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，无限的行为，是相当有限的。这种限制的主要原因是Zermelo-Fraenkel公理系统的集合论（ZFC），这是数学的形式基础的全球标准，以确定有关无限的基本问题的能力。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约束。这个方向取得了巨大成功的领域奇异基数算术新的令人惊讶的原则发现的银和后来的希拉。本项目旨在研究其中的一些原则。第二个方向是ZFC向更强公理系统的扩展与ZFC未解决的陈述之间的微妙相互作用。也许最突出的公理系统是ZFC通过所谓的大基数公理的扩展。该项目有助于开发具有大基数的新结构，并将其用于分析这些大基数公理和未解决语句之间的相互作用。该项目涉及集合论的几个核心领域：（i）强迫理论，特别是Prikry型强迫;（ii）基数算术;（iii）无穷组合学。使用Prikry类型的强制概念进行强制可能是生成具有奇异基数算术的非平凡模式的模型的最重要技术。该项目通过以下几个方面促进这些技术的调查和完善：超滤子组合学的发展，该理论与集合论的其他领域（如内模理论和无限组合学）的新联系的发现，几个Prikry型模型（如Magidor-Radin模型和树Prikry强迫）的中间模型的表征，以及在不可数共尾性的第一奇点的后继处获得静止反射。在这个项目中研究的超滤子的一个特别有趣的组合性质是Galvin性质，由于该性质与正则内模型中的超滤子结构，Tukey阶，Prikry强迫，该奖项反映了NSF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。

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.