Biography
As an undergraduate, I read for an MMath at the University of Oxford (Exeter College). I graduated in 2005 and moved to the University of Exeter where I studied for a PhD in algebraic number theory under the supervision of Nigel Byott. After submitting my thesis I held two short term posts in Exeter, initially as a research fellow and subsequently as a teaching fellow. I finally left Exeter in 2010 to come to Keele.
Research and scholarship
My research interests are in algebra and number theory. Much of my work is in HopfGalois theory, a generalization of classical Galois theory which uses a Hopf algebra in place of the Galois group. I study the structure and properties of the Hopf algebras that occur in this context, and the applications of HopfGalois theory to questions of integral module structure in extensions of local or global fields.
I also study braces, which were introduced as a tool to study settheoretic solutions to the YangBaxter equation. There is a correspondence between skew left braces and HopfGalois 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.
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.
Teaching
 MAT20035 Exploring Algebra and Analysis
 MAT30038 Number Theory and Cryptography
 MAT40016 Field and Galois Theory
 MAT40018 Topics in Group Theory
Selected Publications

Skew left braces and isomorphism problems for HopfGalois structures on Galois extensions. Journal of Algebra and Its Applications. full text>

Hopf algebras and Galois module theory. American Mathematical Society.2021.

Abelian fixed point free endomorphisms and the YangBaxter equation. New York Journal of Mathematics, 14731492, vol. 26. link> full text>2020.
 2019.

HopfGalois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, vol. 224(5). doi> full text>2019.
Full Publications Listshow
Books

Hopf algebras and Galois module theory. American Mathematical Society.2021.
Journal Articles

Skew left braces and isomorphism problems for HopfGalois structures on Galois extensions. Journal of Algebra and Its Applications. full text>

Abelian fixed point free endomorphisms and the YangBaxter equation. New York Journal of Mathematics, 14731492, vol. 26. link> full text>2020.
 2019.

HopfGalois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, vol. 224(5). doi> full text>2019.

The structure of Hopf algebras giving HopfGalois structures on Quaternionic extensions. New York Journal of Mathematics, 219237, vol. 25. full text>2019.

Normality and Short Exact Sequences of HopfGalois Structures. Communications in Algebra. doi> link> full text>2019.

Isomorphism problems for HopfGalois structures on separable field extensions. Journal of Pure and Applied Algebra, 22302245, vol. 223. doi> full text>2019.

Commutative HopfGalois module structure of tame extensions. Journal of Algebra, 389408, vol. 503. doi> link> full text>2018.

Commuting HopfGalois structures on a separable extension. Communications in Algebra, 14201427, vol. 46(4). doi> link> full text>2017.

Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions. Communications in Algebra, 11191130, vol. 44(3). doi> link> full text>2016.

HopfGalois module structure of tame Cp×Cp extensions. Journal de Theorie des Nombres de Bordeaux, 557582, vol. 28(2). doi> full text>2016.

Integral HopfGalois structures for tame extensions. New York Journal of Mathematics, 647655, vol. 19. full text>2013.

HopfGalois module structure of tame biquadratic extensions. Journal de Theorie des Nombres de Bordeaux, 173199, vol. 24(1). doi> full text>2012.

Towards a generalisation of Noether's theorem to nonclassical HopfGalois structures. New York Journal of Mathematics, 799810, vol. 17. full text>2011.
Chapters

The Structure of Hopf Algebras Acting on Dihedral Extensions. In Advances in Algebra: Research from the Southern Regional Algebra Conference 2017. Springer. full text>2019.
Research themes
Pure Mathematics
School of Computing and Mathematics
Keele University
Staffordshire
ST5 5AA
Email: scm.admin@keele.ac.uk
Tel:+44 (0) 1782 733075