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 'leftcoarse structure' of a topological group introduced by C. Rosendal. Associated to this notion are the 'locally bounded Polish groups': those for which the leftcoarse structure is the bounded coarse structure of some compatible, leftinvariant 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 leftcoarse 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. 