Title: Countable Borel equivalence relations are generically hyperfinite
Abstract: One of the most fundamental and well-studied notions in descriptive set theory is the notion of a countable Borel equivalence relation (CBER), that is, an equivalence relation on a standard space X which is Borel as a subset of X^2, and has countable classes. Among these there is a subclass which has been of particular interest, the class of hyperfinite CBERs, which is defined as those which can be approximated by equivalence relations with finite classes. In classical works of Ornstein-Weiss and Connes-Feldman-Weiss, foundational results about this class were proven, and in particular their relation to (measurable) dynamics was demonstrated. Specifically, these works show that the CBER induced by an ergodic pmp action of a non-amenable group is never hyperfinite. In a surprising contrast to this, it was later shown by Sullivan, Weiss and Wright that every countable Borel equivalence relation is hyperfinite when restricted to a comeagre set. In my talk I will introduce the relevant notions and show a beautiful proof of the last result due to Miri Segal which appears in her PhD thesis, "Hyperfinite equivalence relations and forcing". 
This is the most beautiful proof I know.
No prior knowledge in descriptive set theory will be assumed, and despite the title, no forcing will occur.