Logic, dynamics and Ramsey theory

逻辑、动力学和拉姆齐理论

• 批准号: RGPIN-2015-03738
• 负责人: Sabok, Marcin
• 金额: $ 1.02万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=664003
• 关键词: Logic; dynamics; Ramsey; theory

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.**

该计划的目的是将逻辑应用于数学的各个分支，例如拓扑动态，操作员代数或拉姆西理论。长期目标是：****** i。在可计数的鲍尔等效关系的研究中找到新技术，并与几何组理论和符号动力学联系。*** ii。研究本地紧凑型固定组的普遍性问题*** iii。找到来自模型理论的可分离c* - 代数的新示例，尤其是来自连续逻辑。*** iv。 Ramsey理论中的研究组合问题使用了Ramsey空间研究中开发的最新技术。****** I（a）。可计数的鲍尔等效关系的理论与几何群体理论和可数组对度量空间的作用的研究严格相关。该主题的中心主题之一是在成本理论中（等）开发的脱牙性（和强大的树皮性）概念。 Conley，Marks和Tucker-Drob的最新突破导致了一类重要组（表面组）的强烈脱脂性的证据。我们的计划集中于追求这一路径，并研究以几何组理论（例如双曲线组）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可计数的等价关系。几位作者（Gao，Jackson，Seward，Thomas）询问了这种对等关系对各种类别的复杂性。在该计划中，我们计划研究几类最低次要换档的共轭关系。****** ii。各个类别中普遍物体存在的问题导致有趣的新结构或结构理论的发展。对普遍不及离散基团的存在的研究导致了foelner序列的生长（在Erschler的工作中）的深层结构结果。在该计划中，我们计划专注于本地紧凑的符合人群。****** iii。逻辑对操作器代数的应用是一个快速发展的主题，它使用集合理论和模型理论的方法。在首尔的ICM讲话中，法拉提出了几个新的方向，用于将逻辑应用于可分离的C* - 代数研究。在此程序中，我们计划在构建可分离c*-ergebras的新示例中找到连续模型理论的应用。****** iv。拉姆西理论的最新突破导致了关于树木的拉姆西定理的密度版本的几个长期猜想的证据。这包括Dodos，Kanellopoulos，Karagiannis和Tyros的作品，并由Todorcevic最近开发的Ramsey Spaces的框架激励。我们计划扩展和应用这些技术来处理经典密度Ramsey定理的多维版本，并将其方法扩展到其他Ramsey空间。**