Combinatorial Set Theory

组合集合论

• 批准号: 1262019
• 负责人: Justin Moore
• 金额: $ 36.21万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1262019&HistoricalAwards=false
• 关键词: Combinatorial; Set; Theory

The proposed research aims to deepen our understanding of the combinatorics of infinite sets, to develop new techniques, and to explore new applications of infinite combinatorics in other fields of mathematics. One component of the research is foundational: it aims to better understand the relationship between classification problems (such as the basis problem for the uncountable linear orders and the metrization problem for compact convex sets) and the value of the cardinality of the continuum. Currently we have an acceptable understanding of how to produce models of set theory in which the continuum is the second uncountable cardinal; the methods for producing models of set theory in which the continuum has some other value is considerably more limited. Part of the present proposal aims to develop better techniques for producing models of set theory in which the continuum is larger than the second uncountable cardinal, while maintaining the ability to control other combinatorial properties of sets in the model. Additionally, the proposal aims to further develop techniques in Ramsey theory needed to solve problems arising in other fields and in particular arising in the study of the amenability of groups. Specifically the amenability of Thompson's group F - the subject of a longstanding problem in the field - is equivalent to a new type of Ramsey-theoretic statement relating to generalizations of Hindman's theorem to nonassociative operations.Frequently in mathematics it is necessary to find a guiding heuristic in order to gain intuition in a setting where our own human experience falls short. Two examples of this need are provided by the study of the combinatorial properties of infinite sets and the geometry of high dimensional spaces. In both cases, guiding intuition is provided by a classical piece of mathematics known as Ramsey's theorem, concerning colorings of edges in a graph, a mathematical abstraction relating to networks. The proposal aims to develop new techniques in Ramsey theory in order to improve our understanding of foundational issues relating to infinite sets and high dimensional geometric objects.

本研究旨在加深我们对无限集合组合学的理解，开发新的技术，并探索无限组合学在其他数学领域的新应用。该研究的一个组成部分是基础的：它旨在更好地理解分类问题（如不可数线性阶的基问题和紧凸集的度量问题）与连续统的基数值之间的关系。目前我们对如何产生集合论的模型有了一个可接受的理解其中连续统是第二个不可数基数；产生集合论模型的方法，其中连续统有一些其他的值是相当有限的。本提案的部分目的是开发更好的技术来产生集合论模型，其中连续统大于第二个不可数基数，同时保持控制模型中集合的其他组合特性的能力。此外，该提案旨在进一步发展拉姆齐理论中的技术，以解决其他领域出现的问题，特别是在群体适应性研究中出现的问题。具体地说，Thompson群F的适用性——这是该领域一个长期存在的问题的主题——相当于一种新型的ramsey理论陈述，它将Hindman定理推广到非结合运算。在数学中，经常需要找到一个指导性的启发式，以便在我们自己的人类经验不足的情况下获得直觉。无限集的组合性质的研究和高维空间的几何给出了这种需要的两个例子。在这两种情况下，指导性直觉都是由一个被称为拉姆齐定理的经典数学定理提供的，拉姆齐定理是关于图中边的着色的，是一个与网络有关的数学抽象。该提案旨在发展拉姆齐理论的新技术，以提高我们对无限集和高维几何对象的基本问题的理解。