SEMINAR: Groups and Combinatorics Seminar: The Weiss Conjecture and inspirations thereof

Time and place: 16:00 Friday 20 May in Weatherburn LT

Speaker: Luke Morgan (UWA)

Title: The Weiss Conjecture and inspirations thereof

Abstract: In 1979 Richard Weiss made a beautiful conjecture: There exists a function f on the natural numbers such that, for each G-vertex-transitive locally-primitive graph X, the order of a vertex-stabiliser is bounded by f(d), with d the valency X. Although the conjecture remains open in full generality, many mathematicians have been attracted to work on this problem, and in several interesting cases it has been shown that the conjecture is true. As well as proofs of (cases of) the statement, Richard’s conjecture also spurred people to understand vertex-transitive graphs with local actions of certain flavours, such as locally-quasiprimitive graphs. This lead to the Praeger Conjecture  of 2000, and for locally-semiprimitive graphs, the Potocnik-Spiga-Verret Conjecture of 2012. At this point the graphs faded to the background and general questions about semiprimitive permutation groups were raised. These turned out to be quite difficult, but one can say something. I’ll give an overview of some of these recent results and describe what we know on the various conjectures - that is - an overview of the wonderful mathematics that was inspired by Richard’s Conjecture.