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.

工头的作品是两个方向：千古理论和组合塞特里理论。在崇高的理论中，他试图使用描述性集理论的thetools来表征各种各样的千古量度措施的复杂性。在组合套装理论中，他将在一个应用反思原理和Shelah的pcftheory的计划中研究无限组合的经典问题。Foreman研究了本质上是确定性的物理系统中明显随机性的现象。系统可以是确定性的，从某个意义上说，任何特定个体（粒子）系统的行为是完全确定的（也许是通过某些差异化），但是在宏观上，进行的任何测量都显示出随机行为。自然界中的许多系统具有这种矛盾的属性。Foreman的其他研究线认为，像Erdos这样的人在1960年代询问的困难组合问题，在21世纪仍然是主要问题。