Ideals and Equivalence Relations

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。这一驾驶理念包含在由PI合著的《关于波兰空间的典型拉姆齐理论》一书中，该书将于2013年出版。给定一个波兰空间上的Borel或解析等价关系，是否有可能找到一个空间的大子集，在这个空间上的等价是尽可能简单的？在这里，“大”是在空间上的某种理想的意义上解释的，而“简单”是在等价关系的复杂性的Borel可约性等级的意义上解释的。这个问题，用类似正则拉姆齐理论的术语来表述，揭示了一个由sigma理想和等价关系索引的广阔领域，推广了诸如关于拉姆齐立方体上光滑等价关系规范化的Proemel-Voigt定理，或关于完美集上解析图规范化的Blass定理等结果。总的主题是博雷尔等价关系和集合论模型之间的对应关系。其应用包括最强可能命题的不可约性结果和数学分析中遇到的各种sigma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一般主题。数学的大多数领域都是从一类对象和它们之间的相似性概念开始的。对数学分支来说，评估相似性概念的复杂性通常是至关重要的。事实证明，这种相似问题的复杂性有一个自然的等级，它将（无穷）数学的大多数分支联系在一起。为了在这个等级中正确地评估复杂性，除了其他东西之外，还需要一些工具来显示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颖的方法：当一个人被允许将注意力限制在一个较小但仍然重要的对象类别上时，某些相似性概念会大大简化，而其他概念则不会。这种类型的结果可能没有直接的实际应用，但它们确实有助于理解数学方法。作为该领域的典型，它们也将数学的结构更紧密地联系在一起，表明一个领域的关注可能会减少到另一个领域解决的问题。

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.