Logic, dynamics and Ramsey theory

该计划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。长期目标是：****** 1。在可数Borel等价关系的研究中发现新的技术，并与几何群论和符号动力学相联系。研究局部紧可服从群的通用性问题***III。从模型理论，特别是从连续逻辑中找到可分离C*代数的新例子。利用Ramsey空间研究中发展起来的新技术研究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以及可数群在测度空间上的作用研究有着密切的联系。该主题的中心主题之一是可树性（和强可树性）的概念，在成本理论中发展（以及其他）。Conley、Marks和Tucker-Drob最近的突破证明了一类重要群（表面群）的强可树性。我们的课程专注于追求这条道路，并研究由几何群理论（如双曲群）驱动的其他类群。******I(b)符号动力学中的共轭问题可以看作是可数等价关系。几位作者（Gao, Jackson, Seward, Thomas）提出了对于不同类群的这种等价关系的复杂性。在这个项目中，我们计划研究几类极小子位移的共轭关系。******II。在不同范畴中存在普遍对象的问题，要么导致有趣的新结构，要么导致结构理论的发展。对泛可调离散群存在性的研究，对Foelner序列的增长（在Erschler的工作中）产生了深刻的结构结果。在这个项目中，我们计划把重点放在当地紧凑的群体上。******逻辑在算子代数中的应用是一门迅速发展的学科，它运用了集合论和模型论的方法。在他在首尔的ICM演讲中，Farah提出了几个新的方向，将逻辑学应用于可分离C*代数的研究。在这个程序中，我们计划找到连续模型理论在构建可分离C*-代数的新例子中的应用。******IV。最近在拉姆齐理论方面的突破证明了关于树木的拉姆齐定理的密度版本的几个长期存在的猜想。这包括Dodos， Kanellopoulos， Karagiannis和Tyros的作品，并受到Todorcevic最近开发的拉姆齐空间框架的启发。我们计划将这些技术扩展和应用于经典密度拉姆齐定理的多维版本，并将他们的方法扩展到其他拉姆齐空间

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.**