Paul Truman

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 came to Keele in 2010 as a Teaching Fellow; I was promoted to Lecturer (education and research) in 2013 and to Senior Lecturer in 2020.

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 am an organizer of the joint research group The Interplay Between Skew Braces and Hopf-Galois Theory, supported by the London Mathematical Society; this research group holds regular meetings to promote collaboration between researchers working on these topics. 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. I also 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

• MAT-20025 Abstract Algebra
• MAT-20035 Exploring Algebra and Analysis
• MAT-30038 Number Theory and Cryptography
• MAT-30055 Guided Studies in Mathematics 2
• MAT-40016 Field and Galois Theory
• MAT-40018 Topics in Group Theory

Research themes

• Discrete Structures and Computational Modelling

Previous PhD students

• Isabel Martin-Lyons PhD 2025: Skew bracoids: from Hopf-Galois theory to the Yang-Baxter equation
• Lorenzo Stefanello PhD 2024: Hopf-Galois structures, skew braces, and their connection (Joint supervision with Prof. Ilaria Del Corso, University of Pisa)
• George Prestidge PhD 2024: Hopf-Galois module structure of some non-normal extensions
• Stuart Taylor PhD 2020: Hopf-Galois module structure of a class of tame quaternionic fields