Frieze Patterns And Farey Complexes

My preferred project is "Frieze Patterns and Farey Complexes", supervised by Dr. Ian
Short and Dr. Katherine Staden. I am familiar with integers friezes, the Farey complex, and
hyperbolic tilings in general. My knowledge comes from a variety of sources, the nicest of
which is probably A. Kirillov's book A Tale of Two Fractals. I also have some familiarity with
cluster algebras. This problem lies nearby to the topics of lattices, and geometry of continued
fractions, which are considered in my note arXiv:2003.10251. One more reason I think this
project is fascinating is that it is built on the scientific legacy of J. H. Conway, whom I deeply
respect. To approach this project, strategically, after studying in detail what is known about
friezes over the rings Z/nZ, it seems reasonable to me to first look at the rings of low-order
matrices over Fp.
My second-preferred research project is "The Relational Complexity of a Finite Permutation
Group", supervised by Dr. Nick Gill and Dr. Bridget Webb. I have the necessary background
related to finite groups, although it's hard to say which of the research directions listed in
the project description feels most promising. I believe that my experience with computer
algebra systems, in particuar Sage, would be useful for this project, perhaps not directly but by
allowing to get some insights and ideas from computer-aided detailed consideration of particular
examples.