Mathematical Sciences: Applied Descriptive Set Theory

描述集合论是与实数集有关的数理逻辑的一个分支。本文旨在研究应用描述集合论中的三个问题，即描述集论与数学其他部分之间的一些联系，特别是(1)射影层次中点集的分类，(2)描述集论与更精细的拓扑，(3)描述集论与波兰群作用。这三个主题都与分析和拓扑相关。第三个主题也与数理逻辑的其他部分有关，如沃特猜想和可定义基数理论。项目中的一些问题将在强集合论公理的假设下进行考虑，例如确定性公理。考虑到实数可以被认为是小学算术中的普通数字线，令人惊讶的是，可以将多少结构强加给它们，以及关于这种结构可以提出的问题是多么复杂。描述集合论是用现代数理逻辑的所有机制来解决这些问题的理论。例如，这一理论的一个标准构造是Borel集合层，它由两个无限的集合族序列组成，PI集和Sigma集，它们是归纳定义的，因此具有越来越高的复杂性。该理论的一个明显的应用是在这个层次结构中精确地定位在分析中产生的特定集合，例如函数在其上是可微点的集合。事实上，它是描述集合论的一个典型而经典的定理(由于Mazurkiewicz)，任何这样的可微性集都属于Borel族中的PI-One-One族。研究人员专注于这样的问题，这些问题使逻辑与更广泛的数学世界相关。

Descriptive set theory is the branch of mathematical logic concerned with sets of real numbers. The investigator intends to work on three topics in applied descriptive set theory, that is, on some connections which exist between descriptive set theory and some other parts of mathematics, specifically (1) classification of pointsets in the projective hierarchy, (2) descriptive set theory and finer topologies, (3) descriptive set theory and Polish group actions. All three topics are related to analysis and topology. The third topic is also related to other parts of mathematical logic, such as Vaught's Conjecture and the theory of definable cardinality. Some of the problems in the project will be considered under the assumption of strong set theoretic axioms such as the axiom of determinacy. Considering that the real numbers can be thought of as the ordinary number line of grade school arithmetic, it is surprising how much structure can be imposed upon them and how intricate the questions that can be asked about this structure. Descriptive set theory is the theory that addresses these questions with all the machinery of modern mathematical logic. For example, a standard construct of this theory is the Borel hierarchy of sets, consisting of two infinite sequences of families of sets, the Pi sets and the Sigma sets, defined inductively, and hence of increasing complexity. One of the obvious applications of the theory is to locate precisely in this hierarchy a particular set which arises in analysis, say the set of points at which a function is differentiable. In fact, it is a typical and classical theorem of descriptive set theory (due to Mazurkiewicz) that any such set of differentiability belongs to the family Pi-one-one in the Borel hierarchy. The investigator is focussed on questions such as these, which make logic relevant to the wider world of mathematics.