February 7
Assaf Shani,
Concordia University
Generic dichotomies for Borel homomorphisms for the finite Friedman-Stanley jumps
The talk will begin by discussing the basic definitions and general goals behind the theory of Borel equivalence relations. We focus on the Friedman-Stanley jumps $=^{+n}$, for $n=1,2,...$ and $n=\omega$. These Borel equivalence relations represent the notions of being classifiable using invariants which are countable sets of reals, countable sets of countable sets of reals, and so on. We consider the problem of constructing a Borel reduction from $=^{+n}$ to some other equivalence relation.
For $n=1$ the situation is well understood and there are many such results. For example: Marker proved that for a first order theory with an uncountable type space, its isomorphism relation is above $=^{+1}$; Larson and Zapletal characterized the analytic equivalence relations above $=^{+1}$ as those which are 'unpinned' in the Solovay extension.
In this talk we present a new technique for proving that an equivalence relation is above $=^{+n}$, when $n>1$, based on Baire-category methods. As corollaries, we conclude that $=^{+\omega}$ is 'regular' (answering a question of Clemens), and that $=^{+n}$ is 'in the spectrum of the meager ideal' (extending a result of Kanovei, Sabok, and Zapletal for $n=1$).