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.

描述性集合论、动力学和组合学领域是数学的既定领域。近年来，人们对描述动力学这个相对较新的领域越来越感兴趣，在这个领域中，这些领域的工具结合起来研究这些学科共同感兴趣的问题。特别关注的是研究某些自然数学对象的结构，这些对象作为实数上可定义的等价关系出现。这些对象极大地概括了描述集合论中传统研究的对象，也自然地扩展了动力学研究的对象——动力系统。该项目的重点是使用最近开发的跨学科技术来研究这些物体的一些基本属性。对于几个重要的悬而未决的问题，新技术似乎很有前景，我们可以用它们来衡量进展情况。该领域的进一步进展将有助于更好地理解这些数学领域及其联系。该提案的一个主要焦点涉及可数 Borel 等价关系的组合数学。这些是由可数群的 Borel 行为产生的关系。一个自然的研究计划是通过增加生成关系的组的复杂性来进行。即使对于相对简单的群体，他们的群体行为的组合也很有趣。例如，最近的方法允许通过有限生成的阿贝尔群来计算自由伯努利位移的连续色数和博雷尔色数。最近的进展也扩展了提议者的结果，即可数阿贝尔群的所有行动对于可数幂零群（Schneider 和 Seward）以及一些可解的非幂零群（Conley、Jackson、Marks、Seward 和 Tucker-Drob）都是超有限的。证明显示了等价关系的属性与关系中完整部分的组合结构之间有趣的相互作用。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广泛团体的行动。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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

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

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

Hyperfiniteness and Borel combinatorics

超有限性和 Borel 组合学

• DOI: 10.4171/jems/935
• 发表时间: 2020
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Conley, Clinton;Jackson, Steve;Marks, Andrew;Seward, Brandon;Tucker-Drob, Robin
• 通讯作者: Tucker-Drob, Robin