Mathematical Sciences: Polish Group Actions and Descriptive Set Theory

数学科学：波兰群行动和描述集合论

• 批准号: 9505505
• 负责人: Howard Becker
• 金额: $ 6.3万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505505&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Polish; Group; Actions

9505505 Becker Descriptive set theory is the study of definable sets and functions in Polish spaces. A Polish space (group) is a separable, completely metrizable topological space (group). Becker's reqearch concerns the actions of Polish groups, in connection with -- or from the point of view of -- descriptive set theory. The basic classes of definable sets are the classes of Borel, analytic and coanalytic sets. These constitute the main topic of the project, although other classes of definable sets are also considered. A major theme of the project is discovery of dichotomies of the orbit space, i.e. theorems stating that either the orbit space is "small" or else it contains a copy of some specific "large" set. The Vaught Conjecture, a famous open problem in logic concerning the number of countable models of a first order theory, asserts such a dichotomy (either this number is countable or of size the continuum). Much of the project is related to that conjecture. Becker's project employs the standard methods of descriptive set theory together with one new method, whereby the underlying topology is changed. Some of the topics in this project are considered in the presence of strong set theoretic axioms. Becker's work relates to several branches of mathematical logic, as well as to other fields of mathematics, particularly ergodic theory and operator algebras. This research, in the area called descriptive set theory, is of a fundamental kind, and deepens the understanding of exactly those sets which arise in natural ways throughout mathematics. ***

9505505 Becker描述集理论是对波兰空间中可定义的集合和功能的研究。 波兰空间（组）是一个可分离的，完全可分离的拓扑空间（组）。 贝克尔的重新搜索涉及波兰群体的行为，与描述性集理论有关或从描述性集理论的角度而言。 可定义集的基本类别是Borel，分析和共分析集的类别。 尽管还考虑了其​​他类似的可确定集，但这些构成了项目的主要主题。 该项目的一个主要主题是发现轨道空间的二分法，即定理表明轨道空间是“小”的，或者它包含了某些特定“大”集的副本。 Vaught猜想是关于一阶理论的可数模型数量的逻辑数量的著名开放问题，他断言了这种二分法（该数量是可计数的，要么是连续体的大小）。 项目的大部分与该猜想有关。 贝克尔（Becker）的项目采用了描述集理论的标准方法以及一种新方法，从而改变了基础拓扑。 在强大的理论公理存在下，该项目中的某些主题被考虑。 贝克尔的工作涉及数学逻辑的几个分支，以及其他数学领域，尤其是千古理论和操作者代数。 这项研究在称为描述性集合理论的领域是一种基本种类，并加深了对整个数学中自然出现的确切集合的理解。 ***