Mathematical Sciences: Polish Group Actions and Descriptive Set Theory

描述集合论是对波兰空间中可定义集合和函数的研究。波兰空间（群）是一个可分离的、完全可度量的拓扑空间（群）。贝克尔的研究涉及波兰群体的行为，与描述性集合理论有关，或者从描述性集合理论的角度来看。可定义集的基本类是Borel集、解析集和协解析集。这些构成了项目的主要主题，尽管其他类的可定义集合也被考虑。该项目的一个主要主题是发现轨道空间的二分类，即定理表明轨道空间要么是“小”的，要么它包含某个特定“大”集合的副本。沃特猜想（Vaught Conjecture）是一个著名的关于一阶理论的可数模型个数的逻辑开放问题，它断言了这种二分法（这个数要么是可数的，要么是连续统的大小）。这个项目的大部分内容都与这个猜想有关。Becker的项目采用了描述性集合理论的标准方法和一种新方法，从而改变了底层拓扑结构。本计画中的某些主题是在强集合论公理存在的情况下考虑的。贝克尔的工作涉及数学逻辑的几个分支，以及数学的其他领域，特别是遍历理论和算子代数。这项研究，在称为描述性集合理论的领域，是一种基本的研究，它加深了对数学中以自然方式出现的集合的理解。＊＊＊

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. ***