Mathematical Sciences: Applied Descriptive Set Theory

描述集合论是数理逻辑中关于实数集合的分支。研究者打算研究应用描述集合论中的三个主题，即描述集合论与数学其他部分之间存在的一些联系，特别是(1)投影层次中的点集分类，(2)描述集合论与精细拓扑，(3)描述集合论与波兰群作用。这三个主题都与分析和拓扑有关。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（如确定性公理）的假设下考虑。考虑到实数可以被认为是小学算术中的普通数轴，那么令人惊讶的是，它们可以被赋予如此多的结构，以及关于这种结构可以提出的问题是多么复杂。描述性集合理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，这个理论的一个标准构造是Borel集合层次，由两个集合族的无限序列组成，Pi集合和Sigma集合，归纳定义，因此越来越复杂。这个理论的一个明显的应用是精确地在这个层次中找到分析中出现的一个特定的集合，比如一个函数可微的点的集合。事实上，它是描述集合论（由Mazurkiewicz）的一个典型的经典定理，即任何这样的可微集合都属于Borel层次中的pi - 1 - 1族。研究者专注于这些问题，这些问题使逻辑与更广阔的数学世界相关。

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.