Set Theory and Its Applications

一个世纪以前，有一场运动把数学建立在一个严谨、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，人们已经认识到，无限集的性质本身是相当微妙的，并且违背了完全的公理化。此外，这些集合论的复杂性有时在更复杂的数学结构中表现出来，例如在代数、分析和几何中研究的那些。这个项目的目的是进一步发展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。虽然该项目涉及几条调查线，但中心主题将是利用超限序数的镜头，对从单位间隔到其本身的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包括试图证明马修·布林和马克·萨皮尔的以下猜想：如果G是单位区间的分段线性同胚群，那么G要么是初等可服从的，要么G包含Richard Thompson的群f的同构副本。PI的论文不仅证明了这个猜想是正确的，而且它将是对PLoI（单位区间的分段线性同胚群）的子群结构进行更细致分析的结果。这一分析预计会有许多其他的结果：有限生成的F的子群通过可嵌入性是很好的拟有序的；任何有限表示的PLoI子群要么是阿贝尔的，要么包含F的副本；皮亚诺算法不能证明F是可服从的。分析的中心是可数的超限序数。研究项目的第二部分涉及纯集合论中进一步发展的技术：利用强迫公理研究第二不可数基数组合的方法；詹森菱形原理在遗传基数集理论中所起的作用。该项目还支持在其他逻辑领域工作的研究生，包括可计算理论和模型理论。

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. The project also supports graduate students working in other fields of logic, including computability theory and model theory.