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.

这是一个在集合论，组合学和分析的数学主题的接口的研究项目。该研究包含三个项目，涉及两个主要领域的数学：描述集理论和遍历拉姆齐理论。前两个研究项目在于可定义等价关系理论，它为理解数学对象的分类性质提供了一个一般框架，直到一些等价概念;由于其范围广泛，它与许多数学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。第三个项目的特点是一种新的方法，用于获得算术组合学中的语句，其性质类似于著名的Szemeredi定理，它大致说明了任何非-可忽略的整数子集保留了整组整数的大部分加性结构。在波兰的可定义等价关系理论中，空间，一个中心的地方是由可数博雷尔等价关系，其中一个重要的子类是树等价关系。前两个项目调查的问题封闭的子类有限指数扩展在两个不同的情况下：博雷尔和测量理论。前者涉及博雷尔组合学和可能的博雷尔游戏，而后者是紧密相连的遍历理论和等价关系的成本理论，并可能需要非平凡的机器从几何群论。第三个项目是遍历Ramsey理论，其目标是通过非标准分析提供的对应原理，得到顺从群的多重递归结果。这是通过将递归陈述从给定的顺从群转移到更方便的概率群设置来完成的，方法是利用原始群的超幂并为其配备勒布测度。后者是可计数加性的，呈现了概率群相对于仅配备有限加性密度函数的原始顺从群的主要优势，使得能够在群上积分并使用富比尼定理。