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.

这是一个研究项目，在集合理论，组合学和分析的数学主题的界面上。该研究包含三个涉及数学主要领域的项目：描述性集理论和崇高的拉姆西理论。研究项目的前两个在于可确定的等价关系理论，该理论为理解数学对象分类的性质提供了一个一般框架，直到某种等效概念。由于其广泛的范围，它与许多数学领域具有自然相互作用。这两个项目致力于研究可确定的等效关系的重要子类，以及该子类成员的轻微扩展是否仍然属于它。第三个项目具有一种新的方法，该方法在算术组合中获得的陈述本质上类似于著名的szemeredi定理，该定理大致表明，整数的任何不可忽略的子集都保留了整个整数集的大部分添加效应。可观的等价关系。前两个项目研究了在有限指数扩展下在两个不同情况下关闭该子类的问题：Borel和Measure Theoretic。前者涉及Borel Compinatorics和可能的Borel游戏，而后者与Ergodic理论紧密相连和等效关系成本理论，并且可能需要从几何组理论中进行非平凡的机械。第三个项目在于崇高的拉姆西理论，其目标是通过非标准分析提供的对应原理为正面组获得多个复发结果。这是通过将复发性语句从给定的木制组转移到更方便的概率组设置来完成的，并通过将原始组的超赋和装备为loeb度量来完成。后者具有次数加性，它比只有有限添加性密度函数的原始amenable组提供了概率组的主要优点，从而使整个组的集成和使用Fubini定理。