Set Theory in Topology and Dynamics

拓扑空间是抽象的数学结构，其几何性质不受形状或大小的连续变化的影响。从一开始，集合论就在拓扑空间的研究中起着至关重要的作用。随着时间的推移，集合论和拓扑学之间的关系越来越深入。这个项目涉及在拓扑学和拓扑动力学中持续使用集合论方法。该项目还为研究生提供研究培训机会。集合论方法是基本子模型统一主题的各种表现形式，是本课题的重点。这个主题中包括所谓的“L-like”原则，如Jensen’s square及其变体，它们源于对内部模型的初等性和反射的考虑。在拓扑学中使用这些原理通常需要假设集合论宇宙在某种程度上接近Gödel的可构造宇宙L。该项目旨在探索这些类l原理的拓扑结果，以及在它们缺席的情况下可能存在的奇异可能性。后者需要对大基数进行拓扑研究，迫使从大基数构建模型，以及由此产生的奇异基数组合。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topological spaces are abstract mathematical structures whose geometric properties are not affected by the continuous change of shape or size. From early on, set theory has played a crucial part in study of topological spaces. The relationship between set theory and topology has grown deeper over time. This project concerns the ongoing use of set-theoretic methods in topology and topological dynamics. The project also provides research training opportunities for graduate students.The set-theoretic methods forming the focal point of this project are various manifestations of the unifying theme of elementary submodels. Included in this theme are so-called "L-like" principles, like Jensen's square and its variants, that arise from considerations of elementarity and reflection in inner models. Making use of such principles in topology typically requires the assumption that the set-theoretic universe be close to Gödel's constructible universe L in some way. This project aims to explore the topological consequences of these L-like principles, as well as the sometimes exotic possibilities that can exist in their absence. The latter entails a topologically-minded investigation of large cardinals, forcing models built from large cardinals, and the singular cardinal combinatorics arising therein.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.