Inverse semigroups algebraically model partial symmetry, generalizing the way groups model global symmetry. They were introduced in the 1950's as a correct algebraic setting for studying pseudogroups (of partial homeomorphisms between open subsets of a topological space). The association of pseudogroups with ordered groupoids goes back to Ehresmann in the 1950's. Later, the influential monograph of J. Renault (1980) introduced the close connections between inverse semigroups, étale groupoids, and C*-algebras. More recently, in the 1990's, Kellendonk developed connections of inverse semigroups to aperiodic tilings and C*algebras, inspired in part by Shechtman's work on quasicrystals in the 1980's. Paterson's monograph (1999) advanced the study of the connections between pseudogroups and topological groupoids; it was further analyzed by Lenz (2008) and Resende (2007). In particular, Resende found a nice connection between étale groupoids, complete inverse semigroups and involutive quantales. Lawson and Lenz (2010) clarified the relationships between the two works. Recently, in joint work of Kudryavtseva, Lawson, Lenz, and Resende, a global understanding of the relationship between inverse semigroups and étale groupoids has been put in a categorical setting: a non-commutative generalization of Stone duality between inverse semigroups and étale groupoids, mediated by quantales (non-commutative spaces).

This proposal is a follow-up to the successful Fields Workshop Semigroups and Categories held at the University of Ottawa, May 2-May 4, 2010 (Organizers: J. Funk, P. Hofstra, B. Steinberg). In addition, there was a working group Virtual Spaces and Hidden symmetries: applications of non-commutative Stone dualities in the theory of non-commutative spaces with Kudryavtseva, Lawson, Lenz, and Resende at the ICMS, Edinburgh (2014).

Location

All talks in Desmarais (Room 1140), 55 Laurier Ave. E.