Combinatorial Set Theory

本研究旨在加深我们对无限集组合数学的理解，开发新技术，并探索无限组合数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性的：它旨在更好地理解分类问题（例如不可数线性阶的基础问题和紧凸集的度量问题）与连续统基数值之间的关系。 目前，我们对如何生成集合论模型有一个可以接受的理解，其中连续统是第二个不可数基数。产生连续统具有其他价值的集合论模型的方法相当有限。 本提案的一部分旨在开发更好的技术来生成集合论模型，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属性的能力。 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，特别是群体顺从性研究中出现的问题。 具体来说，汤普森群 F 的适用性（该领域长期存在的问题的主题）相当于一种新型拉姆齐理论陈述，涉及将欣德曼定理推广到非结合运算。在数学中，经常需要找到一种指导启发式方法，以便在我们自己的人类经验不足的情况下获得直觉。 对无限集的组合性质和高维空间的几何的研究提供了这种需求的两个例子。 在这两种情况下，指导直觉都是由称为拉姆齐定理的经典数学提供的，该定理涉及图形中边缘的颜色，这是与网络相关的数学抽象。 该提案旨在开发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相关基础问题的理解。

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.