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

这主要是在数理逻辑中提出的，涉及到它最活跃的一些领域，如描述集合论和同质结构的模型理论。该提案延续了国际和平研究所的一项研究计划。本课程集中于近年来该学科的几个中心主题，如Borel组合学、可测群论、Hrushovski性质及其与无限群论的联系以及自动连续现象。 第一部分以顺从的团体为单位。这部分将集中于发现度量空间上的顺从群作用中的结构现象。1990年S提出的Gardner猜想指出：如果G是以保测的方式作用在概率测度空间上的可约群，则当空间的两个可测子集是G-等重组合时，则它们是G-等重组合的。这是对近期突破性成果的深远概括。 第二部分，超有限等价关系的结构。超有限等价关系出现在遍历理论和描述集合论的界面上，是由整数群Z的Borel作用诱导的等价关系。这门学科中一个臭名昭著的悬而未决的问题是，关于每个概率度量几乎处处超有限的等价关系是否一定是超有限的。 第三部分：树状等价关系的结构。有限生成群的p.m.p(保持概率度量)作用的结构是测群论的中心问题之一。遍历维和强遍历维是根据群的概率度量保持作用的结构定义的群的不变量。可树(强可树)群是那些遍历(强遍历)维度等于1的群。一个著名的公开问题是，是否每个可树群都是强可树的。 第四部分：自同构和准有限拓扑的扩张性质。S在1990年代证明了一个关于有限图的部分自同构扩张的基本定理：对每个有限图G，都存在一个有限图G‘，它包含G作为导出子图，使得G的所有部分自同构都扩张到G’的自同构。从那时起，了解哪些Frisse类有限结构共享这一性质一直是扩展研究的焦点。这一领域最有趣的问题之一是Herwig和Lascar长期存在的问题，即有限竞赛类是否具有这一性质。 第五部分，自动延续现象。自动连续性是拓扑群的性质，即从该群到可分群的任何同态都是连续的。最近，许多无限维群通过与Frisee理论的联系证明了这一点。这个项目的目的是开发证明同胚群的自动连续性的技术。

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.