# Marcin Sabok: A dichotomy in canonization of analytic equivalence relations

Monday, January 7, 2013, 16.30
Seminar room 1.007, Mathematical Institute, University of Bonn

Speaker: Marcin Sabok (IMPAN Warsaw)

Title: A dichotomy in canonization of analytic equivalence relations

Abstract:

I will show and discuss a dichotomy that occurs in attempts of canonization of analytic equivalence relations. The dichotomy says that under suitable minimality condition for P_I, if E is an analytic equivalence relation on a Polish space, then – either there is a Borel I-positive set which consists of E-independent elements (i.e. E canonizes to the identity) – or there is a Borel I-positive set such that E|B is I-ergodic (this means the saturations of every two Borel I-positive subsets of B have nonempty intersection ) I will present some typical examples for both types of behavior and sketch a proof of the dichotomy. The proof will use suitably chosen generic ultrapowers. This is joint work with Vladimir Kanovei and Jindra Zapletal.