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Wednesday, October 23, 2019

**Abstract:** An equivalence relation is essentially countable when it is Borel reducible to the orbit equivalence relation of some action of a countable group. In a 2005 paper, G. Hjorth introduced the notion of a stormy Polish G-space and showed that this condition is the fundamental obstruction to essential countability for orbit equivalence relations. Specifically, he proved that if a Polish group, G, acts continuously on a Polish space, X, with a Borel orbit equivalence relation then this relation fails to be essentially countable if and only there is a continuous G-equivariant embedding of a stormy action into X. In the first part of this talk we recall the basic concepts and results in Hjorth's work on stormy actions. In the second part, we apply these to the constructions considered in recent joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos for actions of non-Archimedian groups and separable Banach spaces.