Mathematical Sciences: Problems in Descriptive Set Theory, Ergodic Theory and Set Theory

数学科学：描述集合论、遍历理论和集合论中的问题

• 批准号: 9500494
• 负责人: Matthew Foreman
• 金额: $ 7.64万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500494&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Descriptive; Set

9500494 Foreman Foreman will work on two types of problems in this research project. The first kind lies within ergodic theory and has to do with conjugacy of invertible measure-preserving transformations. In particular, Foreman will use the tools of descriptive set theory to attempt to determine whether every invertible ergodic transformation with finite entropy is conjugate to a smooth transformation on a compact smooth manifold. The second kind of problem concerns the strength and consequences of very large ideals on cardinals such as the second uncountable cardinal. Foreman intends to use these ideals to investigate partition relations. In the early twentieth century, ergodic theory arose as a technique for studying complicated models of physical phenomenon by looking at the statistical behavior of the associated flows. Dynamical systems arising in physics with otherwise intractible behavior were amenable to this kind of analysis. A prominent problem in ergodic theory deals with the extent to which the general statistical behavior of measure preserving systems models the behavior of physically realistic situations. Foreman will continue to investigate these problems using the tools of descriptive set theory, an unconventional approach which brings logic to bear on the problem. ***

9500494 Foreman 福尔曼将在该研究项目中解决两类问题。 第一种属于遍历理论，与可逆测度保持变换的共轭有关。 特别是，福尔曼将使用描述性集合论的工具来尝试确定每个具有有限熵的可逆遍历变换是否与紧致平滑流形上的平滑变换共轭。 第二类问题涉及非常大的理想对基数（例如第二个不可数基数）的强度和后果。 福尔曼打算利用这些理想来研究分区关系。 二十世纪初，遍历理论作为一种通过观察相关流的统计行为来研究复杂物理现象模型的技术而出现。 物理学中出现的具有其他棘手行为的动力系统适合这种分析。 遍历理论中的一个突出问题涉及测度保持系统的一般统计行为对物理现实情况的行为进行建模的程度。 福尔曼将继续使用描述性集合论工具来研究这些问题，这是一种非常规方法，可以为问题带来逻辑。 ***