Descriptive Set Theory

描述性集合论

• 批准号: 9619880
• 负责人: Alexander Kechris
• 金额: $ 17.38万
• 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: 1997
• 资助国家: 美国
• 起止时间: 1997-04-15 至 2000-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9619880&HistoricalAwards=false
• 关键词: Descriptive; Set; Theory

Abstract: The general aim of this project is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions with other areas of mathematics, such as the theory of topological and transformation groups, topological dynamics and ergodic theory, model theory, and recursion theory. One of the fundamental questions that arises in many fields of mathematics is that of classifying a given collection of objects that has been studied in this field. This amounts to providing a "catalog" or "listing" of these objects, in principle not unlike that of cataloging species in biology or 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 very important to understand under what circumstances a classification is possible. This difficult foundational question is further complicated by the fact that what constitutes an acceptable classification is very much dependent on the particular field of mathematics studied, so the criteria for a "good" classification in one area might not be appropriate in another. At its basic level, this project aims to develop a general quantitative theory, which in many situations can precisely measure the complexity of a classification problem and thus provide objective means by which one can decide, in any given field, whether a satisfactory classification of the objects in question is possible. This is achieved by associating with each collection of objects to be studied an appropriate concept of "magnitude" or "size", which in a precise sense measures the difficulty of its classification problem. This new theory of "size" is investigated in this project.

摘要：该项目的总体目标是发展波兰群的可定义作用的理论，其轨道空间的结构和分类，以及与可定义等价关系密切相关的研究。 这项工作的动机是基本的基础问题，如理解的性质，完全分类的数学对象的一些概念等价的不变量，并创建一个数学框架来衡量这种分类问题的复杂性。这个理论是在描述集合论的背景下发展起来的，描述集合论提供了基本的概念和方法。另一方面，鉴于这个理论的广泛范围，与其他数学领域有自然的相互作用，如拓扑和变换群理论，拓扑动力学和遍历理论，模型理论和递归理论。在数学的许多领域中出现的一个基本问题是对在这个领域中已经研究过的给定对象集合进行分类。这相当于提供了这些物体的“目录”或“列表”，原则上与生物学中的物种或天文学中的恒星和星系的编目没有什么不同。如果这样的分类是可能的，那么人们就对所涉及的数学结构有了“完整”的理解。 否则，或多或少的“混乱”行为是预期的。因此，了解在什么情况下可以进行分类非常重要。这个困难的基本问题进一步复杂化的事实是，什么构成一个可接受的分类是非常依赖于特定的数学领域的研究，所以标准的“好”的分类在一个领域可能不适合在另一个。 在其基本层面上，该项目旨在发展一种通用的数量理论，在许多情况下可以精确地衡量分类问题的复杂性，从而提供客观的手段，使人们可以在任何特定领域确定是否有可能对有关物体进行令人满意的分类。这是通过将待研究的每个物体集合与“大小”或“大小”的适当概念相关联来实现的，这在精确意义上衡量了其分类问题的难度。这个新的“尺寸”理论在这个项目中进行了研究。