Set Theory and Its Applications

集合论及其应用

• 批准号: 2153975
• 负责人: Justin Moore
• 金额: $ 36万
• 依托单位: Cornell University
• 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: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153975&HistoricalAwards=false
• 关键词: Set; Theory; Its; Applications

A century ago, there was a movement to put mathematics on a rigorous, unified foundation. Because the notion of a set is among the most primitive in mathematics, it was used as the basic fabric with which to build the more complicated objects of mathematics. Since that time, it has been realized that the properties of infinite sets are themselves quite subtle and defy a complete axiomatization. Moreover, these set-theoretic complexities sometimes manifest themselves in more complex mathematical structures, such as those studied in algebra, analysis, and geometry. The aim of this project is to further develop both our understanding of set-theoretic methods and also how they can be applied to problems arising in fields of mathematics such as algebra, analysis, and topology. While the project involves several lines of investigation, a central theme will be to develop a deeper understanding of the structure of the algebra of all piece-wise linear functions from the unit interval to itself using the lens of transfinite ordinal numbers, compactness, and large cardinals. This project includes the training of graduate students. The first part of the research project involves using set-theoretic tools to study groups of piecewise linear and piecewise projective homemorphisms. This includes attempting to prove the following conjecture of Matthew Brin and Mark Sapir: if G is a group of piece-wise linear homeomorphims of the unit interval, then either G is elementary amenable or else G contains an isomorphic copy of Richard Thompson's group F. It is the PI's thesis that not only is this conjecture true, but that it will be a consequence of a much finer analysis of subgroup structure of PLoI, the group of piece-wise linear homeomophisms of the unit interval. This analysis is expected to have other consequences: that the finitely generated subgroups of F are well quasi-ordered by embeddability; that any finitely presented subgroup of PLoI is either abelian or contains a copy of F; that Peano Arithmetic does not prove that F is amenable. Central to the analysis will be the countable transfinite ordinals. This part of the research project also concerns use of set-theoretic tools such as compactness and the algebra of elementary embeddings to study the amenability problem for F. The second part of the research project concerns further developing techniques in pure and applied set theory: methods for studying the vanishing of higher derived limits in homological algebra; the role that Jensen's diamond principle plays in the theory of the sets of hereditary cardinality at most aleph1.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.

一个世纪前，有一项运动将数学置于严格的统一基础上。由于集合的概念是数学上最原始的概念，因此它被用作构建数学更复杂对象的基本结构。从那时起，已经意识到，无限集的特性本身是非常微妙的，并且无视完整的公理化。此外，这些设定的理论复杂性有时在更复杂的数学结构中表现出来，例如在代数，分析和几何形状中研究的结构。该项目的目的是进一步发展我们对设定理论方法的理解，以及如何将它们应用于在代数，分析和拓扑等数学领域产生的问题。虽然该项目涉及几行调查，但中心主题将是对所有方面线性函数的结构进行更深入的了解，从单位间隔到本身的所有零件线性函数的结构，使用跨足的序数，紧凑型和大型的红衣主教。该项目包括对研究生的培训。研究项目的第一部分涉及使用固定理论工具来研究分段线性和分段投影型自杀式的组。 这包括试图证明Matthew Brin和Mark Sapir的以下猜想：如果G是单位间隔的一组线性同源物，那么G进行G是基本的，或者G包含Richard Thompson F群的同构副本。 Ploi，单位间隔的一组零件线性同源论。预计该分析将带来其他后果：F的有限生成的亚组通过嵌入性很好地排序；任何有限的Ploi子组都是Abelian或包含F的副本； Peano算术并不能证明F是可正常的。分析的中心将是可数的跨足序。 研究项目的这一部分还涉及使用集合理论工具（如紧凑性和基本嵌入的代数）来研究F的舒适性问题。研究项目的第二部分涉及在纯和应用集合理论中进一步开发技术：研究用于研究同源词汇中较高衍生限制的方法；詹森（Jensen）的钻石原理在最多杀手中的遗传基础性理论中所扮演的角色。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来支持的。