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 Hopf-Galois 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 Hopf-Galois 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 set-theoretic solutions to the Yang-Baxter equation. There is a correspondence between skew left 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.

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 Algebraic Number Theory.

Teaching

  • MAT-20035 Exploring Algebra and Analysis
  • MAT-30038 Number Theory and Cryptography
  • MAT-40016 Field and Galois Theory
  • MAT-40018 Topics in Group Theory

Selected Publications

  • Truman PJ and Koch A. 2019. Opposite Skew Left Braces and Applications. Journal of Algebra, vol. 546, 218-235. doi> full text>
  • Truman PJ. 2019. Hopf-Galois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, vol. 224(5). doi> full text>
  • Truman PJ, Koch A, Kohl T, Underwood R. 2019. The Structure of Hopf Algebras Acting on Dihedral Extensions. In Advances in Algebra: Research from the Southern Regional Algebra Conference 2017. Springer. full text>
  • Truman PJ and Taylor S. 2019. The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions. New York Journal of Mathematics, vol. 25, 219-237. full text>
  • Koch A, Kohl T, Truman PJ, Underwood R. 2019. Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra. doi> link> full text>

Full Publications List show

Journal Articles

  • Truman PJ and Koch A. 2019. Opposite Skew Left Braces and Applications. Journal of Algebra, vol. 546, 218-235. doi> full text>
  • Truman PJ. 2019. Hopf-Galois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, vol. 224(5). doi> full text>
  • Truman PJ and Taylor S. 2019. The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions. New York Journal of Mathematics, vol. 25, 219-237. full text>
  • Koch A, Kohl T, Truman PJ, Underwood R. 2019. Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra. doi> link> full text>
  • Truman PJ, Koch A, Kohl T, Underwood R. 2019. Isomorphism problems for Hopf-Galois structures on separable field extensions. Journal of Pure and Applied Algebra, vol. 223, 2230-2245. doi> full text>
  • Truman PJ. 2018. Commutative Hopf-Galois module structure of tame extensions. Journal of Algebra, vol. 503, 389-408. doi> link> full text>
  • Truman PJ. 2017. Commuting Hopf-Galois structures on a separable extension. Communications in Algebra, vol. 46(4), 1420-1427. doi> link> full text>
  • Truman PJ. 2016. Canonical Nonclassical Hopf–Galois Module Structure of Nonabelian Galois Extensions. Communications in Algebra, vol. 44(3), 1119-1130. doi> link> full text>
  • Truman PJ. 2016. Hopf-Galois module structure of tame Cp×Cp extensions. Journal de Theorie des Nombres de Bordeaux, vol. 28(2), 557-582. doi> full text>
  • Truman PJ. 2013. Integral Hopf-Galois structures for tame extensions. New York Journal of Mathematics, vol. 19, 647-655. full text>
  • Truman PJ. 2012. Hopf-Galois module structure of tame biquadratic extensions. Journal de Theorie des Nombres de Bordeaux, vol. 24(1), 173-199. doi> full text>
  • Truman PJ. 2011. Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois structures. New York Journal of Mathematics, vol. 17, 799-810. full text>

Chapters

  • Truman PJ, Koch A, Kohl T, Underwood R. 2019. The Structure of Hopf Algebras Acting on Dihedral Extensions. In Advances in Algebra: Research from the Southern Regional Algebra Conference 2017. Springer. full text>

Research themes

Pure Mathematics