### $\nabla $-model and distributivity in Boolean algebras

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A Banach space X with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if X/Y is hereditarily indecomposable for any infinite-codimensional subspace Y with a successive finite-dimensional decomposition on the basis of X. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.

We give a classical proof of the theorem stating that the $\sigma $-ideal of meager sets is the unique $\sigma $-ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the ${L}_{\omega \u2081\omega}\left(Q\right)$ logic obtained by adding predicates for Borel sets.

In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

We study the isomorphism relation of invariant Borel classes of countable mono-unary algebras and prove a strong dichotomy theorem.

The algebraic counterpart of the Wagner hierarchy consists of a well-founded and decidable classification of finite pointed ω-semigroups of width 2 and height ωω. This paper completes the description of this algebraic hierarchy. We first give a purely algebraic decidability procedure of this partial ordering by introducing a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees. The Wagner degree of any ω-rational language can therefore be computed...

The algebraic study of formal languages shows that ω-rational sets correspond precisely to the ω-languages recognizable by finite ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the ω-rational language to the ω-semigroup context. More precisely, we first show that the Wagner degree is indeed a syntactic invariant. We then define a reduction relation on...

We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the modulus of continuity.

We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let denote the Urysohn sphere and let Mod(,) be the space of metric -structures supported on . Then for any Iso()-invariant Borel function f: Mod(,) → [0,1], there exists a sentence ϕ of ${}_{\omega \u2081\omega}$ such that for all M ∈ Mod(,) we have $f\left(M\right)={\varphi}^{M}$. This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group...

A class of Banach spaces, countably determined in their weak topology (hence, WCD spaces) is defined and studied; we call them strongly weakly countably determined (SWCD) Banach spaces. The main results are the following: (i) A separable Banach space not containing ℓ¹(ℕ) is SWCD if and only if it has separable dual; thus in particular, not every separable Banach space is SWCD. (ii) If K is a compact space, then the space C(K) is SWCD if and only if K is countable.