Ideals and Equivalence Relations

理想与等价关系

• 批准号: 1161078
• 负责人: Jindrich Zapletal
• 金额: $ 13.59万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161078&HistoricalAwards=false
• 关键词: Ideals; Equivalence; Relations

The project aims to further develop the connection between the fast developing area of Borel equivalence relations and forcing theory within the broader context of set theory and mathematical analysis. The driving idea is contained in the book "Canonical Ramsey Theory on Polish Spaces", coauthored by the PI, to appear in 2013. Given a Borel or analytic equivalence relation on a Polish space, is it possible to find a large subset of the space on which the equivalence is as simple as possible? Here, the largeness is to be interpreted in the sense of some sigma-ideal on the space, and the simplicity in the sense of the Borel reducibility rating of complexity of equivalence relations. This question, stated in terms similar to canonical Ramsey theory, uncovers a broad landscape indexed by sigma-ideals and equivalence relations, generalizing such results as the Proemel-Voigt theorem on canonization of smooth equivalence relations on Ramsey cubes, or Blass theorem on canonization of analytic graphs on perfect sets. The overarching general theme is a correspondence between Borel equivalence relations and models of set theory. The applications include nonreducibility results with the strongest possible statements and Silver type theorems for various sigma-ideals encountered in mathematical analysis. The project offers numerous other offshoots as well.The project follows the general theme of classification of equivalence problems in mathematics. Most areas of mathematics start with a class of objects and a notion of similarity between them. It is often of paramount importance to that branch of mathematics to evaluate the complexity of that notion of similarity. It turns out that there is a natural rating of complexity of such similarity problems that ties together most branches of (infinitary) mathematics. In order to evaluate the complexity correctly within this rating, one needs, among other things, tools for showing that one such notion of similarity is not reducible to another. The project offers a novel way for doing that: certain notions of similarity greatly simplify when one is allowed to restrict attention to a smaller, but still significant, class of objects--while others do not. Results of this type may not have immediate practical applications, but they do help with the understanding of methodology of mathematics. As is typical for this field, they also tie the fabric of mathematics closer together, showing that concerns of one field may reduce to questions solved by another field.

该项目旨在进一步发展Borel等效关系的快速发展领域与迫使理论在集合理论和数学分析的更广泛背景下的联系。驾驶想法包含在2013年由PI合着的《波兰空间的法典拉姆西理论》中。鉴于在波兰空间上存在Borel或分析等价关系，是否有可能找到尽可能简单的空间子集？在这里，应从空间上的某些西格玛理想的意义上解释宽敞，以及在borel降低等价关系复杂性级别的含义上的简单性。这个问题用类似于规范的拉姆西理论的术语说明，发现了一个由Sigma-Ideals和等效关系索引的广阔景观，概述了诸如Proemel-Voigt voigt Theorem关于在Ramsey立方体上平滑等效关系的义务的结果，或在Ramsey Ciotes上，或对完美集合的分析图上的Blass Theorem on Perfects canonization。总体一般主题是鲍尔等效关系与集合理论模型之间的对应关系。这些应用程序包括最强的可能陈述和银型定理的可还原性结果，用于数学分析中遇到的各种Sigma-Ideal。该项目也提供了许多其他分支。该项目遵循数学等效问题分类的一般主题。数学的大多数领域都始于一类对象和它们之间的相似性概念。对于数学分支，评估相似性概念的复杂性通常至关重要。事实证明，这种相似性问题的复杂性自然评级与（无限）数学的大多数分支联系在一起。为了在此评级中正确评估复杂性，一个需要的工具可以表明一种这种相似性的概念不能降低另一个相似性。该项目提供了一种新颖的方法：当允许人们将注意力限制在较小但仍然很重要的对象类别时，某些相似性的概念极为简化，而其他对象则没有。这种类型的结果可能没有立即的实际应用，但是它们确实有助于理解数学方法论。正如该领域的典型情况一样，它们还将数学的结构紧密结合在一起，表明一个领域的担忧可能会减少到另一个领域解决的问题。