Measurable group theory, descriptive set theory and model theory of homogeneous structures

可测群论、描述集合论和齐次结构模型论

• 批准号: RGPIN-2020-05445
• 负责人: Sabok, Marcin
• 金额: $ 2.26万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759115
• 关键词: Measurable; group; theory; descriptive; set

This is a proposal primarily in mathematical logic, concerning some of its most active areas, such as descriptive set theory and model theory of homogeneous structures. The proposal continuous a research program of the PI. This program concentrates on several themes that have been central in the discipline in the recent years, such as Borel combinatorics, measurable group theory, the Hrushovski property with its connections to profinite group theory and the automatic continuity phenomenon. Part I. Flows in amenable groups. This part will focus on discovering structural phenomena in actions of amenable group on measure spaces. A well-known conjecture of Gardner from the 1990's states that if G is an amenable group acting on a probability measure space in a measure preserving way, then whenever two measurable subsets of the space are G-equidecomposable, then they are G-equidecomposable using measurable pieces. This is a far-reaching generalization of recent breakthrough results. Part II. Structure of hyperfinite equivalence relations. Hyperfinite equivalence relations appear on the interface of ergodic theory and descriptive set theory as equivalence relations induced by Borel actions of the group Z of the integers. A notorious open problem in this discipline asks whether an equivalence relation which is almost everywhere hyperfinite with respect to every probability measure must be hyperfinite.  Part III. Structure of treeable equivalence relations. Structure of p.m.p (probability measure preserving) actions of finitely generated groups is one of the central themes in measured group theory. Ergodic dimension and strong ergodic dimension are invariants of a group defined in terms of the structure of its probability measure preserving actions. Treeable (strongly treeable) groups are those of ergodic (strong ergodic) dimension equal to 1. One of the notorious open questions asks if every treeable group is strongly treeable. Part IV. Extension properties for automorphisms and profinite topology. In the 1990's Hrushovski proved a fundamental theorem about extensions of partial automorphisms of finite graphs: for every finite graph G there exists a finite graph G' containing G as an induced subgraph such that all partial automorphisms of G extend to automorphisms of G'. Since then it has been a focus of extended study to understand which Fraisse classes of finite structures share this property. One of the most interesting problems in this area is a long-standing question of Herwig and Lascar asking whether the class of finite tournaments has this property. Part V. The automatic continuity phenomenon. Automatic continuity is the property of a topological group which says that any homomorphism from that group into a separable group is continuous. It has been recently proved for many infinite-dimensional groups via connections to Fraisee theory. This project aims at developing techniques for proving the automatic continuity for homeomorphism groups.

这是数学逻辑中的一项建议，涉及其一些最活跃的领域，例如描述性集合理论和均匀结构的模型理论。该提案继续进行了PI的研究计划。该计划集中在近年来该学科中核心的几个主题上，例如Borel组合学，可测量的群体理论，Hrushovski属性及其与Profinite群体理论的联系和自动连续性现象。第一部分。在可机械组中流动。该部分将着重于在可机械组对测量空间的作用中发现结构现象。 1990年代的Gardner的一个众所周知的猜想指出，如果G是一个可机械的群体，以一种测量方法来维护概率测量空间，那么每当空间的两个可测量子集都可以g- equidecompososoble上，那么它们是可使用可测量的零件的g- equecososososos。这是最近突破性结果的深远概括。第二部分。高铁与等效关系的结构。高限量的等效关系出现在沿阵行理论的界面和描述性集理论的界面上，作为整数Z组的Borel作用引起的等效关系。该学科中的一个臭名昭著的开放问题询问，对于每种概率度量，几乎到处都有过度限制的等价关系是否必须是高限度。第三部分。可预性关系的结构。最终产生的组的P.M.P（概率测量保存）作用是测量组理论中的中心主题之一。千古维度和强层面维度是根据其概率测量保护作用的结构定义的组的不变性。可观的（强大的）组是等于1的千古（强烈）维度。第四部分。自动形态和拓扑拓扑的扩展特性。在1990年代，Hrushovski提供了一个基本理论，讲述了有限图的部分自动形态的扩展：对于每个有限图G，存在一个有限的图G'，它包含G作为诱导的子图形，以使G'G'的所有局部自动形态扩展到G'的自动形态。从那时起，它一直是扩展研究的重点，以了解哪些有限结构共享该属性。在这一领域，最有趣的问题之一是赫维格（Herwig）和拉斯卡（Lascar）的一个长期问题，询问有限锦标赛是否具有此财产。第V部分。自动连续性现象。自动连续性是拓扑组的属性，该拓扑群体说，该组从该组到单独组的任何同态性都是连续的。最近，通过与Fraisee理论的联系，许多无限二维群体已被证明。该项目旨在开发用于证明同构群体自动连续性的技术。