Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

本项目的研究领域是集合论，特别是大基数和强迫。集合论包含了经典数学，因为经典数学的每一个定理都可以被表述为关于集合的陈述，然后从集合论公理的标准集合ZFC正式推导出来。从哥德尔（1936）和科恩（1963）的工作开始，集合理论家已经表明数学家所研究的一些问题是独立的，即既不能证明也不能否定ZFC。然而，当ZFC被断言存在大无限基数的公理扩充时，这些独立陈述中的一些是可证明的。即时项目有三个部分：第一，继续研究一类非常大的基数；第二，继续研究这些基数与群和有限组合问题的关系；第三，研究了强迫与无穷组合学中一些不相关的开放性问题。这个项目的主要课题与集合论、代数和拓扑学都有联系。在代数/拓扑方面，要研究的主题之一是经典辫群，它在许多情况下出现，包括统计力学（例如，参见考夫曼，结理论中的新不变量，AMS Monthly， 1988年3月，195-241）。这一领域与本项目所研究的代数的联系导致了关于辫群的新结果。此外，该项目在集合论方面的结果提出了一个问题，即结和辫的某些性质是否需要集合论的大型基本工具来解决。

The research area of this project is set theory, in particular, large cardinals and forcing. Set theory contains classical mathematics in the sense that every theorem of classical mathematics can be formulated as a statement about sets, and then formally derived from the standard collection ZFC of axioms for set theory. Beginning with the work of Godel (1936) and Cohen (1963), set theorists have shown that some problems on which mathematicians have worked are independent, i.e. neither provable nor disprovable from ZFC. Some of these independent statements are nevertheless known to be provable when ZFC is augmented by axioms asserting the existence of large infinite cardinal numbers. The instant project has three parts: first, to continue the study of a class of very large cardinals; second, to continue the study of the relation of these cardinals to problems in groups and finite combinatorics; and third, to study some unrelated open questions in forcing and infinite combinatorics. The main subject of this project is connected both to set theory and to algebra and topology. On the algebra/topology side, one of the topics to be studied is the classical braid groups, which arise in many contexts, including statistical mechanics (see for instance Kauffman, New invariants in the theory of knots, AMS Monthly, March 1988, 195-241). The connection of this area with the algebras studied in this project has led to new results about the braid groups. Moreover, results on the set theory side of the project have opened the question whether some properties of knots and braids will require the large cardinal tools of set theory for their solution.