Mathematical Sciences: Polish Group Actions and Descriptive Set Theory

点击翻译按钮获取中文摘要

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. ***