Pure Maths Colloquium: Simon Smith

In topological group theory, the study of locally compact groups naturally splits into looking at those that are connected and those that are totally disconnected. The connected case is broadly understood, following the solution to Hilbert’s Fifth Problem. The totally disconnected case (“tdlc groups”) was considered intractable until breakthrough work by George Willis in the 1990s. In 2002, work by Rögnvladur Möller showed that Willis’ theory could be recovered via the theory of infinite permutation groups acting on locally-finite graphs.

In this talk I will discuss some of the interplay between tdlc theory and the theory of infinite permutation groups. I will try to highlight places where the application of permutation groups has lead to breakthroughs in our understanding of tdlc groups. I will also present some high-profile open problems on tdlc groups that I think could potentially be solved using permutation groups.