Descriptive Set Theory

描述性集合论

• 批准号: 9987437
• 负责人: Alexander Kechris
• 金额: $ 26.33万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9987437&HistoricalAwards=false
• 关键词: Descriptive; Set; Theory

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.

DMS-998-7437本项目的总体目标是发展可定义等价关系理论，并对波兰群的可定义作用及其轨道空间的结构和分类进行密切相关的研究。这一理论发生在描述性集合理论的背景下，它提供了基本的基本概念和方法。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存在着自然的相互作用，如模型理论、递归理论、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基本问题之一是对正在某个特定领域研究的给定对象集合进行分类。这相当于提供了这些天体的“目录”或“清单”，原则上与对生物中的物种或天文学中的恒星和星系进行分类没有什么不同。如果这样的分类是可能的，人们就对所涉及的数学结构有了“完全”的理解。否则，或多或少会有一种“混乱”的行为。因此，了解在什么情况下分类是可能的是非常重要的。这个困难的基本问题因以下事实而变得更加复杂：什么构成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，因此在一个领域的“好”分类的标准可能不适用于另一个领域。在其基本层面上，这个项目的目的是发展一个通用的量化理论，它在许多情况下可以准确地衡量分类问题的复杂性，从而提供客观的手段，在任何给定的领域中，人们可以通过它来判断所讨论的对象是否可能进行满意的分类。这是通过将适当的“大小”或“大小”概念与每个要分类的对象集合相关联来实现的，准确地说，这一概念衡量了其分类问题的难度。这个新的“震级”理论就是在这个项目中被研究的。