Noncommutative Stone dualities and their applications
24/06/2014 June 24, 2014, Tuesday, 11:00 – 12:00, Room B201; June 26, 2014, Thursday, 14:00 – 15:00, Room B201; July 1, 2014, Tuesday, 14:00 – 15:00, Room A225.
Mark V Lawson (HeriotWatt University, Edinburgh)
Institute for Interdisciplinary Research  University of Lisbon, 23 june  2 july, 2014
Classical Stone duality, relating Boolean algebras to topological spaces, can be generalized to a noncommutative 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 noncommutative 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.
