Research

I am currently writing my Master's research project on algebraic topology, with a focus on the Seifert-Van Kampen theorem. More details on that below.

My research interests, in more detail

• Algebraic topology. Most classical topological methods (connectedness, compactness, Hausdorfness etc.) fail for a surpringly large number of cases. Algebraic topology fixes this problem by looking at the structure of the space, and tries to figure out information that way. I am writing my Master's project on homotopy theory, which uses the fact that circles embedded in a topological space form a group. This does also work for all possible $n$-spheres, but gets rather fiddly to calculate quite quickly. We are still currently yet to find out the number of ways an $n$-sphere can be embedded inside an $m$-sphere for anything larger than a small value of $m$ or $n$.
• Geometric group theory. This area has had an abundance of research in the last century, and measures how groups act on graphs and other objects (and vice versa) to classify them.
• Algebraic automata theory. I am interested in seeing how automata theory relates to algebraic ideas and proofs. Given a group presentation $\langle S | R \rangle$, what words are easy to show are trivial? It has been shown that in general this is not possible. However, if we restrict this to certain groups, then nice results do occur, for example in Coxeter groups through the Brink-Howlett automata. We can ask this for other groups, which has applications in both group theory and automata theory.
• Computer algebra. In the summer of 2025 I worked towards developing ATLAS v4 with Professor David Craven. There have been many recent developments in this area; recently, a package was written that could make calculations on the Monster group in a practical time on a standard computer, which has put us a step closer towards finding out most of the information for all finite simple groups. There are also a range of applications beyond group theory: packages that have been created to work on homological algebra are now being used for analysis of big data through topological data analysis.