The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e. sets, and more specifically, on various classes of c.e. structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e. graphs or on computably presented groups. Here, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remains.

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if $A$ is a separable algebra which is either simple or stable, then the corona of $A$ has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit

We introduce an analog to the notion of Polish space for spaces of weight $\leq\kappa$, where $\kappa$ is an uncountable regular cardinal such that $\kappa^{<\kappa}=\kappa$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for $\kappa$ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^\kappa$ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size $>\kappa$ are isomorphic by a $\kappa$-Borel function. We then consider a dynamic version of the Choquet game and show that in this case the existence of a winning strategy for player II implies the existence of a winning tactic, that is, a strategy that depends only on the most recent move. We also study a generalization of Polish ultrametric spaces where the ultrametric is allowed to take values in a set of size $\kappa$. We show that in this context, there is a family of universal Urysohn-type spaces, and we give a characterization of such spaces which are hereditarily $\kappa$-Baire

Abstract. We show that a version of López-Escobar's theorem holds in the setting of model theory for metric structures. More precisely, let U denote the Urysohn sphere and let Mod(L, U) be the space of metric L-structures supported on U. Then for any Iso(U)-This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group action is Borel isomorphic to the isomorphism relation on the set of models of a given Lω 1 ω -sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture. §1. Background and statement of main result A well-known theorem of López-Escobar [LE] says roughly that every Borel class of countable structures can be axiomatized by a sentence in the logic where countable conjunctions and disjunctions are allowed. The theorem has been generalized to apply to wider classes of structures, using sentences from a variety of logics (see for example [T, V]).To state López-Escobar's theorem more precisely, let L be a countable first-order language consisting of the relational symbols {R i } where each R i has arity n i . The space Mod(L) of countably infinite L-structures is given byand we note it is compact in the product topology. The space carries a natural S ∞ -action by left-translation on each factor, and the S ∞ -orbits are precisely the isomorphism classes. Next, recall that L ω 1 ω denotes the extension of first-order logic in which countable conjunctions and disjunctions are allowed (formulas are still only allowed to have finitely many free variables). If φ is a sentence of L ω 1 ω then the subset Mod(φ) ⊂ Mod(L) consisting just of the models of φ is clearly S ∞ -invariant (isomorphism invariant), and it is easy to see that 2000 Mathematics Subject Classification. Primary 03C95, 03E15; Secondary 54E50.

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely ∆ 1 2 reductions, and hence to the infinite time computable reductions. 30

ABSTRACT. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

ABSTRACT. We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.