of classical Galois theory which uses a Hopf algebra in place of the Galois group. I have studied the structure and properties of the Hopf algebras that occur in this context, and the applications of Hopf-Galois theory to questions of integral module structure in extensions of local or global fields.
I also study skew braces, which were introduced as tools to study set-theoretic solutions to the Yang-Baxter equation. There is a correspondence between skew braces and Hopf-Galois structures on Galois field extensions, and I am interested in how this correspondence can be used to translate structural results from one topic to the other. In a project funded by the EPSRC I introduced skew bracoids, a generalization of skew braces that correspond with Hopf-Galois structures on separable, but potentially non-normal, field extensions.
I was the corresponding author and chief copy editor for the research monograph Hopf algebras and Galois module theory, which was published by the American Mathematical Society in 2021, and I maintain an archive of the talks given at the annual conference Hopf algebras and Galois module theory.
I am interested in hearing from potential PhD students in Algebra and Algebraic Number Theory.