Set Theory and Its Applications

一个世纪前，有一场将数学建立在严格、统一的基础上的运动。因为集合的概念是数学中最原始的概念之一，所以它被用作构建更复杂的数学对象的基本结构。从那时起，人们已经意识到无限集的性质本身是相当微妙的，并且无视完全的公理。此外，这些集合论的复杂性有时表现在更复杂的数学结构中，例如那些在代数、分析和几何中研究的数学结构。这个项目的目的是进一步加深我们对集合论方法的理解，以及如何将它们应用于数学领域中出现的问题，如代数、分析和拓扑学。虽然这个项目涉及几条研究路线，但一个中心主题将是利用超限序数、紧性和大基数的镜头，更深入地理解从单位区间到单位区间的所有分段线性函数的代数的结构。该项目包括研究生的培养。研究项目的第一部分涉及使用集合论工具来研究分段线性和分段投射同态的群。这包括试图证明Matthew Brin和Mark Sapir的以下猜想：如果G是单位区间的分段线性同胚群，则G要么是初等服从的，要么G包含Richard Thompson群F的同构副本。PI的命题不仅证明了这一猜想是正确的，而且它将是对PLoI的子群结构进行更精细分析的结果。PLoI是单位区间的分段线性同态群。这一分析预计还会有其他结果：F的有限生成的子群是按可嵌入性很好地拟序的；PLoI的任何有限表示的子群要么是交换的，要么包含F的副本；Peano算法不能证明F是服从的。分析的中心将是可数的超限序数。研究项目的这一部分还涉及集合论工具的使用，如紧致性和初等嵌入代数，以研究F的适应性问题。研究项目的第二部分涉及纯集合论和应用集合论的进一步发展技术：研究同调代数中更高派生极限的消失的方法；詹森钻石原理在遗传基数集理论中所起的作用1。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.