158 research outputs found

    Model theoretic properties of the Urysohn sphere

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    We characterize model theoretic properties of the Urysohn sphere as a metric structure in continuous logic. In particular, our first main result shows that the theory of the Urysohn sphere is SOPn\text{SOP}_n for all nβ‰₯3n\geq 3, but does not have the fully finite strong order property. Our second main result is a geometric characterization of dividing independence in the theory of the Urysohn sphere. We further show that this characterization satisfies the extension axiom, and so forking and dividing are the same for complete types. Our results require continuous analogs of several tools and notions in classification theory. While many of these results are undoubtedly known to researchers in the field, they have not previously appeared in publication. Therefore, we include a full exposition of these results for general continuous theories.Comment: 23 pages, some proofs shortened, appendix adde

    Neostability in countable homogeneous metric spaces

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    Given a countable, totally ordered commutative monoid R=(R,βŠ•,≀,0)\mathcal{R}=(R,\oplus,\leq,0), with least element 00, there is a countable, universal and ultrahomogeneous metric space UR\mathcal{U}_\mathcal{R} with distances in R\mathcal{R}. We refer to this space as the R\mathcal{R}-Urysohn space, and consider the theory of UR\mathcal{U}_\mathcal{R} in a binary relational language of distance inequalities. This setting encompasses many classical structures of varying model theoretic complexity, including the rational Urysohn space, the free nthn^{\text{th}} roots of the complete graph (e.g. the random graph when n=2n=2), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of Th(UR)\text{Th}(\mathcal{U}_\mathcal{R}) by algebraic properties of R\mathcal{R}, many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOPn_n-hierarchy. Using the submonoid of idempotents in R\mathcal{R}, we also characterize superstability, supersimplicity, and weak elimination of imaginaries. Finally, we give necessary conditions for elimination of hyperimaginaries, which further develops previous work of Casanovas and Wagner.Comment: 32 page

    Distance structures for generalized metric spaces

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    Let R=(R,βŠ•,≀,0)\mathcal{R}=(R,\oplus,\leq,0) be an algebraic structure, where βŠ•\oplus is a commutative binary operation with identity 00, and ≀\leq is a translation-invariant total order with least element 00. Given a distinguished subset SβŠ†RS\subseteq R, we define the natural notion of a "generalized" R\mathcal{R}-metric space, with distances in SS. We study such metric spaces as first-order structures in a relational language consisting of a distance inequality for each element of SS. We first construct an ordered additive structure Sβˆ—\mathcal{S}^* on the space of quantifier-free 22-types consistent with the axioms of R\mathcal{R}-metric spaces with distances in SS, and show that, if AA is an R\mathcal{R}-metric space with distances in SS, then any model of Th(A)\text{Th}(A) logically inherits a canonical Sβˆ—\mathcal{S}^*-metric. Our primary application of this framework concerns countable, universal, and homogeneous metric spaces, obtained as generalizations of the rational Urysohn space. We adapt previous work of Delhomm\'{e}, Laflamme, Pouzet, and Sauer to fully characterize the existence of such spaces. We then fix a countable totally ordered commutative monoid R\mathcal{R}, with least element 00, and consider UR\mathcal{U}_\mathcal{R}, the countable Urysohn space over R\mathcal{R}. We show that quantifier elimination for Th(UR)\text{Th}(\mathcal{U}_\mathcal{R}) is characterized by continuity of addition in Rβˆ—\mathcal{R}^*, which can be expressed as a first-order sentence of R\mathcal{R} in the language of ordered monoids. Finally, we analyze an example of Casanovas and Wagner in this context.Comment: 30 page

    A L\'opez-Escobar theorem for metric structures, and the topological Vaught conjecture

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    We show that a version of L\'opez-Escobar's theorem holds in the setting of logic for metric structures. More precisely, let U\mathbb{U} denote the Urysohn sphere and let Mod(L,U)\mathrm{Mod}(\mathcal{L},\mathbb{U}) be the space of metric L\mathcal{L}-structures supported on U\mathbb{U}. Then for any Iso(U)\mathrm{Iso}(\mathbb{U})-invariant Borel function f ⁣:Mod(L,U)β†’[0,1]f\colon \mathrm{Mod}(\mathcal{L}, \mathbb{U})\rightarrow \lbrack 0,1], there exists a sentence Ο•\phi of LΟ‰1Ο‰\mathcal{L}_{\omega_{1}\omega} such that for all M∈Mod(L,U)M\in \mathrm{Mod}(\mathcal{L},\mathbb{U}) we have f(M)=Ο•Mf(M)=\phi ^{M}. At the same time we introduce a variant LΟ‰1Ο‰βˆ—\mathcal{L}_{\omega_1\omega}^\ast of LΟ‰1Ο‰\mathcal{L}_{\omega_1\omega} in which the usual quantifiers are replaced with category quantifiers, and establish the analogous theorem for LΟ‰1Ο‰βˆ—\mathcal{L}_{\omega_1\omega}^\ast. 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 LΟ‰1Ο‰\mathcal{L}_{\omega_{1}\omega}-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.Comment: 17 page

