Descriptive Dynamics and Borel Combinatorics of Group Actions

群体行为的描述动力学和 Borel 组合学

• 批准号: 1800323
• 负责人: Stephen Jackson
• 金额: $ 27万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800323&HistoricalAwards=false
• 关键词: Descriptive; Dynamics; Borel; Combinatorics; Group

The fields of descriptive set theory, dynamics, and combinatorics are established areas of mathematics. In recent years there has been increasing interest in the relatively new field of descriptive dynamics in which tools from these areas combine to study problems of common interest to these subjects. A particular focus is to study the structure of certain natural mathematical objects which occur as definable equivalence relations on the reals. These objects greatly generalize the objects of traditional study in descriptive set theory, and also extend naturally the objects, dynamical systems, studied in dynamics. The focus of this project is to use recently developed, cross-disciplinary techniques to study some of the fundamental properties of these objects. There are several important open questions for which the new techniques seem promising, and with which we can benchmark progress. Further progress in this area will lead to a better understanding of these areas of mathematics and their connections.A major focus of the proposal concerns the combinatorics of countable Borel equivalence relations. These are the relations generated by the Borel actions of countable groups. A natural plan of study is to proceed by increasing complexity of the group generating the relation. Even for relatively simple groups, the combinatorics of their group actions is interesting. As an example, recent methods allow for a computation of both the continuous and Borel chromatic numbers of free Bernoulli shifts by finitely generated abelian groups. There have also been recent developments to extend the proposers' result that all actions of countable abelian groups are hyperfinite to countable nilpotent groups (Schneider and Seward) and to some solvable, non-nilpotent groups (Conley, Jackson, Marks, Seward, and Tucker-Drob). The proofs show an interesting interplay between the properties of the equivalence relations and the combinatorial structure of complete sections in the relation. The project seeks to explore this interplay and to extend the current methods and results to actions of more general groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

描述性集理论，动力学和组合学领域是数学的建立领域。近年来，人们对相对较新的描述性动态领域的兴趣越来越大，其中这些领域的工具结合了研究这些主题的共同兴趣问题。一个特定的重点是研究某些自然数学对象的结构，这些对象是对真实物的可定性对等关系。这些对象在描述性集理论中极大地概括了传统研究的对象，并且也自然地扩展了动态研究的对象，动力学系统。该项目的重点是使用最近开发的跨学科技术来研究这些对象的一些基本属性。有几个重要的开放问题，新技术似乎很有希望，我们可以通过这些问题进行基准进步。该领域的进一步进步将使人们更好地了解这些数学领域及其联系。该提案的主要重点涉及可数鲍勒等效关系的组合。这些是可数组的Borel行动产生的关系。一个自然的研究计划是通过增加产生关系的群体的复杂性来进行。即使对于相对简单的小组，他们的小组行动的组合也很有趣。例如，最近的方法允许计算有限生成的Abelian群体的连续和鲍尔色的自由伯诺利移动。最近的事态发展是为了扩展提议者的结果，即可计数的阿贝尔群体的所有行动都超过了可计数的nilpotent群体（Schneider和Seward）以及一些可解决的，可解决的非智力组（Conley，Jackson，Jackson，Marks，Seward，Seward，Seward和Tucker-Drob）。证明显示了等效关系的属性与关系中完整部分的组合结构之间的有趣相互作用。该项目旨在探索这种相互作用，并将当前的方法和结果扩展到更多一般小组的行动。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来获得支持的。

项目成果

Definable combinatorics at the first uncountable cardinal

第一个不可数基数的可定义组合

• DOI: 10.1090/tran/8281
• 发表时间: 2021
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Chan, William;Jackson, Stephen
• 通讯作者: Jackson, Stephen

Some complexity results in the theory of normal numbers

正规数理论产生了一定的复杂性

• DOI: 10.4153/s0008414x20000723
• 发表时间: 2022
• 期刊: Canadian Journal of Mathematics
• 作者: Airey, Dylan;Jackson, Steve;Mance, Bill
• 通讯作者: Mance, Bill

Cardinality of wellordered disjoint unions of quotients of smooth equivalence relations

光滑等价关系商的良序不相并并的基数

• DOI: 10.1016/j.apal.2021.102988
• 发表时间: 2021
• 期刊: Annals of Pure and Applied Logic
• 影响因子: 0.8
• 作者: Chan, William;Jackson, Stephen
• 通讯作者: Jackson, Stephen

Borel complexity of sets of normal numbers via generic points in subshifts with specification

通过具有规范的子移位中的通用点计算正常数集的 Borel 复杂性

• DOI: 10.1090/tran/8001
• 发表时间: 2020
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Airey, Dylan;Jackson, Steve;Kwietniak, Dominik;Mance, Bill
• 通讯作者: Mance, Bill

The complexity of the classification problems of finite-dimensional continua

有限维连续体分类问题的复杂性

• DOI: 10.1016/j.topol.2019.106876
• 发表时间: 2019
• 期刊: Topology and its Applications
• 影响因子: 0.6
• 作者: Chang, Cheng;Gao, Su
• 通讯作者: Gao, Su