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：该项目的总体目的是发展波兰群体的可定义行动理论，其轨道空间的结构和分类以及对可定性等效关系的紧密相关研究。 这项工作是由基本的基础问题激励的，例如了解数学对象的完整分类的性质，直到不变性对等效性的某些概念，并创建一个数学框架来测量此类分类问题的复杂性。该理论是在描述性集理论的背景下发展的，该理论提供了基本的基本概念和方法。另一方面，鉴于该理论的广泛范围，与其他数学领域存在自然相互作用，例如拓扑和转型群体的理论，拓扑动力学和千古理论，模型理论和递归理论。在许多数学领域中出现的基本问题之一是对在该领域进行了研究的对象集合进行分类。这相当于提供这些物体的“目录”或“清单”，从原则上讲，与生物学或天文学中的恒星和星系中的分类物种不同。如果可以进行这样的分类，则对所涉及的数学结构有一个“完全”的理解。 否则，预计或多或少会产生“混乱”行为。因此，了解在什么情况下可能会出现分类非常重要。这个困难的基础问题进一步复杂化，即构成可接受的分类的事实很大程度上取决于所研究的特定数学领域，因此在一个领域中“良好”分类的标准可能在另一个领域不合适。 从基本层面上讲，该项目旨在开发一种一般的定量理论，在许多情况下，该理论可以精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，是否可以在任何给定的领域中决定是否令人满意地分类有问题的对象是可能的。这是通过与每个对象集合相关联来实现的，该集合要研究一个适当的“大小”或“大小”的概念，从而确切的意义上可以衡量其分类问题的难度。这个项目中研究了这种“大小”的新理论。