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.

该项目建议将一些最近开发的技术带入对可数鲍尔等效关系的研究。这些技术在过去几年中开发，涉及对等效关系的新类型标记结构。例如，这些新结构导致了一个证据，证明了可数的阿贝尔小组行动的所有轨道等效关系都是高限度的。一个重要的目标是将结果扩展到较大的群体，并描述过度丰富性的程度。第二种新方法涉及在任意可数组上的标记结构，也称为``蓝图''。可以在这些行为引起的等效关系上存在的蓝图和标记结构，预计沿这些行的进展将提高我们对特殊类型的群体的理解，而鲍尔行动的性质以及一般可数的群体的性质。作为动力学，千古理论和几何群体理论。因此，在这一领域的工作涉及逻辑以及动态，组合学和其他领域的技术。考虑古希腊人研究的等效关系，即可相当的性：如果其比率是理性的，则两个正实数是等效的。直到最近才知道这个简单关系的一个非常基本的问题，即是否可以有效地将其描述为有限关系的增加。这个问题的几种自然概括仍然开放。该项目旨在进一步开发一些新技术，以进一步研究这些基本问题。预计这项研究还将与其他数学领域建立新的联系。