Logic Workshop

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 $n=1$).

Expansions of the ordered additive group of the reals (or more generally definably complete expansions of ordered Abelian groups) of finite dp-rank are a class of reasonably well-behaved ordered structures that generalize the class of o-minimal structures. In this talk I will give a survey of ongoing work with John Goodrick on exploring the properties of definable sets in this class of structures.

In recent work with Luca Motto Ros we prove that under analytic determinacy there exists an analytic relation that is not class-wise Borel embeddable into any orbit equivalence relation. The result builds on an unpublished result of Becker from 2001 and fits in the area of invariant descriptive set theory. I will mainly discuss our result and how it is related to a major conjecture in the field known as the '$E_1$ conjecture'.

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