My summer research project tackles a problem in pure mathematics, specifically, in graph theory - the abstract study of networks and how information is connected. So what is pure mathematics? We often talk about 'pure' mathematics in contrast to 'applied' mathematics, which is mathematics developed to solve a real world problem, like in physics, engineering and computer science.
So why study pure mathematics, if it is, by definition, mathematics which is not made with a real world use in mind? Many problems are not solvable without abstracting them, stripping away all of the unnecessary noise surrounding them, and focusing only on the important details. Pure mathematicians are interested in making abstract logical connections, sometimes with the hope that someone may one day come along and make real world use of their nice logic, and sometimes purely for the sake of making fuzzy ideas more precise.
As I said, my research is in graph theory. Graph theory occupies a place both in pure and applied mathematics: graphs are everywhere in fields like computer science, and are also fascinating mathematical objects of study for their own sake. I’m interested in the latter point, and specifically I'm studying pairs of ‘pseudo-similar’ vertices in graphs. These are vertices arising in pairs which posses an ‘almost symmetry’ which has frustrated efforts to prove a central conjecture in graph theory: the Graph Reconstruction Conjecture.
My research has focused on finding out how large a set of mutually pseudo-similar vertices can be packed into a graph of a given size. The way I’ve done this is rather peculiar: I’ve had to expand my definition of ‘symmetry’ in a graph to include far more things than I would traditionally call a symmetry. The first goal of my research was to reach a theorem classifying pseudo-similarity in terms of this 'partial symmetry'. That goal has been completed, and now I’m trying to use my techniques to arrive at an answer as to exactly how many pseudo-similar vertices can be fit into a graph.
Something that stands out to me about completing a research project is just how much new mathematics I’ve learned over the course of my research so far, and how much this contrasts with the pace and style of a lecture course. This project has been the perfect opportunity to get to grips with some mathematical tools that I otherwise would have waited until the 5th year modules in Semigroups and Advanced Combinatorics to experience. Learning mathematics in order to apply it to research demands a much greater depth of understanding than to sit an exam, and I've enjoyed this change of pace very much.
However the research process has not always been smooth. A couple of weeks ago, I discovered that a theorem my supervisor and I thought was airtight actually contained a hole. After spending the past two weeks trying, and not succeeding, to patch this hole, I'm now realising that the story is probably much more complicated than we first thought. Unexpectedly, I've come to see this as a good thing. What was once a straightforward uniqueness theorem about pseudo-similar vertices in directed graphs has now turned into a string of quite useful lemmas about which partial symmetries of the graph can actually be extended to full symmetries. What I'm learning from this is that sometimes, messy and partial results tell a much bigger story than clean, complete results do. This gives an interesting contrast to the way mathematics is presented in our lecture notes.
So far, I've found the research process quite enthralling. Going into this project, I already thought I wanted to complete a PhD in mathematics, and this has definitely affirmed that ambition!