Mathematical logic seminar – May 1 2018

Time: 3:30pm – 4:30 pm

Room: Wean Hall 8220

Speaker: Joseph Zielinski

Department of Mathematical Sciences

CMU

Title: Roelcke precompact sets in Polish groups

Abstract:

In these talks we first recall the uniform structures associated to a topological group. We then present J. Roe’s notion of a coarse space, and consider compatible coarse structures on groups with emphasis on the ‘left-coarse structure’ of a topological group introduced by C. Rosendal. Associated to this notion are the ‘locally bounded Polish groups’: those for which the left-coarse structure is the bounded coarse structure of some compatible, left-invariant metric.

Next, we introduce the Roelcke precompact subsets of a Polish group, which admit equivalent natural definitions both in terms of the lower uniformity on the group and as a subideal of the bounded sets in the left-coarse structure. Through this we define the ‘locally Roelcke precompact Polish groups’ — a subfamily of the locally bounded Polish groups — and present various examples, applications, and several characterizations of these groups.