Descriptive Set Theory

本课题的总体目标是发展可定义等价关系理论，并密切相关地研究波兰群的可定义作用及其轨道空间的结构和分类。这个理论发生在描述性集合论的背景下，它提供了基本的潜在概念和方法。另一方面，鉴于这一理论的广泛范围，它与逻辑和数学的其他领域有自然的相互作用，如模型论、递归论、拓扑和变换群理论、拓扑动力学、遍历理论和算子代数。在许多数学领域中出现的一个基本问题是对某一特定领域中正在研究的一组给定对象进行分类。这相当于为这些天体提供一份“目录”或“清单”，原则上与天文学上为生物学家或恒星和星系编目没有什么不同。如果这样的分类是可能的，那么一个人就对所涉及的数学结构有了“完整”的理解。否则，预计会出现或多或少的“混乱”行为。因此，了解在什么情况下分类是可能的是非常重要的。这个困难的基础问题因以下事实而进一步复杂化：什么是可接受的分类，在很大程度上取决于所研究的特定数学领域，因此，在一个领域“好”分类的标准可能不适用于另一个领域。在其基本层面上，该项目旨在发展一种通用的定量理论，在许多情况下可以精确地测量分类问题的复杂性，从而提供客观的手段，让一个候选人决定，在任何给定的领域，是否有可能对问题中的对象进行令人满意的分类。这是通过将每个要分类的对象集合关联到一个适当的“量级”或“大小”概念来实现的，这在精确的意义上衡量了其分类问题的难度。这个新的“量级”理论在这个项目中进行了研究。

DMS-998-7437ABSTRACTThis general aim of this project is the development of the theory ofdefinable equivalence relations, and the closely related study of definableactions of Polish groups, and the structure and classification of theirorbit spaces. This theory takes place within the context of descriptive settheory, which provides the basic underlying concepts and methods. On theother hand, in view of the broad scope of this theory, there are naturalinteractions with other areas of logic and mathematics, such as modeltheory, recursion theory, the theory of topological and transformationgroups, topological dynamics, ergodic theory, and operator algebras.One of the fundamental questions that arises in many areas of mathematics isthat of classifying a given collection of objects that is being studied insome particular field. This amounts to providing a "catalog" or "listing" ofthese objects, in principle not unlike that of cataloging species in biologyor stars and galaxies in astronomy. If such a classification is possible,one has a "complete" understanding of the mathematical structures involved.Otherwise a more or less "chaotic" behavior is expected. It is thus veryimportant to understand under what circumstances a classification ispossible. This difficult foundational question is further complicated by thefact that what constitutes an acceptable classification is very muchdependent on the particular field of mathematics studied, so the criteriafor a "good" classification in one area might not be appropriate in another.At its basic level, this project aims to develop a general quantitativetheory, which in many situations can precisely measure the complexity of aclassification problem and thus provide objective means by which one candecide, in any given field, whether a satisfactory classification of theobjects in question is possible. This is achieved by associating with eachcollection of objects to be classified an appropriate concept of "magnitude"or "size", which in a precise sense measures the difficulty of itsclassification problem. This new theory of "magnitude" is investigated inthis project.