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.

该项目旨在在集合论和数学分析的更广泛背景下，进一步发展快速发展的 Borel 等价关系领域与强迫理论之间的联系。其驱动思想包含在由 PI 合着的 2013 年出版的《关于波兰空间的规范拉姆齐理论》一书中。给定波兰空间上的 Borel 或解析等价关系，是否有可能找到等价性尽可能简单的空间大子集？这里，“大”被解释为空间上某种西格玛理想的意义，“简单”被解释为等价关系复杂性的波莱尔可约性等级的意义。这个问题以类似于规范拉姆齐理论的术语表述，揭示了由西格玛理想和等价关系索引的广阔前景，概括了诸如拉姆齐立方体上平滑等价关系规范化的普罗梅尔-沃伊特定理或完美集上解析图规范化的布拉斯定理等结果。总体主题是 Borel 等价关系和集合论模型之间的对应关系。这些应用程序包括具有最强可能陈述的不可约性结果以及数学分析中遇到的各种西格玛理想的 Silver 型定理。该项目还提供了许多其他分支。该项目遵循数学中等价问题分类的总体主题。大多数数学领域都是从一类对象以及它们之间的相似性概念开始的。对于该数学分支来说，评估相似性概念的复杂性通常至关重要。事实证明，此类相似问题的复杂性有一个自然的评级，它将（无限）数学的大多数分支联系在一起。为了正确评估此评级中的复杂性，除其他外，我们还需要工具来证明这种相似性概念不能还原为另一种相似性概念。该项目提供了一种新颖的方法来做到这一点：当人们被允许将注意力限制在较小但仍然重要的一类物体上时，某些相似性的概念就会大大简化——而其他的则不然。此类结果可能不会立即产生实际应用，但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，它们还将数学结构紧密地联系在一起，表明一个领域的关注点可能会简化为另一个领域解决的问题。