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.

拟议的研究旨在加深我们对无限集合组合的理解，开发新技术，并探索在其他数学领域中无限组合的新应用。 研究的一个组成部分是基本的：它旨在更好地了解分类问题之间的关系（例如，无数线性订单的基础问题和紧凑型凸组集的计量问题）和连续体的基数价值。 目前，我们对如何产生连续体是第二个不可数的红衣主教的集合理论模型有一个可接受的理解。产生集合理论模型的方法，其中连续体具有其他值的限制更为有限。 本提案的一部分旨在开发更好的技术来生成集合理论的模型，其中连续体大于第二个不可数的基础主教，同时保持控制模型中集合的其他组合特性的能力。 此外，该提案旨在进一步开发拉姆齐理论中所需的技术，以解决在其他领域中引起的问题，尤其是在研究群体不适的研究中引起的问题。 特别是，汤普森（Thompson）的F组（在该领域长期存在问题的主题）相当于与Hindman定理对非缔约性操作的概括有关的新型Ramsey理论陈述。在数学方面，有必要在我们的经验中找到直觉，从而在数学上获得直觉，从而有必要在数学上获得直觉。 通过研究无限集的组合特性和高维空间的几何形状提供了两个例子。 在这两种情况下，指导性直觉都是由称为Ramsey定理的经典数学片段提供的，关于图中边缘的颜色，与网络相关的数学抽象。 该建议旨在开发拉姆齐理论中的新技术，以提高我们对与无限集和高维几何对象有关的基础问题的理解。