Mathematical Sciences: Applied Descriptive Set Theory

描述集合论是数理逻辑的分支 与真实的数的集合有关。 研究者打算 在应用描述集合论中的三个主题上工作，即， 描述集合论与 数学的一些其他部分，特别是（1）分类 射影层次中的点集，（2）描述集合论 （3）描述集合论与波兰群 行动 这三个主题都与分析和拓扑学有关。 第三个主题也与数学的其他部分有关 逻辑，如沃特的猜想和可定义的理论 基数 该项目中的一些问题将是 在强集合论公理的假设下， 作为确定性的公理。 考虑到真实的数可以被认为是 普通数行小学算术，令人吃惊 有多少结构可以强加给他们，以及如何复杂的， 关于这个结构的问题。 描述集合 理论是用所有的理论来解决这些问题的理论。 现代数理逻辑的机器。 例如标准 这个理论的结构是集合的Borel层次，包括 两个集合族的无限序列，Pi集合和 Sigma集合，归纳定义，因此 复杂性 该理论的一个明显的应用是 在这个层次中精确地定位一个特定的集合， 分析，比如说一个函数在其上的一组点， 可微的 事实上，它是一个典型的和经典的定理， 描述性集合论（由于Mazurkiewicz），任何这样的集合， 可微性属于Borel方程中的Pi-one-one族 等级制度。 调查人员关注的问题包括 这使得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.