Cardinal Characteristics and Related Topics

连续统基本特征理论的最新研究表明，人们可以通过研究与它们相关的某些关系而不是基本特征本身来获得更详细的信息。 这尤其适用于这些关系是波莱尔集的情况。部分计划的研究涉及的问题“哪些基本特征与博雷尔关系？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，答案还不知道是否定的。 研究人员希望通过证明某些特定特征与Borel关系无关来缩小这一差距。 计划研究的第二个方面涉及关系的顺序组成，这发生在许多定理和证明的基本特征。 先验的，顺序组成不能博雷尔关系，但调查人员打算扩展理论的博雷尔关系，使用工具从拓扑理论，以涵盖顺序组成。 这项研究还包括与可计算性理论的基本特征有关的问题。 最后，调查还计划研究使用groupwise密度数（一个特点介绍了几年前由Laflamme和调查）在partition theorems.Cardinal特征的连续构成一个重要领域的当代研究集理论。不仅是他们的利益为自己的缘故，但在过去的二十年里，他们发挥了作用，应用集理论的方法，一般拓扑结构。 最近，已经有应用程序（包括一些由于调查）的其他部分的数学，特别是代数。 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理论的联系，处理涉及真实的数的相对容易定义的关系。 第二个方向将这个理论与递归理论联系起来，递归理论研究什么是（什么不是）原则上可计算的。 第三个方向建立了与无限集合和结构的组合信息的联系。

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.