# Dr Gabriel Verret

(Department of Mathematics, The University of Auckland)

# Local actions in vertex-transitive graphs

A graph is vertex-transitive if its group of automorphism acts transitively on its vertices. A very important concept in the study of these graphs is that of local action, that is, the permutation group induced by a vertex-stabiliser on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalise it to the case of directed graphs.

# Prof Michael Giudici

(School of Mathematics and Statistics, University of Western Australia)

# The synchronisation hierarchy for permutation groups

The concept of a synchronising permutation group was introduced nearly 15 years ago as a possible way of approaching The \v{C}ern\'y Conjecture. Such groups must be primitive. In an attempt to understand synchronising groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, $\mathbb{Q}$I-groups, spreading, separating, synchronsing, almost synchronising and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where.