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工头将在该研究项目中处理两种类型的问题。 第一种在于崇高的理论，这与可逆性衡量标准转换的结合有关。 特别是，工头将使用描述性集理论的工具来尝试确定是否有有限熵的每一个可逆的千古转换都与紧凑的光滑歧管上的平滑转换结合在一起。 第二种问题涉及在第二个不可数的红衣主教等红衣主教上非常大的理想的强度和后果。 工头打算使用这些理想来研究分区关系。 在20世纪初期，通过查看相关流的统计行为来研究复杂的物理现象模型，成为一种研究复杂模型的技术。 在具有其他可悲行为的物理学中产生的动力系统可以接受这种分析。 崇高理论中的一个突出问题涉及测量系统的一般统计行为在多大程度上建模物理上现实情况的行为。 工头将继续使用描述性集理论的工具来研究这些问题，这是一种非常规的方法，它为问题带来了逻辑。 ***