    Universal and homogeneous structures on the Urysohn and Gurarij spaces

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    Using Fra\" iss\' e theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, LL-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm LL on the Lispchitz-free space of the Urysohn space. Moreover, we enrich the Gurarij space by a universal and homogeneous closed subspace and norm one projection onto a 11-complemented subspace. We construct the Gurarij space by the classical Fra\" iss\' e theoretic approach.Comment: This paper contains new proofs and extends the results of the earlier draft arXiv:1305.0501. In the version 2, some arguments were improved. The third version contains updated information about the author's gran

    Definable Functions in Urysohn's Metric Space

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    Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function from U^n to U is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the definable function in case it is compact.Comment: 11 page

    The Ascoli property for function spaces and the weak topology of Banach and Fr\'echet spaces

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    Following [3] we say that a Tychonoff space XX is an Ascoli space if every compact subset K\mathcal{K} of Ck(X)C_k(X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every kRk_\mathbb{R}-space, hence any kk-space, is Ascoli. Let XX be a metrizable space. We prove that the space Ck(X)C_{k}(X) is Ascoli iff Ck(X)C_{k}(X) is a kRk_\mathbb{R}-space iff XX is locally compact. Moreover, Ck(X)C_{k}(X) endowed with the weak topology is Ascoli iff XX is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of β„“1\ell_1, we show that the following assertions are equivalent for a Banach space EE: (i) EE does not contain isomorphic copy of β„“1\ell_1, (ii) every real-valued sequentially continuous map on the unit ball BwB_{w} with the weak topology is continuous, (iii) BwB_{w} is a kRk_\mathbb{R}-space, (iv) BwB_{w} is an Ascoli space. We prove also that a Fr\'{e}chet lcs FF does not contain isomorphic copy of β„“1\ell_1 iff each closed and convex bounded subset of FF is Ascoli in the weak topology. However we show that a Banach space EE in the weak topology is Ascoli iff EE is finite-dimensional. We supplement the last result by showing that a Fr\'{e}chet lcs FF which is a quojection is Ascoli in the weak topology iff either FF is finite dimensional or FF is isomorphic to the product KN\mathbb{K}^{\mathbb{N}}, where K∈{R,C}\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}

    Math-Selfie

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    This is a write up on some sections of convex geometry, functional analysis, optimization, and nonstandard models that attract the author.Comment: 8 page

    Generic representations of countable groups

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    The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups Ξ“\Gamma in Polish groups GG, i.e. those elements in the Polish space Rep(Ξ“,G)\mathrm{Rep}(\Gamma,G) of all representations of Ξ“\Gamma in GG, whose orbit under the conjugation action of GG on Rep(Ξ“,G)\mathrm{Rep}(\Gamma,G) is comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or KnK_n-free graphs, and we show its connections with Ribes-Zalesski-like properties of the acting groups. We prove that N\mathbb{N} has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes-Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser and Melleray characterizing groups with a generic permutation representation. We also investigate representations of infinite groups Ξ“\Gamma in automorphism groups of metric structures such as the isometry group Iso(U)\mathrm{Iso}(\mathbb{U}) of the Urysohn space, isometry group Iso(U1)\mathrm{Iso}(\mathbb{U}_1) of the Urysohn sphere, or the linear isometry group \mbox{LIso}(\mathbb{G}) of the Gurarii space. We show that the conjugation action of Iso(U)\mathrm{Iso}(\mathbb{U}) on Rep(Ξ“,Iso(U))\mathrm{Rep}(\Gamma,\mathrm{Iso}(\mathbb{U})) is generically turbulent answering a question of Kechris and Rosendal.Comment: The main change is Theorem 1.6 which replaces Theorem 1.1 in the previous version. Referee's comments taken into account. Accepted to Transactions of the American Mathematical Societ

    Games and elementary equivalence of II1\rm II_1 factors

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    We use Ehrenfeucht-Fra\"iss\'e games to give a local geometric criterion for elementary equivalence of II1_1 factors. We obtain as a corollary that two II1_1 factors are elementarily equivalent if and only their unitary groups are elementarily equivalent as Z4\mathbb Z_4-metric spaces.Comment: 13 pages; final version to appear in the Pacific Journal of Mathematic
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