Non-commutative Stone dualities and their applications

Classical Stone duality, relating Boolean algebras to topological spaces, can be generalized to a non-commutative setting where Boolean algebras may be replaced by certain types of semigroup and topological spaces by topological (and ultimately, localic) categories. This theory can be seen as part of non-commutative frame theory and combines ideas from semigroup theory and the theory of quantales. In my talks, I intend to focus on one example of such a duality: namely, that between a class of inverse monoids, the Boolean inverse $\wedge$-monoids, and a class of Hausdorff étale topological groupoids. This example helped motivate the general theory but, as I shall argue in my talks, is also of independent interest. In particular, the problem of classifying the countable inverse monoids of this type seems to be both meaningful and and interesting with connections to group theory, particularly to groups belonging to the Thompson family, and to the theory of dynamical systems. I plan to develop everything from scratch so no prior knowledge of inverse semigroups or étale topological groupoids is necessary.