Joseph Paul MacManus

I was awarded a grant by the James Pantyfedwen Foundation to pursue an MSc in Mathematics and Foundations of Computer Science (known locally as the MFoCS) at Balliol College, Oxford. After such a unique year full of new experiences, I cannot say thank you enough to the Foundation for their financial support through this period.

During my time at Mathematical Institute here in Oxford, I have developed in a strong interest in geometric group theory and surrounding fields. In short, a “group” is an algebraic structure which encodes the set of symmetries of some object (e.g. the mirror/rotational symmetries of a polygon). With this in mind, “geometric group theory” is an area of mathematics where we study the interplay between the algebraic properties of these groups and the geometric properties of the shapes they act upon. It is a very active area of research, with strong links to many other areas, such as low-dimensional topology.

My dissertation, titled “Splitting Theory of Groups”, was focused on introducing the three cornerstone results from this area of group theory, due to Grushko, Stallings, and Dunwoody respectively. These results shed light on the structure of finitely generated groups, in terms of their free/finite splittings. This dissertation also showcased our progress on simplifying the construction of an object known as the “Fujiwara-Papasoglu slender JSJ decomposition”. We hope that further work on this project could lead to a new exposition of this theory.

I have since received a fully funded scholarship to pursue a DPhil in Mathematics here at the University of Oxford. This opportunity is a dream come true, and I am incredibly grateful to the Foundation for their support in this step of my education. Their generosity has set me on a path towards a career as an academic mathematician, and it is difficult to say what I would currently be doing without the Foundation's help.