Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

该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 经典数学中的每一个定理 经典数学可以表述为关于 集合，然后从标准集合ZFC正式派生 集合论的公理 从哥德尔的作品开始 （1936）和科恩（1963），集合理论家已经表明，一些 数学家们所研究的问题是相互独立的， 即既不能从ZFC证明也不能从ZFC证明。 其中一些 然而，独立陈述是已知的可证明的，当 ZFC是由公理断言存在大 无穷基数 本项目有三个部分： 第一，继续研究一类非常大的枢机主教; 二是继续研究这些枢机之间的关系 群和有限组合学中的问题;第三， 研究了强迫和无穷大中一些无关的开放性问题 组合学 该项目的主要课题是连接既要设置 理论以及代数和拓扑学。 关于代数/拓扑 侧，要研究的主题之一是经典的辫子 在许多情况下出现，包括统计 力学（参见例如Kauffman，New Invariants in the 纽结理论，AMS月刊，1988年3月，195-241）。 的 这一领域与本项目中研究的代数的联系 导致了关于辫群的新结果 此外，结果 在集合论方面， 结和编织物的某些特性是否需要 大型基数工具集理论为他们的解决方案。

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.