不变泛解析拟序

Universality of group embeddability

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].

Invariant universality for quandles and fields

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.

Invariant universality for projective planes

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second author to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in c…

大规模模型上可嵌入关系的描述集合论复杂性

The descriptive set-theoretical complexity of the embeddability relation on models of large size

我们证明了如果κ是弱紧基数，那么大小为κ的（广义）树上的可嵌入关系是不变泛的。这意味着对于广义康托空间2^κ上的每个解析拟序R，存在一个L_{κ^+ κ} -语句集，使得在其大小为κ的模型上的可嵌入性关系，这些模型都是树，是Borel双可约的（事实上，是类Borel同构的）到R。特别地，这意味着大小为κ的树的可嵌入性关系对于解析拟序是完备的。这些结果推广了文献[LR 05，FMR 11]中关于κ=ω的类似结果，也部分推广了文献[Bau 76]中关于尺寸为κ的线性阶上可嵌入关系的结构的结果.

Universal covers of commutative finite Morley rank groups

We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of "universal covers" of rigid divisible commutative finite Morley rank groups.

作为解析Zeberki结构的阿贝尔簇的覆盖

Covers of Abelian varieties as analytic Zariski structures

我们使用数理逻辑的工具来分析复代数簇上的路的概念，并导致制定特定的代数簇的基本群的“刚性”性质，以及定义一个真正的拓扑结构密切相关的etale拓扑结构。这些出现作为标准的不可数范畴，或者更确切地说，稳定性和同质性，正式可数语言，我们建议描述同伦类的道路上的品种，或等价地说，其万有覆叠空间。我们还表明，与此拓扑结构，万有覆叠空间的品种是一个分析的Zagliki结构。在技术上，我们给出了一类解析Zeroki结构的可数$L_{omega_1ω}$ -语句公理化，该结构包含数域上代数簇的万有覆叠空间，并对代数簇作了一定的假设.