Longyun Ding
Let $(X_n,d_n),\,n\in\Bbb N$ be a sequence of pseudo-metric spaces, $p\ge 1$. For $x,y\in\prod_{n\in\Bbb N}X_n$, let $(x,y)\in E((X_n)_{n\in\Bbb N};p)\Leftrightarrow\sum_{n\in\Bbb N}d_n(x(n),y(n))^p<+\infty$. For Borel reducibility between equivalence relations $E((X_n)_{n\in\Bbb N};p)$, we show it is closely related to finitely Hölder($α$) embeddability between pseudo-metric spaces.
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We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called Hölder$(α)$ embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between $E(L_r,p)$'s and $E(c_0,p)$'s for $r,p\in[1,+\infty)$. We also answer a problem presented by Kanovei in the affirmative by showing that $C({\Bbb R}^+)/C_0({\Bbb R}^+)$ is Borel bireducible to ${\Bbb R}^{\Bbb N}/c_0$.
Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the isometric embeddability between separable groups with a bounded bi-invariant complete metric are all invariantly universal analytic quasi-orders. This strengthens some results from [Wil14] and [FLR09].
In this paper we study the Borel reducibility of Borel equivalence relations, including some orbit equivalence relations, on the generalised Baire space $κ^κ$ for an uncountable $κ$ with the property $κ^{<κ}=κ$. The theory looks quite different from its classical counterpart where $κ=ω$, although some basic theorems do generalise.
Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order
本文证明了无 torsion 的 Abel 群是忠实 Borel 完整的，并展示了纯可嵌入性是一个完全的解析准序。
We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions.
We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to those discovered in descriptive set theory. The Borel reducibility structure of Borel sets and (partially) CD reducibility structure of CD sets in *N is described. We prove that all CD ERs with countable equivalence classes are CD-smooth, but not all are B-smooth, for instance, the ER of having finite difference on *N. Similarly to the Silver dichotomy theorem …
Degree spectra of analytic complete equivalence relations
本文研究图上的双可嵌入与初等双可嵌入关系，通过Borel归约证明初等双可嵌入为解析完全等价关系，并揭示其度数谱与跳跃谱的联系。
We prove that for every uncountable cardinal $κ$ such that $κ^{<κ}=κ$, the quasi-order of embeddability on the $κ$-space of $κ$-sized graphs Borel reduces to the embeddability on the $κ$-space of $κ$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $κ$-space of $κ$-sized $R$-modules, for every $\mathbb{S}$-cotorsion-free ring $R$ of cardinality less than the continuum. As a consequence we get that all the previous are complete $\boldsymbolΣ^1_1$ quasi-orders.
We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of $D$. When $D$ contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If $D$ is well-ordered the situation is more…
Shelah's Main Gap and the generalized Borel-reducibility
本文证明了 Shelah 主间隙与广义 Borel 可归约性之间的对应关系，在特定基数条件下不可分类理论的模型同构严格高于可分类理论，并力迫证明可数一阶理论的模型同构具有解析余解析或解析完全的二分类性质。
Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel function $f:X\to Y$ that is both a reduction from $E$ to $E'$ and a reduction from $F$ to $F'$. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs $E\subseteq F$ of smooth countable Borel equivalence relations up to simultaneous Borel bireducibility and biembeddability, and a significant portion of such pairs up to simultaneous Borel i…
We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order.
We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.
We study the Borel reducibility of isomorphism relations in the generalized Baire space $κ^κ$. In the main result we show for inaccessible $κ$, that if $T$ is a classifiable theory and $T'$ is stable with OCP, then the isomorphism of models of $T$ is Borel reducible to the isomorphism of models of $T'$.
We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation $\cong^\ast_{ω+1,0}$ is strictly below $\cong^\ast_{ω+1,<ω}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{ω+1,<ω}$ provides invariants for $Σ^0_{ω+1}$ equivalence relations induced by actions of $S_\infty$, while $\cong^\ast_{ω+1,0}$ provides invariants for $Σ^0_{ω+1}$…
We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.
We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras.
If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,τ)$ is called the \emph{Wadge quasi-order} for $(X,τ)$. We further show that this quasi-order, restricted to…
Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L_{ω_1 ω}) is far from complete (see [5, 2]). In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivale…
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…
We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes …
Embeddability of graphs and Weihrauch degrees
本文利用有效Wadge可归约性与Weihrauch可归约性，研究固定可数图G的（导出）子图同构判定与查找任务的计算复杂性，并回答了BeMent等人提出的若干公开问题。
Analogues of the countable Borel equivalence relations in the setting of computable reducibility
本文研究可计算归约下可数Borel等价关系的两种类似物，证明基于Feldman-Moore有效化的概念存在严格二分法，而基于Lusin-Novikov的概念则不存在，并存在介于相等与几乎相等关系之间的链和反链，从而否定了Glimm-Efros二分法的一般类似物。
We study a fine hierarchy of Borel-piecewise continuous functions, especially, between closed-piecewise continuity and $G_δ$-piecewise continuity. Our aim is to understand how a priority argument in computability theory is connected to the notion of $G_δ$-piecewise continuity, and then we utilize this connection to obtain separation results on subclasses of $G_δ$-piecewise continuous reductions for uniformization problems on set-valued functions with compact graphs. This method is also applicable for separating various non-constructive principles in the Weihrauch lattice.
