Tyler Arant (UCLA)

Title: Borel graphings of analytic equivalence relations.

Abstract: Every Borel graph has a connectedness equivalence relation which is (at worst) analytic. In this talk, we explore the other direction: which analytic equivalence relations are the connectedness equivalence relations of Borel graphs, i.e., which analytic equivalence relations are Borel graphable? The talk will present interesting examples and non-examples of Borel graphable equivalence relations; indeed, these examples come from many areas: model theory (isomorphism of countable structures), computability theory (countable admissible ordinals), and Borel group actions. We will also discuss some results and open problems about the possible diameters (the least  such that all connected nodes have a path of length  connecting them) of Borel graphings. This work is joint with Alekos Kechris and Patrick Lutz