## Frontiers Talk by Sam van Gool

Sam van Gool gave a *Frontiers talk* today at our department.
Frontiers talks provide a venue for researchers to present bleeding-edge work to a more specialized audience than what one finds at a colloquium.
Sam took full advantage of the format and delivered an amazing presentation on the connections between formal language theory and abstract algebra.
His topic intersects with recent work on subregular languages spearheaded by Jeffrey Heinz here at Stony Brook.
Needless to say, the post-talk dinner had a very lively discussion.

The title and abstract of Sam’s talk are shown below.

## Language hierarchies and profinite monoids via Stone duality

Finite monoids are useful algebraic invariants for regular languages: logical properties of a regular language correspond to algebraic properties of the finite monoids that recognize it. Profinite monoids figure prominently in this theory as the appropriate inverse limit objects for well-behaved classes of finite monoids. While the theory of (pro)finite monoids and regular languages is very classical, many problems in it remain open. In particular, there are notoriously difficult unanswered questions about the decidability of hierarchies of regular languages defined by restricting quantifier depth. In this talk, I will show how, in the past few years, the mathematical theory of Stone duality has helped to shed new light on some of these problems.