NYLogic

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 $n=1$).

This first meeting will be partially devoted to organizing for the semester. But, I will also begin talking about Jamshid Derakhshan' survey paper on the model theory of the adeles entitled 'Model Theory of the Adeles and Number Theory'.

This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.

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This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.

Slides

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This talk, in the Mathematics Department Colloquium of the CUNY Graduate Center, will be aimed at a broad mathematical audience.
Traditionally, computability theory has been restricted to countable structures (such as groups or rings). We explain how digital computation by Turing machines can be applied to continuum-sized structures, with particular attention to the real numbers and the absolute Galois group of the rationals, and present some natural and intriguing questions regarding each.

We will discuss the paper of Cluckers, Derakhshan, Leeknegt and Macintyre on uniformly defining valuation rings in Henselian valued fields with finite or pseudofinite residue fields.

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.

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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'.

I will continue talking about Derakhsan's survey article 'Model Theory of Adeles and Number Theory'.

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We prove that a random group, in Gromov's density model with $d\lt 1/2$, satisfies an AE sentence (in the language of groups) if and only if this sentence is true in a nonabelian free group. This is a joint work with R. Sklinos.

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