Set Theory and its Applications

一个世纪前，有一场运动把数学建立在一个严格的、统一的基础上。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，人们意识到无限集合的性质本身是非常微妙的，并且不符合完整的公理化。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，例如代数，分析和几何中的那些。该项目的目的是进一步发展我们对集合论方法的理解，以及它们如何应用于数学领域（如代数，分析和拓扑）中出现的问题。虽然该项目涉及几条研究路线，但一个中心主题将是使用超限序数的透镜对所有分段线性函数的代数结构进行更深入的理解。研究项目的第一部分涉及试图证明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要么是初等顺从的，要么G包含Richard Thompson群F的同构副本。这是PI的论文，不仅是这个猜想是真实的，但它将是一个更精细的分析的子群结构的PLoI，组的分段线性同胚的单位区间。这个分析预计会有一些其他的结果：F的非线性生成子群是很好的准有序的嵌入性; PLoI的任何非线性表示子群要么是阿贝尔的，要么包含F的副本;皮亚诺算术不能证明F是顺从的。分析的中心是可数超限序数。研究计划的第二部分涉及进一步发展纯集合论的技术：通过强迫公理研究第二不可数基数的组合学的方法;詹森的钻石原则在遗传基数集理论中所起的作用。该奖项反映了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。

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.The first part of the research project involves 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 a number of 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. The second part of the research project concerns further developing techniques in pure set theory: methods for studying the combinatorics of the second uncountable cardinal via forcing axioms; 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.