Topological groups: where to from here?
by
Vladimir Pestov
Victoria University of Wellington and Australian National University

For a randomly chosen mathematician outside of the field of general topology (or rather `general topological algebra'), the words `topological group' most probably sound synonimous with `locally compact group, ' and indeed, the depth, beauty, and importance of theory of locally compact groups are overwhelming. What are the perspectives of developing a coherent new theory of topological groups in the absence of local compactness? In this talk we will argue that the right path to follow is to isolate new classes of examples of topological groups of importance and study their properties, rather than to try and generalize known results for locally compact groups to a wider setting. Such classes of groups include free topological groups on compacta and also `massive' groups (groups of homeomorphisms, unitary groups, groups of isometries, groups of automorphisms of measure spaces, etc.) Indeed, it turns out that in some respects their properties are `orthogonal' to those of locally compact groups. Our survey talk is `coordinatized' by such notions as embedding theorems for topological groups, actions of topological groups on compacta, and links with outside disciplines, especially various parts of functional analysis.

Date received: July 20, 1999