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 The result builds on an unpublished result of Becker from 2001 and fits in the area of invariant descriptive set theory.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'.

Polish spaces, the objects of study of descriptive set theory, are completely metrizable topological spaces that have a countable dense subset. For example, the reals - the first Polish space in the world. We will look at 'sigma-ideals' of compact subsets of a Polish space. Think of a sigma-ideal as being a collection of 'small' compact sets, under some notion of smallness -- so for example, your Polish space could be the interval $[0,1]$ and your sigma-ideal could be the collection of all its compact sets of Lebesgue measure $0$. The descriptive-set-theoretic study of these objects yields rich results for the following reason. If you look at the collection of all the compact subsets of a Polish space, that too, topologized and metrized in a natural way, turns out to be a Polish space. This means that you can look at your sigma-ideal of compact sets in two places: in the original space, say $E$, and in the `hyperspace' of all compact sets of $E$. In this talk we will deal with sigma-ideals that can be represented in a very nice way inside this hyperspace, and we will examine the behaviour of so-called G-delta subsets of $E$ with respect to this representation.

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