Model theoretic properties of the Urysohn sphere

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 $\text{SOP}_n$ for all $n\geq 3$, 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

Given a countable, totally ordered commutative monoid $\mathcal{R}=(R,\oplus,\leq,0)$, with least element $0$, there is a countable, universal and ultrahomogeneous metric space $\mathcal{U}_\mathcal{R}$ with distances in $\mathcal{R}$. We refer to this space as the $\mathcal{R}$-Urysohn space, and consider the theory of $\mathcal{U}_\mathcal{R}$ 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 $n^{\text{th}}$ roots of the complete graph (e.g. the random graph when $n=2$), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of $\text{Th}(\mathcal{U}_\mathcal{R})$ by algebraic properties of $\mathcal{R}$, many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOP$_n$-hierarchy. Using the submonoid of idempotents in $\mathcal{R}$, 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

Let $\mathcal{R}=(R,\oplus,\leq,0)$ be an algebraic structure, where $\oplus$ is a commutative binary operation with identity $0$, and $\leq$ is a translation-invariant total order with least element $0$. Given a distinguished subset $S\subseteq R$, we define the natural notion of a "generalized" $\mathcal{R}$-metric space, with distances in $S$. We study such metric spaces as first-order structures in a relational language consisting of a distance inequality for each element of $S$. We first construct an ordered additive structure $\mathcal{S}^*$ on the space of quantifier-free $2$-types consistent with the axioms of $\mathcal{R}$-metric spaces with distances in $S$, and show that, if $A$ is an $\mathcal{R}$-metric space with distances in $S$, then any model of $\text{Th}(A)$ logically inherits a canonical $\mathcal{S}^*$-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 $\mathcal{R}$, with least element $0$, and consider $\mathcal{U}_\mathcal{R}$, the countable Urysohn space over $\mathcal{R}$. We show that quantifier elimination for $\text{Th}(\mathcal{U}_\mathcal{R})$ is characterized by continuity of addition in $\mathcal{R}^*$, which can be expressed as a first-order sentence of $\mathcal{R}$ 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

We show that a version of L\'opez-Escobar's theorem holds in the setting of logic for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let $\mathrm{Mod}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\mathrm{Iso}(\mathbb{U})$-invariant Borel function $f\colon \mathrm{Mod}(\mathcal{L}, \mathbb{U})\rightarrow \lbrack 0,1]$, there exists a sentence $\phi$ of $\mathcal{L}_{\omega_{1}\omega}$ such that for all $M\in \mathrm{Mod}(\mathcal{L},\mathbb{U})$ we have $f(M)=\phi ^{M}$. At the same time we introduce a variant $\mathcal{L}_{\omega_1\omega}^\ast$ of $\mathcal{L}_{\omega_1\omega}$ in which the usual quantifiers are replaced with category quantifiers, and establish the analogous theorem for $\mathcal{L}_{\omega_1\omega}^\ast$. 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 $\mathcal{L}_{\omega_{1}\omega}$-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

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, $L$-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm $L$ 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 $1$-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

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

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

Math-Selfie

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

The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups $\Gamma$ in Polish groups $G$, i.e. those elements in the Polish space $\mathrm{Rep}(\Gamma,G)$ of all representations of $\Gamma$ in $G$, whose orbit under the conjugation action of $G$ on $\mathrm{Rep}(\Gamma,G)$ is comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or $K_n$-free graphs, and we show its connections with Ribes-Zalesski-like properties of the acting groups. We prove that $\mathbb{N}$ 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 $\Gamma$ in automorphism groups of metric structures such as the isometry group $\mathrm{Iso}(\mathbb{U})$ of the Urysohn space, isometry group $\mathrm{Iso}(\mathbb{U}_1)$ of the Urysohn sphere, or the linear isometry group \mbox{LIso}(\mathbb{G}) of the Gurarii space. We show that the conjugation action of $\mathrm{Iso}(\mathbb{U})$ on $\mathrm{Rep}(\Gamma,\mathrm{Iso}(\mathbb{U}))$ 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

We use Ehrenfeucht-Fra\"iss\'e games to give a local geometric criterion for elementary equivalence of II$_1$ factors. We obtain as a corollary that two II$_1$ factors are elementarily equivalent if and only their unitary groups are elementarily equivalent as $\mathbb Z_4$-metric spaces.Comment: 13 pages; final version to appear in the Pacific Journal of Mathematic