Cardinal Characteristics and Related Topics

连续体基本特征理论的最新工作表明，人们可以通过工作获得更详细的信息，而不是与基本特征本身有关，而是与它们相关的某些关系。这尤其适用于这些关系是Borel集的情况。计划中的部分研究涉及这样一个问题：“哪些基本特征与波雷尔的关系有关？”对于许多特征，答案是肯定的；对于其他特征，答案似乎是否定的，但对于任何特定的特征，还不知道是否定的。研究人员希望通过证明某些特定特征与博雷尔关系无关来弥合这一差距。计划研究的第二个方面涉及关系的顺序组成，它出现在许多关于基数特征的定理和证明中。先验地，顺序合成不可能是Borel关系，但研究者打算利用Topos理论的工具来扩展Borel关系的理论，以便涵盖顺序合成。这项研究还包括与可计算性理论和基本特征相关的问题。最后，研究人员还计划研究GroupWise密度数(由Laflamme和研究人员在几年前引入的一个特征)在划分定理中的应用。连续统的基本特征构成了集合论当代研究的一个重要领域。它们不仅是为了它们自己的利益，而且在过去的二十年里，它们在集合论方法在一般拓扑学中的应用中发挥了作用。最近，数学的其他部分，特别是代数，也得到了应用(包括研究者的一些应用)。研究人员和研究生将在几个方向上扩展这一理论。一个方向涉及与经典描述集合论的联系，处理涉及实数的相对容易定义的关系。第二个方向将这一理论与递归理论联系起来，递归理论研究原则上什么是可计算的，什么是不可计算的。第三个方向与关于无限集合和结构的组合信息建立联系。

ABSTRACTRecent work in the theory of cardinal characteristics of the continuum has indicated that one can get more detailed information by working, not with the cardinal characteristics themselves, but with certain relations associated with them. This applies especially in cases where these relations are Borel sets. Part of the planned research concerns the question "Which cardinal characteristics are associated with Borel relations?" For many characteristics the answer is known to be positive; for others it appears to be negative, but it isn't yet known to be negative for any particular characteristic. The investigator hopes to close this gap by proving that certain specific characteristics are not associated with Borel relations. A second aspect of the planned research concerns the sequential composition of relations, which occurs in many theorems and proofs about cardinal characteristics. A priori, sequential compositions cannot be Borel relations, but the investigator intends to extend the theory of Borel relations, using tools from topos theory, so as to cover sequential composition. The research also includes questions relating computability theory to cardinal characteristics. Finally, the investigator also plans to study the use of the groupwise density number (a characteristic introduced some years ago by Laflamme and the investigator) in partition theorems.Cardinal characteristics of the continuum constitute a significant area of contemporary research in set theory. Not only are they of interest for their own sake, but for the last two decades they have played a role in applications of set-theoretic methods to general topology. More recently, there have been applications (including some due to the investigator) to other parts of mathematics, particularly algebra. The investigator and graduate students will extend this theory in several directions. One direction concerns connections with classical descriptive set theory, dealing with relatively easily definable relations involving real numbers. A second direction connects this theory with recursion theory, the study of what is (and what is not) computable in principle. A third direction establishes connections with combinatorial information about infinite sets and structures.