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. ***

小行星9505505 描述集合论是研究波兰空间中的可定义集合和函数。 一个波兰空间（群）是一个可分的，完全的 度量化拓扑空间（群） 贝克尔的要求涉及到 波兰团体，在连接-或从的观点- 描述集合论 可定义集合的基本类是Borel类、解析集类和协解析集类。 这些构成了该项目的主要主题，虽然其他类的可定义集也被考虑。 该项目的一个主要主题是发现轨道空间的二分法，即指出轨道空间要么“小”，要么 它包含某个特定“大”集合的副本。 关于Vaught Conjecture，a 一个著名的逻辑公开问题，关于一个 一阶理论，断言这样一个二分法（要么这个数是可数的，要么是连续统的大小）。 该项目的大部分内容都与这一猜想有关。 贝克尔的项目采用了描述性集合论的标准方法和一种新方法，从而改变了底层拓扑结构。 在这个项目中的一些主题被认为是在强集理论公理的存在。 贝克尔的工作涉及数理逻辑的几个分支，以及其他数学领域，特别是遍历理论和 算子代数 这项研究，在该领域称为描述集 理论，是一种基本的，并加深了理解， 这些集合在数学中以自然的方式出现。 ***