Descriptive Dynamics and Borel Combinatorics of Group Actions

描述集合论、动力学和组合学等领域都是数学的既定领域。近年来，人们对描述动力学这一相对较新的领域越来越感兴趣，在这个领域中，来自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。一个特别的重点是研究某些自然数学对象的结构，这些自然数学对象在实数上以可定义的等价关系出现。这些对象极大地推广了传统描述集合论的研究对象，也自然地扩展了动力学研究的对象--动力系统。这个项目的重点是使用最近发展起来的跨学科技术来研究这些物体的一些基本属性。有几个重要的悬而未决的问题，新技术似乎很有希望，我们可以用它们来衡量进展。这一领域的进一步进展将导致对这些数学领域及其联系的更好理解。该提案的一个主要焦点涉及可数Borel等价关系的组合学。这些关系是由可数群的Borel作用生成的。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群体，他们的群体行为的组合也是有趣的。例如，最近的方法允许计算有限生成的阿贝尔群的自由Bernoulli移位的连续数和Borel色数。最近的发展也推广了提出者的结果，即可数交换群的所有作用对可数幂零群(Schneider和Seward)和一些可解的非幂零群(Conley，Jackson，Marks，Seward和Tucker-Drob)都是超有限的。证明表明等价关系的性质与该关系中完整截面的组合结构之间存在着有趣的相互作用。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一般群体的行动。这一奖项反映了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。

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.