Investigations into Set Theory and Ergodic Theory

集合论和遍历理论的研究

• 批准号: 0400887
• 负责人: Matthew Foreman
• 金额: $ 20万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400887&HistoricalAwards=false
• 关键词: Investigations; into; Set; Theory; Ergodic

Foreman will investigate the descriptive set theoreticaspects of the classification problem for ergodic measure preservingsystems. He has partial results showing, for example, that they cannot beclassified by countable structures. He conjectures that the isomorphismproblem is provably complex, e.g. analytic but not Borel. The secondaspect of Foreman's proposal is to study ``canonical structure" in settheory. This is structure that requires the Axiom of Choice for itsexistence, but is independent of the choices made in it construction. Ofparticular interest are PCF objects and relations with squares anddiamonds.Foreman will investigate two areas of mathematics. The first has to dowith the evolution of complex systems in time. These systems appear toshow random behaviour, but of qualitatively different sorts. Foreman'sproject is to show that various behavior types are not distinguishable byconcrete observations. The second project has to do with the assumptionsthat the study of mathematics is based on. The usual assumptions are notstrong enough to be able to find the answers to all mathematicalquestions. For this reason it is necessary to investigate strengtheningsof the ordinary assumptions. Foreman proposes to do this by focussing on``canonical structure" in set theory.

福尔曼将调查的描述集theoreticaspects分类问题的遍历措施scheduling系统。例如，他的部分结果表明，它们不能被可数结构分类。他认为同构问题是可证明的复杂问题，例如解析问题而不是波莱尔问题。福尔曼建议的第二个方面是研究集合论中的“规范结构”。这是一种结构，它的存在需要选择公理，但它独立于在它的构造中所做的选择。特别感兴趣的是PCF对象和与正方形和菱形的关系。福尔曼将研究两个数学领域。第一个与复杂系统的时间演化有关。这些系统似乎表现出随机行为，但性质不同。福尔曼的项目是要表明，各种行为类型是无法通过具体的观察来区分的。第二个项目与数学研究所基于的假设有关。通常的假设不足以找到所有数学问题的答案。因此，有必要研究普通假设的强化。福尔曼建议通过关注集合论中的“规范结构”来做到这一点。