Ramsey Theory, Set Theory, and Tukey Order

拉姆齐理论是一门研究在看似混乱中寻找秩序的理论。经典的例子是拉姆齐定理，该定理指出，对于将所有自然数对染成两种颜色的任何颜色，存在一组无限的数字，其中每一对自然数具有相同的颜色。将这一定理推广到更复杂的结构，而不仅仅是成对的数字，已经并将继续导致数学上的突破。寻找复杂结构的精确副本，其中所有某种形式的小结构的行为都很简单，是对数学中的结构结果进行分类的一种手段。拓扑拉姆齐空间理论将该领域中的许多重要定理统一到一个一般框架中。拉姆齐理论、集合论和图基秩序之间的相互关系推动了这些领域的每一个领域的进步。该项目旨在继续发展拓扑拉姆齐空间理论及其在映射由自然数构成的一些基本拓扑空间中的精确结构的应用；促进对数学公理基础的更好理解；并在这些复杂结构之间找到分界线，例如网络，其中所有小结构的行为简单，而那些没有大副本的复杂结构。拉姆齐理论和集合论在有趣的问题和证明方法上都有重叠。这在拓扑拉姆齐空间的理论中尤其明显，其经典的例子包括Ellentuck空间、等价关系的Carlson-Simpson空间和无限块序列的Milliken空间。当同构是一个太精细的概念而没有用处时，偏序之间的Tukey约简是一种对偏序进行分类的方法。自然数上的超滤子的Tukey结构正是自然数的Stone-Cech紧化中邻域基的结构。该项目的目标是继续发展拓扑拉姆齐空间理论及其与集合论和Tukey结构的联系，并将其应用于超齐次关系结构和分析。该项目的目标是多方面的，但都是相互关联的。其中包括将超滤子的确切Tukey结构映射到自然数上，以及一般的布尔代数上，以及证明与超滤子相关的新的鸽子原理。在强迫理论中，该项目旨在通过刻画那些强迫等价于拓扑Ramsey空间的偏序来简化某些强迫领域，其中一个特别的焦点是生物强迫，这也涉及到新的鸽子原理。另一项工作是大基数的Ramsey理论，将经典的关于可数结构的Ramsey定理推广到不可数结构。最后，该项目旨在解决关于超齐次关系结构上的Ramsey理论的问题，并通过拓扑Ramsey空间理论来构造新型的Banach空间。

Ramsey theory is the study of finding order within seeming chaos. The classic example of this is Ramsey's Theorem, which states that for any coloring of all pairs of natural numbers into two colors, there is an infinite set of numbers from which every pair has the same color. Extensions of this theorem to more complex structures, rather than just pairs of numbers, have led to and continue to lead to breakthroughs in mathematics. The finding of exact copies of complex structures in which all small structures of some form behave simply is a means for sorting structural results in mathematics. Topological Ramsey space theory unifies many of the important theorems in the area into one general framework. The interrelations between in Ramsey theory, set theory, and Tukey order fuels progress in each of these areas. The project aims to continue development of topological Ramsey space theory and its applications to mapping exact structures in some fundamental topological spaces constructed from the natural numbers; to promote better understanding of the axiomatic foundations of mathematics; and to find dividing lines between those complex structures, for instance networks, which have large copies in which all small structures behave simply and those which do not. Ramsey theory and set theory overlap both in problems of interest and in methods of proof. This is seen in particular in the theory of topological Ramsey spaces, classic examples of which include the Ellentuck space, the Carlson-Simpson space of equivalence relations, and the Milliken space of infinite block sequences. Tukey reduction between partial orderings is a means for classifying partial orderings when isomorphism is too fine a notion to be useful. The Tukey structure of ultrafilters on the natural numbers is exactly the structure of the neighborhood bases in the Stone-Cech compactification of the natural numbers. The project's goals are to continue developing topological Ramsey space theory and its connections with set theory and Tukey structure, with applications to ultrahomogeneous relational structures and analysis. The aims of the project are several-fold but all interrelated. These include mapping the exact Tukey structure of ultrafilters on the natural numbers, and on Boolean algebras in general, and proving new pigeonhole principles relevant to the ultrafilters. In forcing theory, the project aims to streamline some areas of forcing by characterizing those partial orderings which are forcing equivalent to a topological Ramsey space, one particular focus being on creature forcings, again involving new pigeonhole principles. Another line of work is Ramsey theory at large cardinals, extending classical Ramsey theorems on countable structures to the uncountable. Finally, the project aims to solve problems regarding Ramsey theory on ultrahomogeneous relational structures, and to construct new types of Banach spaces via topological Ramsey space theory.