Highlights from Carnegie PhD Scholars’ Gathering 2019
Project Title: Low dimensional homotopy for inverse semigroups
Emma’s research is concerned with different ways to describe algebraic structures that model symmetry. The structures are groups and inverse semigroups, and they can be described concisely by presentations — in effect, a short list of rules for carrying out computations. However, the same structure can be described by quite different sets of rules, and the question then arises of how we compare different presentations, and perhaps decide on the selection of an optimal one. The relevant properties of a particular presentation are encoded in its “Squier complex”, which records the effects of all the computational rules when applied in succession. The Squier complex itself has an involved structure that requires more mathematical tools to describe and understand, and in effect we can use groups to understand group presentations, and structures called ordered groupoids — also models of symmetry — to understand inverse semigroup presentations. One important property that we look for is asphericity, which shows that a presentation has an efficient set of rules. Aspherical group presentations have been studied for a long time, and we know a lot — but not yet everything — about them. Emma’s work is currently focussed on how we might extend this study, using ordered groupoids, to inverse monoid presentations.
Awarded: Carnegie-Caledonian PhD Scholarship
University: Heriot-Watt University