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： I.在可数Borel等价关系的研究中找到新的技术，与几何群论和符号动力学有关。 二.研究局部紧顺从群的普适性问题 三.寻找新的例子，可分离的C*-代数，来自模型论，特别是从连续逻辑。 四.研究组合问题的拉姆齐理论使用最近的技术发展的研究拉姆齐空间。 一（a）.可数Borel等价关系理论与几何群论和可数群在测度空间上作用的研究密切相关。本课程的中心主题之一是在成本理论中发展起来的收入（和强收入）的概念。最近的突破康利，马克斯和塔克-德罗布导致证明强treeconomy的一类重要的群体（表面集团）。我们的计划集中在追求这条道路，并研究由几何群论（如双曲群）动机的其他类别的群体。 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（高，杰克逊，苏厄德，托马斯）问过这个复杂的等价关系的各种类别的群体。在这个程序中，我们计划研究几类极小子移位的共轭关系。 二.在各种范畴中普遍对象的存在问题，要么导致有趣的新构造，要么导致结构理论的发展。研究普遍顺从离散群的存在导致了关于Foelner序列增长的深层结构结果（在Erschler的工作中）。在这个项目中，我们计划把重点放在当地紧凑的顺从群体。 三.逻辑在算子代数中的应用是一个迅速发展的学科，它使用了集合论和模型论的方法。在他的ICM地址在首尔，法拉提出了几个新的方向应用逻辑的研究可分的C*-代数。在这个程序中，我们计划找到连续模型理论在构建可分C*-代数的新例子中的应用。 四.最近在拉姆齐理论的突破导致证明了几个长期存在的命题密度版本的拉姆齐定理的树木。这包括Dodos，Kanellopoulos，Karagiannis和Tyros的工作，并受到Todorcevic最近开发的Ramsey空间框架的启发。我们计划扩展和应用这些技术来研究多维版本的经典密度Ramsey定理，并将其方法扩展到其他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.