I have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “standard grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “Cherlin’s conjecture for finite primitive binary permutation groups”.
The case for support document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it.
Excitingly, the university has agreed to fund a PhD student as part of this research. I’ll post a brief description of what the PhD will focus on below. If you are interested, please get in touch!
This programme of doctoral research is within the study of finite permutation group theory. Motivated by questions in model theory, about 20 years ago Cherlin introduced the notion of the relational complexity of a permutation group G; this is a positive integer which, roughly speaking, gives an indication of how easily the group G can act homogeneously on a relational structure. Cherlin’s conjecture concerns binary primitive permutation groups, i.e. primitive permutation groups which have relational complexity equal to 2. It is hoped that this conjecture might be proved in the next couple of years.
In light of this one naturally asks, next, whether we can classify groups with larger relational complexity, or whether we can calculate the relational complexity of important families of permutation groups. Calculating the relational complexity of a permutation group can be surprisingly tricky, so these sorts of questions can hide many mysteries!
In the process of working in this area, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.