Automorphisms of the Rado Complex

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Under the advice of my supervisor, Peter Cameron, I have decided to change my original project slightly, so instead of looking at the Rado graph, I have instead been focusing on the Rado complex. Simplicial complexes can be seen as generalisations of graphs, where instead of just looking at edges between points, we can consider triangles between sets of 3 points, tetrahedrons between sets of 4 points, and so on to any n-dimensional shape, where all lower dimensional faces are contained within the shape. The Rado Complex is such an object, which has similar properties to the Rado Graph, so I have been investigating which properties of the graph carry across to this more general object. In particular, I have been studying the regular automorphisms of the Rado complex, which in effect means I have been studying its symmetries.

These are the first 4 simplexes; the next one is 4-dimensional!
Image: Accessed 26/6/20, 18:31

I have thoroughly enjoyed my research so far, but it has been different to how I envisioned it. Far from being boring, it has been intense in a way that I failed to anticipate before starting. When it has been going well, it has been hard to pull myself away from it; but when there has been a long time since a breakthrough, I have found it difficult to motivate myself to continue to search for ways to attack the problem. I think this has been compounded by the situation we have all found ourselves in where, working in isolation with no one to be held physically accountable to, I have had to actively try to be a self-leader in order to complete my work to the best of my ability.

Despite these challenges, I do recognise that compared to many of my fellow scholars I have faced relatively few impediments to my research, especially since all of the resources I need are available readily on the internet and I have no need for access to special equipment. I also feel extremely fortunate that I have had this project to focus on at a time when many of my peers have had similar opportunities cancelled; and so I would like to thank everybody at the Laidlaw programme for giving us this opportunity, in spite of all of the difficulties that must stem from having to run the scholarship remotely!

Poster image is of a construction of the Rado Graph: Accessed 26/6/20, 18:34

Ole Jorgensen

Laidlaw Scholar - Mathematics, St. Andrews