My research is in the area of finite group theory. Informally, this means that I study the symmetry of objects, although a mathematician's notion of symmetry is a little more involved than the "looking in a mirror" notion that non-mathematicians are familiar with.
Broadly speaking, my research can be grouped into three main themes:
- groups acting on geometries: this was the subject of my PhD (supervised by Jan Saxl at Cambridge), where I studied finite projective planes which admit an automorphism group that acts transitively on lines. I wrote a number of papers on this (and related topics), with the main focus being a proof of the Ostrom-Wagner conjecture. That conjecture remains unproved, but I made some progress.
- growth in groups: I became interested in this as a postdoctoral researcher (working with Harald Helfgott at Bristol). Roughly speaking, this area of research focuses on what happens when one multiplies a subset of a group by itself: is the ensuing set of products much larger than the original set, or not? This simple question has profound implications, with applications in additive combinatorics, number theory, graph theory, computer programming, and the like. I held an EPSRC grant on this subject.
- groups generated using geometries: this is work inspired by Conway's famous construction of the sporadic Mathieu group, M_12, using "moves" on the projective plane of order 3. This construction can be generalized, and one discovers that many other groups can be generated in similar ways from rather unexpected geometries.
- relational complexity: This is my current focus, and is motivated by important work in model theory. Greg Cherlin, one of the world's leading model theorists, introduced the notion of "the relational complexity of a permutation group" in the 1990's, building on work of Lachlan. He formulated an important conjecture on this topic around the same time, and I have spent much of the last 3 or 4 years working on that conjecture. I currently hold an EPSRC grant which is aimed at resolving this conjecture.
I have had the pleasure of supervising a number of students, including around 20 undergraduates. At Masters level, I have supervised Sam Hughes (USW) and Victor Tomno (Moi University, Kenya). My current PhD students are Bianca Loda (USW), Scott Hudson (USW), Margaret Stanier (Open University), Kimeu Arphaxad Ngwava (Moi University, Kenya) and Stephen Kibet Kemboi (Moi University, Kenya).
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