Logic, dynamics and Ramsey theory

该程序旨在将逻辑应用于数学的各个分支，如拓扑动力学、算子代数或拉姆齐理论。我们的长远目标是： I.在可数Borel等价关系的研究中发现了与几何群论和符号动力学有关的新方法。 二、研究局部紧顺从群的普适性问题 从模型理论，特别是从连续逻辑中寻找可分C*-代数的新例子。 利用在Ramsey空间研究中发展起来的最新技术研究Ramsey理论中的组合问题。 I(A)。可数Borel等价关系理论与几何群论和可数群在度量空间上的作用的研究有着密切的联系。这门课的中心主题之一是可树(和强可树)的概念，这是(除其他外)在成本理论中发展起来的。Conley，Marks和Tucker-Drob最近的突破证明了一类重要的群(曲面群)具有很强的树性。我们的计划集中于探索这条道路，并研究几何群论所激发的其他类群(如双曲群)。 I(B)符号动力学中的共轭问题可以看作是一个可数等价关系。几位作者(Gao，Jackson，Seward，Thomas)曾询问过不同类别群的这种等价关系的复杂性。在这个程序中，我们计划研究几类极小子移位的共轭关系。 各种范畴中普遍存在的物体的问题要么导致有趣的新构式，要么导致结构理论的发展。对泛从性离散群的存在性的研究导致了关于Foelner序列增长的深刻的结构性结果(在Erschler的工作中)。在这项计划中，我们计划专注于当地紧凑的顺从团体。 3.逻辑在算子代数中的应用是一门迅速发展的学科，它同时使用了集合论和模型论的方法。在首尔的ICM演讲中，Farah提出了几个新的方向，将逻辑应用于可分C*-代数的研究。在这个项目中，我们计划发现连续模型理论在建立可分C*-代数的新例子方面的应用。 最近Ramsey理论的突破证明了几个长期存在的关于树的Ramsey定理的密度版本的猜想。这包括Dodos、Kanellopoulos、Karagiannis和Tyros的工作，并受到Todorcevic最近开发的Ramsey空间框架的激励。我们计划将这些技巧扩展和应用到经典密度Ramsey定理的多维版本上，并将它们的方法扩展到其他Ramsey空间。

The program aims at applications of logic to various branches of mathematics such as topological dynamics, operator algebras or Ramsey theory. The long-term objectives are: I. Find new techniques in the study of countable Borel equivalence relation, with connections to geometric group theory and symbolic dynamics. II. Study the universality question for locally compact amenable groups III. Find new examples of separable C*-algebras that come from model theory and, in particular from continuous logic. IV. Study combinatorial problems in Ramsey theory using recent techniques developed in the study of Ramsey spaces. I(a). The theory of countable Borel equivalence relation is strictly connected with geometric group theory and the study of actions of countable groups on measure spaces. One of the central themes in the subject is the notion of treeability (and strong treeability), developed (among others) in the theory of cost. Recent breakthroughs by Conley, Marks and Tucker-Drob have lead to proofs of strong treeability of an important class of groups (surface groups). Our program concentrates on pursing this path and studying other classes of groups motivated by geometric groups theory (such as hyperbolic groups). I(b) The conjugacy problem in symbolic dynamics can be viewed as a countable equivalence relation. Several authors (Gao, Jackson, Seward, Thomas) have asked about the complexity of this equivalence relation for various classes of groups. Within this program we plan to study the conjugacy relation of several classes of minimal subshifts. II. The question of the existence of universal objects in various categories leads either to interesting new constructions, or to the development of structural theory. The study of the existence of universal amenable discrete groups resulted in deep structural results on the growth of Foelner sequences (in the work of Erschler). Within this program we plan to focus on locally compact amenable groups. III. The applications of logic to operator algebras is a rapidly developing subject that uses methods from both set theory and model theory. During his ICM address in Seoul, Farah proposed several new directions for applications of logic to the study of separable C*-algebras. Within this program we plan to find applications of continuous model theory in building new examples of separable C*-algebras. IV. Recent breakthroughs in Ramsey theory have lead to proofs of several long-standing conjectures on density versions of Ramsey theorems for trees. This includes the work of Dodos, Kanellopoulos, Karagiannis and Tyros and is motivated by the framework of Ramsey spaces developed recently by Todorcevic. We plan to extend and apply these techniques to work on multi-dimensional versions of classical density Ramsey theorems, as well as to extend their methods to other Ramsey spaces.