Some Problems in Set Theory and Ergodic Theory

集合论和遍历论中的一些问题

• 批准号: 0101155
• 负责人: Matthew Foreman
• 金额: $ 16.5万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101155&HistoricalAwards=false
• 关键词: Some; Problems; Set; Theory; Ergodic

Foreman's work is in two directions: ergodic theory and combinatorial settheory. In ergodic theory he seeks to characterise the complexity ofvarious classes of ergodic measure preserving transformations, using thetools of descriptive set theory. In combinatorial set theory he iscontinuing a program of applying reflection principles and Shelah's PCFtheory to study classical problems in infinite combinatorics.Foreman investigates the phenomenon of apparent randomness in physicalsystems that are deterministic in nature. A system can be deterministic inthe sense that the behavior of any particular individual (particle) inthat system is completely determined (perhaps by some differentialequations), but on a macro level, any measurement made shows randombehavior. Many systems in nature have this kind of paradoxical properties.Foreman's other line of research considers difficult combinatorialproblems asked by people such as Erdos in the 1960's that remain majorquestions in the 21st century.

福尔曼的工作是在两个方向：遍历理论和组合集合论。在遍历理论中，他试图利用描述性集合论的工具来证明各类遍历测度保持变换的复杂性。在组合集理论，他继续应用反射原理和谢拉的PCF理论研究经典问题的无限combinatorics.Foreman调查现象的明显随机性在physicalsystems是确定性的性质。一个系统可以是确定性的，在这个意义上，系统中任何特定个体（粒子）的行为都是完全确定的（也许是由一些微分方程决定的），但在宏观层面上，任何测量都显示出随机行为。自然界中的许多系统都具有这种自相矛盾的性质。福尔曼的另一条研究路线考虑了像鄂尔多斯这样的人在20世纪60年代提出的困难的组合问题，这些问题在21世纪世纪仍然是主要问题。