In late 2021, I had the absolute pleasure to do an independent study as a part of the “Honours Programme” at my university. My topic of choice was Combinatorial Group Theory which, roughly speaking, studies groups from their presentations. In the paper, I formally define what a group presentation is (based off of the Free Group on a set) and give several other important definitions. I gave particular emphasis to a nice set of tools called Tietze Transformations, which I used later in the paper. The end goal was to use this theory to solve the “Word Problem” for groups, which boils down to determining when two expressions are equivalent as group elements (very roughly speaking). It turns out that the framework of CGT is very helpful for finding algorithms that solve the Word Problem.
An area of application of CGT is algebraic topology, where it is at times useful to consider certain groups (e.g. the fundamental group) via a given presentation.
Contact me if you are interested in the paper!