Set Theory in Topology and Dynamics

拓扑和动力学中的集合论

• 批准号: 2154229
• 负责人: Will Brian
• 金额: $ 18万
• 依托单位: University of North Carolina at Charlotte
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154229&HistoricalAwards=false
• 关键词: Set; Theory; Topology; Dynamics

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.

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。 从很早开始，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。这个项目涉及拓扑学和拓扑动力学中集合论方法的持续使用。该项目还为研究生提供了研究培训机会。形成该项目焦点的集合论方法是基本子模型统一主题的各种表现形式。在这个主题中包括所谓的“类L”原则，如詹森的平方和它的变体，它产生于对内在模型中的元素性和反射的考虑。在拓扑学中使用这些原理通常需要假设集合论的宇宙在某种程度上接近哥德尔的可构造宇宙L。该项目旨在探索这些L-样原则的拓扑后果，以及在没有它们的情况下有时可能存在的奇异可能性。后者需要一个拓扑思想的调查大基数，迫使模型建立从大基数，和奇异基数组合产生于此。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。