Equivalence Relations, Symbolic Dynamics, and Descriptive Set Theory

等价关系、符号动力学和描述集合论

• 批准号: 1201290
• 负责人: Su Gao
• 金额: $ 24万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201290&HistoricalAwards=false
• 关键词: Equivalence; Relations; Symbolic; Dynamics; Descriptive

The project proposes to bring some recently developed techniques to the study of countable Borel equivalence relations. These techniques, developed over the last several years, involve new types of marker structures on the equivalence relations. For example, these new structures have led to a proof that all orbit equivalence relations of countable abelian group actions are hyperfinite. An important goal is to extend the results to larger classes of groups, and to delineate the extent of hyperfiniteness. A second new method concerns marker structures on arbitrary countable groups, also referred to as ``blueprints". The bluprints, for example, give a proof that the Bernoulli shift action of every countable group has a free subflow. They have also been used to give other results for general actions, such as results on the complexity of the topological conjugacy relation. Another goal of this project is to explore the connections between the possible blueprints that can exist on groups and the marker structures on the equivalence relations induced by actions of these groups. It is expected that progress along these lines will improve our understanding both of actions by special types of groups, and of the nature of Borel actions for general countable groups.Countable Borel equivalence relations are fundamental mathematical objects which occur in many mathematical contexts. Aside from their intrinsic interest, their theory interacts with other important areas of mathematics such as dynamics, ergodic theory, and geometric group theory. Thus, work in this area involves techniques from logic as well as dynamics, combinatorics, and other areas. Consider an equivalence relation studied by the ancient Greeks, that of commensurability: two positive real numbers are equivalent if their ratio is rational. A very basic question about this simple relation was not known until recently, namely whether it can be described in an effective way as an increasing union of finite relations. Several natural generalizations of this question are still open. This project seeks to further develop some of the new techniques to further the study of these fundamental questions. It is expected that this study will also make new connections with other areas of mathematics.

该项目提议将一些最近开发的技术用于可数 Borel 等价关系的研究。这些技术是在过去几年中发展起来的，涉及等价关系上的新型标记结构。例如，这些新结构证明了可数阿贝尔群作用的所有轨道等价关系都是超有限的。一个重要的目标是将结果扩展到更大的群体类别，并描绘超有限性的范围。第二种新方法涉及任意可数群上的标记结构，也称为“蓝图”。例如，蓝图证明了每个可数群的伯努利移位动作都有一个自由子流。它们还被用于给出一般动作的其他结果，例如拓扑共轭关系的复杂性结果，该项目的另一个目标是探索群上可能存在的蓝图与动作引起的等价关系的标记结构之间的联系。预计沿着这些方向的进展将提高我们对特殊类型群的作用以及一般可数群的 Borel 作用的性质的理解。可数 Borel 等价关系是许多数学中出现的基本数学对象。除了其内在的兴趣之外，他们的理论还与动力学、遍历理论和几何群论等其他重要的数学领域相互作用。因此，该领域的工作涉及逻辑以及动力学、组合学和其他领域的技术。考虑古希腊人研究的等价关系，即可通约性：如果两个正实数的比率是有理数，则它们是等价的。关于这个简单关系的一个非常基本的问题直到最近才为人所知，即它是否可以有效地描述为有限关系的递增并集。这个问题的几个自然概括仍然有待解决。该项目旨在进一步开发一些新技术来进一步研究这些基本问题。预计这项研究还将与数学的其他领域建立新的联系。