PhD Studentship: Geometric and combinatorial inverse monoid theory

This PhD project is in the area of combinatorial algebra which is the study of algebraic objects like groups, and more generally monoids, defined in terms of a generating set, and a set of defining relations holding among those generators. For groups this topic is part of geometric group theory where ideas from geometry, topology, combinatorics and theoretical computer science (e.g. automata and formal language theory) are used to study infinite groups.

The main objects of study in this PhD project are a generalisation of groups called inverse monoids. While groups are an algebraic abstraction of permutations, and semigroups of arbitrary mappings, inverse monoids correspond 2to partial bijections and provide a framework for studying partial symmetries. Combinatorial and geometric inverse monoid theory is a much newer area of research than the corresponding theory for groups, and the main aim of this PhD project will be to build new theory and apply it to prove results about finitely presented inverse monoids.

In this project building on results of Gray, Silva and Szakács (2022), you will establish results that relate the geometry of Cayley graphs of inverse monoids to the algorithmic and algebraic properties of the monoid. You will also develop

new tools for the analysis of Schutzenberger graphs of inverse monoids and apply them to construct families of finitely presented inverse monoids with interesting and unexpected properties. In addition, developing ideas from Gray and Kambites (2025), you will investigate the relationship between properties of finitely presented inverse monoids and properties of groups associated with them and, building on work of Mangioni and Tesolin (2026), you will study the action of inverse monoids on directed metric spaces (and related ordered spaces) and prove results that relate properties of the inverse monoid to those of the space and the action.

Entry requirements

The standard minimum entry requirement is 2:1 in Mathematics.

Start date

1 October 2027

Mode of study

Full-time

Additional Funding Information

This PhD project is in a competition for a Faculty of Science funded studentship. Funding is available to UK applicants and comprises ‘home’ tuition fees and an annual stipend for 3 years. The current UEA / institutional PhD stipend rate is £21,805 and will be subject to any agreed inflationary uplift for 2027/28.

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