Ph.D. Public Lecture (Math) - Michal Cizek

"Profinite group actions on sets and Graphs with application to Galois theory"

In this talk we will explore relations between Galois theory and Graph theory. By the theorem of Frucht every group is an automorphism group of an undirected connected loopless graph. We will expand on this theorem by introducing the group action Cayley graphs and show that in fact not only a finite group, but also the action of a finite group on a given finite set can be represented by a graph. We will then show how this result could be used in representing finite separable field extensions and more generally etale algebras. Finally, we will generalize this result to the action of a profinite group on a profinite set by introducing the profinite group action Cayley graphs.