Ideals and Equivalence Relations

该项目旨在进一步发展之间的联系快速发展领域的博雷尔等价关系和强迫理论在更广泛的背景下，集理论和数学分析。驱动思想包含在由PI合著的《波兰空间的经典拉姆齐理论》一书中，该书将于2013年出版。给定一个波兰空间上的博雷尔或解析等价关系，是否有可能找到一个大的空间子集，使等价关系尽可能简单？在这里，大是在空间上的一些sigma理想的意义上解释的，简单性是在等价关系的复杂性的Borel归约的意义上解释的。这个问题，在类似于典型的拉姆齐理论，揭示了一个广泛的景观指数西格玛理想和等价关系，推广等结果的Proemel-Voigt定理规范的光滑等价关系的拉姆齐立方体，或布拉斯定理规范的解析图的完美集。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。应用程序包括nonreductionary结果与最强的可能的声明和银型定理的各种西格玛理想中遇到的数学分析。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。对于数学的那个分支来说，评估相似性概念的复杂性往往是至关重要的。事实证明，有一个自然的复杂性等级，这种相似性问题，联系在一起的大多数分支（无穷）数学。为了在这个等级内正确地评估复杂性，除其他外，人们需要一些工具来表明一个这样的相似性概念不能简化为另一个。该项目提供了一种新颖的方法：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，某些相似性概念大大简化了，而其他人则没有。这种类型的结果可能不会有直接的实际应用，但它们确实有助于理解数学方法论。正如这一领域的典型情况一样，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.