Combinatorics of Filters, Large Cardinal and Prikry-Type Forcing

我们对数学宇宙的理解，或者更准确地说，对无限的行为的理解是相当有限的。造成这种限制的主要原因是，作为数学形式基础的全球标准的集合论Zermelo-Fraenkel公理系统(ZFC)无法确定关于无限的基本问题。这将现代集合论的研究引向了两个主要方向。第一个方向是寻找ZFC对无限行为施加的新限制。这个方向在奇异基数算术领域取得了巨大的成功，西尔弗和后来的谢拉发现了新的令人惊讶的原理。本项目旨在研究其中的一些原则。第二个方向是ZFC对更强公理系统的扩展与ZFC未解决的语句之间的微妙交互。也许最突出的公理系统是所谓的大型基数公理对ZFC的扩展。这个项目有助于开发具有大基数的新结构，实现它们来分析这些大基数公理和未确定语句之间的相互作用。这个项目涉及集合论的几个核心领域：(I)强迫理论，更具体地说，Prikri型强迫；(Ii)基数算术；(Iii)无限组合。用Prikry类型的强迫概念来强迫可能是用非平凡的奇异基数运算模式来生成模型的最重要的技术。这个项目通过以下几个方面对这些技术的研究和完善做出了贡献：超滤组合学的发展，该理论与集合论其他领域如内模型理论和无限组合学的新联系的发现，几个Prikri型模型的中间模型的表征，如Magidor-Radin模型和树Prikry强迫，以及在不可数余项式的第一个奇点的后继处获得静止反射。在这个项目中研究的超滤器的一个特别有趣的组合性质是Galvin性质，它最近重新引起了人们的兴趣，因为这个性质与规范内部模型中的超滤器的结构、Tukey序、Prikry作用力、分配关系等令人惊讶地联系在一起。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Our understanding of the mathematical universe, or more precisely, the behavior of the infinite, is quite limited. The main reason for this limitation is the incapability of the Zermelo-Fraenkel axiomatic system of set theory (ZFC), which is the global standard for the formal foundations of mathematics, to determine basic questions about the infinite. This channeled the research in modern set theory to mainly two directions. The first direction is the search for new restraints imposed by ZFC on the behavior of the infinite. This direction had great success in the realm of singular cardinal arithmetic where new surprising principles were discovered by Silver and later by Shelah. This project aims to study some of these principles. The second direction is the subtle interaction between extensions of ZFC to stronger axiomatic systems and statements which are unsettled by ZFC. Perhaps the most prominent axiomatic systems are extensions of ZFC by the so-called large cardinal axioms. This project contributes to the development of new constructions with large cardinals, implementing them to analyze the interaction between those large cardinal axioms and unsettled statements. The PI will lean of his extensive experiences of mentoring underprivileged students in his teaching activities.This project deals with several central areas in set theory: (i) forcing theory and more particularly Prikry-type forcing; (ii) cardinal arithmetic; and (iii) infinitary combinatorics. Forcing with a Prikry-type forcing notion is perhaps the most important technique to generate models with non-trivial patterns of singular cardinal arithmetic. This project contributes to the investigation and sophistication of these techniques through several aspects: the development of combinatorics of ultrafilters, the discovery of new connections of this theory with other areas of set theory such as inner model theory and infinite combinatorics, the characterization of intermediate models of several Prikry-type models such as the Magidor-Radin model and the tree Prikry forcing, and obtaining stationary reflection at the successor of the first singulars of uncountable cofinalities. One particularly interesting combinatorial property of ultrafilters which is investigated in this project is the Galvin property which recently gained renewed interest due to the surprising connections of this property to the structure of ultrafilters in canonical inner models, the Tukey order, Prikry forcing, partition relations and more.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.