Treeable Equivalence Relations and the Use of Probability Groups in Arithmetic Combinatorics

可树化的等价关系和概率群在算术组合中的使用

• 批准号: 1501036
• 负责人: Anush Tserunyan
• 金额: $ 13.77万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-05-15 至 2018-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501036&HistoricalAwards=false
• 关键词: Treeable; Equivalence; Relations; Use; Probability

This is a research project at the interface of the mathematical topics of set theory, combinatorics, and analysis. The research contains three projects involving two main areas of mathematics: descriptive set theory and ergodic Ramsey theory. The first two of the research projects lie in the theory of definable equivalence relations, which provides a general framework for understanding the nature of classification of mathematical objects up to some notion of equivalence; due to its broad scope, it has natural interactions with many areas of mathematics. These two projects are devoted to studying an important subclass of definable equivalence relations and whether slight extensions of the members of this subclass still belong to it. The third project features a new method for obtaining statements in arithmetic combinatorics similar in nature to a celebrated theorem of Szemeredi, which roughly states that any non-negligible subset of integers retains much of the additive structure of the entire set of integers.In the theory of definable equivalence relations on Polish spaces, a central place is occupied by countable Borel equivalence relations, an important subclass of which is that of treeable equivalence relations. The first two projects investigate the question of closure of this subclass under finite index extensions in two different contexts: Borel and measure-theoretic. The former involves Borel combinatorics and possibly Borel games, whereas the latter is tightly connected with ergodic theory and the theory of cost of equivalence relations, and may require nontrivial machinery from geometric group theory. The third project lies in ergodic Ramsey theory and its goal is to obtain multiple recurrence results for amenable groups via a correspondence principle provided by nonstandard analysis. This is done by transferring recurrence statements from a given amenable group to a more convenient setting of probability groups by taking the ultrapower of the original group and equipping it with Loeb measure. The latter, being countably additive, presents the main advantage of the probability group over the original amenable group equipped with only a finitely additive density function, enabling integration over the group and the use of Fubini's theorem.

这是集合论、组合学和分析等数学主题的交叉研究项目。该研究包含三个项目，涉及数学的两个主要领域：描述性集合论和遍历拉姆齐理论。前两个研究项目涉及可定义的等价关系理论，该理论为理解数学对象的分类本质以及某种等价概念提供了一个总体框架；由于其广泛的范围，它与数学的许多领域有着自然的相互作用。这两个项目致力于研究可定义等价关系的一个重要子类以及该子类成员的轻微扩展是否仍然属于它。第三个项目的特色是一种在算术组合中获得陈述的新方法，其本质上类似于著名的 Szemeredi 定理，该定理粗略地指出，任何不可忽略的整数子集都保留了整个整数集的大部分加法结构。在波兰空间上可定义的等价关系中，中心位置被可数的 Borel 等价关系占据，其中一个重要的子类是可树等价关系。前两个项目研究了两个不同背景下有限索引扩展下该子类的闭包问题：Borel 和测度理论。前者涉及 Borel 组合学和可能的 Borel 博弈，而后者与遍历理论和等价关系成本理论紧密相关，并且可能需要几何群论中的重要机制。第三个项目是遍历拉姆齐理论，其目标是通过非标准分析提供的对应原理获得服从群的多重递推结果。这是通过将递归语句从给定的服从组转移到更方便的概率组设置来完成的，方法是采用原始组的超幂并为其配备勒布测度。后者是可数相加的，与仅配备有限​​相加密度函数的原始服从群相比，呈现出概率群的主要优点，从而能够对群进行积分并使用 Fubini 定理。