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快速发展的Borel等价关系领域与更广泛的集合论和数学分析背景下的强迫理论之间的联系。驾驶理念包含在由PI合著的《关于波兰空间的经典拉姆齐理论》一书中，该书将于2013年出版。给定波兰空间上的一个Borel等价关系或解析等价关系，是否有可能找到该空间的一个大子集，使其等价关系尽可能简单？这里，广度被解释为空间上的某个sigma-理想意义上的简单性，以及等价关系复杂性的Borel可约性评级意义下的简单性。这个问题的表述类似于典型的Ramsey理论，揭示了以sigma-理想和等价关系为指标的广泛的图景，推广了关于Ramsey立方体上光滑等价关系的典范的Proemel-Voigt定理，或关于完全集上解析图的典范的Blass定理。总体主题是Borel等价关系和集合论模型之间的对应关系。其应用包括具有最强可能陈述的不可约性结果和数学分析中遇到的各种西格玛理想的Silver类型定理。该项目还提供了许多其他分支。该项目遵循数学中等价性问题分类的总主题。大多数数学领域都是从一类对象和它们之间的相似性概念开始的。对这一数学分支来说，评估相似性概念的复杂性往往是极其重要的。事实证明，将(无穷)数学的大多数分支联系在一起的这类相似问题的复杂性有一个自然的等级。为了在这一评级范围内正确地评估复杂性，除其他外，人们需要工具来证明一个这样的相似性概念不能简化为另一个概念。该项目提供了一种新的方法来做到这一点：当一个人被允许将注意力限制在较小但仍然重要的一类物体上时，某些相似的概念会大大简化--而另一些则不是。这种类型的结果可能不会立即有实际应用，但它们确实有助于理解数学的方法论。就像这个领域的典型情况一样，它们还将数学的结构更紧密地联系在一起，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。