In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders. In this paper we are concerned with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We…
We present a selection of basic results on Borel reducibility of Borel ideals and equivalence relations, especially those with comparably short proofs. The focal point are reducibility/irreducibility results related to some special equivalences like $E_0, E_1, E_2, E_3, E_\infty, Z_0,$ and Banach-induced equivalences $l_p,$ in particular several dichotomy theorems. The bulk of results included in the book were obtained in the 1990s, but some rather recent theorems are presented as well, like Rosendal's proof that Borel ideals are cofinal within Borel equivalences of general form.
On unsuperstable theories in GDST
本文在GDST框架下用着色树研究一阶理论同构关系的$κ$-Borel归约，证明在适当基数假设下，非超稳定理论的模型同构严格高于可分类理论的模型同构。
Torsion-free abelian groups of finite rank and fields of finite transcendence degree
本文比较有限秩无挠阿贝尔群类与有限超越度域类的描述复杂性，给出函子性图灵可计算嵌入，并证明在可计算可数可归约性下 $\operatorname{TFAb}_1$ 的同构关系处于最顶端。
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bmΔ^1_2$ in a satisfying way.
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from…
For $f \colon [0,1] \rar \real^{+}$, consider the relation $\mathbf{E}_{f}$ on $[0,1]^ω$ defined by $(x_{n}) \mathbf{E}_{f} (y_{n}) \Leftrightarrow \sum_{n < ω} f(|y_{n} - x_{n}|) < \infty.$ We study the Borel reducibility of Borel equivalence relations of the form $\mathbf{E}_{f}$. Our results indicate that for every $1 \leq p < q < \infty$, the order $\leq_{B}$ of Borel reducibility on the set of equivalence relations $\{\bE \colon \bE_{\Id^{p}} \leq_{B} \bE \leq_{B} \bE_{\Id^{q}}\}$ is more complicated than expected, e.g. consistently every linear order of cardinality continuum embeds into …
We prove that the Borel space of torsion-free Abelian groups with domain $ω$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.
We study the Borel-reducibility of isomorphism relations in the generalized Baire space $κ^κ$. In the main result we show for inaccessible $κ$, that if $T$ is a classifiable theory and $T'$ is superstable with the strong dimensional order property (S-DOP), then the isomorphism of models of $T$ is Borel reducible to the isomorphism of models of $T'$. In fact we show the consistency of the following: If $κ$ is inaccessible and $T$ is a superstable theory with S-DOP, then the isomorphism of models of $T$ is $Σ_1^1$-complete.
We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that the relation $E_{K_σ}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or $\ell_p, 1 \leq p <2$. We show that the relation $E_{K_σ} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$.
A selection of basic results on Borel reducibility of ideals and equivalence relations, especially those with comparably short proofs. This is an unfinished text as yet. Some proofs have missing parts and loose ends. kanovei@mccme.ru and vkanovei@math.uni-wuppertal.de are my contact addresses.
Answering one of the main questions of [FHK14, Chapter 7], we show that there is a tight connection between the depth of a classifiable shallow theory $T$ and the Borel rank of the isomorphism relation $\cong^κ_T$ on its models of size $κ$, for $κ$ any cardinal satisfying $κ^{< κ} = κ> 2^{\aleph_0}$. This is achieved by establishing a link between said rank and the $\mathcal{L}_{\infty κ}$-Scott height of the $κ$-sized models of $T$, and yields to the following descriptive set-theoretical analogue of Shelah's Main Gap Theorem: Given a countable complete first-order theory $T$, either $\cong^κ_…
We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in $A$, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that $A$ must have in order that all Polish spaces with distances in that set belong to a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complex…
We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_σ$ equivalence relation is Borel reducible to the uniform homeomorphism relation, and we also determine the possible complexities of the relation when restricted to some small classes of Banach spaces. Moreover, we determine the exact complexity of the local equivalence relation between Banach spaces, namely that it is bireducible with $K_σ$. Finally, we construct a class …
In this article, we study an analogue of $tt$-reducibility for points in computable metric spaces. We characterize the notion of the metric $tt$-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.
Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance
本文研究标准Borel空间上伪度量的Borel可归约性与轨道伪度量概念，证明波兰度量空间的Gromov-Hausdorff距离与某类轨道伪度量的泛元双可归约。
Let $X$ be a Polish space, $d$ a pseudo-metric on $X$. If $\{(u,v):d(u,v)<δ\}$ is ${\bfΠ}^1_1$ for each $δ>0$, we show that either $(X,d)$ is separable or there are $δ>0$ and a perfect set $C\subseteq X$ such that $d(u,v)\geδ$ for distinct $u,v\in C$. Granting this dichotomy, we characterize the positions of $\ell_p$-like and $c_0$-like equivalence relations in the Borel reducibility hierarchy.
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