Measurable dynamics of Polish groups and Ramsey theory

波兰群体的可测量动态和拉姆齐理论

• 批准号: 1266189
• 负责人: Slawomir Solecki
• 金额: $ 31.37万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266189&HistoricalAwards=false
• 关键词: Measurable; dynamics; Polish; groups; Ramsey

Solecki will study measure preserving actions of certain large Polish groups. It is hoped that this study will uncover analogies between measurable dynamics of such groups and that of a single measure preserving transformation. This study is also expected to shed light on the structure of generic measure preserving transformations. In the second part of the project, Solecki will investigate certain combinatorial phenomena (Ramsey phenomena) that come up in connection with topological dynamics. Building on his abstract approach to Ramsey theory, he will attempt proving certain concrete Ramsey statements that have been proposed in the past. He will also attempt classification of concrete Ramsey theorems in certain limited contexts. Finally, in the third part of the project, he will work on very general, set theoretic methods (Tukey reductions) that are used to compare partial orders coming up in various areas of mathematics. Solecki will investigate problems that involve interactions of various areas of mathematics: logic, ergodic theory, topological dynamics, and combinatorics. Three connected themes serve as both motivation for and areas of applications of the research funded by the project: study of groups of broadly understood symmetries of mathematical objects; study of generic, that is, exhibiting all possible random behavior, symmetries; and study of spaces of objects exhibiting strong forms of homogeneity.

Solecki将研究测量某些大型波兰群体的保护行动。希望这项研究能够发现此类群体的可测量动力学与保留转换的单一衡量动力之间的类比。预计这项研究还可以阐明一般度量保留转换的结构。在项目的第二部分中，Solecki将研究与拓扑动态有关的某些组合现象（Ramsey现象）。在他对拉姆齐理论的抽象方法的基础上，他将尝试证明过去提出的某些具体的拉姆齐陈述。他还将在某些有限的情况下尝试对具体的拉姆齐定理进行分类。最后，在项目的第三部分中，他将研究非常普遍的，设置理论方法（Tukey降低），这些方法用于比较在数学的各个领域中的部分订单。 Solecki将研究涉及数学各个领域相互作用的问题：逻辑，千古理论，拓扑动态和组合学。三个相关的主题既是该项目资助的研究的动机和应用的动机：研究数学对象的广泛理解的对称性组的研究；对通用的研究，即表现出所有可能的随机行为，对称性；并研究表现出强烈同质性的物体空间。