Measurable dynamics of Polish groups and Ramsey theory

Solecki将研究某些大型波兰团体的措施保存行动。希望这项研究将揭示这类群体的可测动力学与单一度量保持变换之间的相似之处。这项研究也有望揭示一般度量保持变换的结构。在该项目的第二部分，Solecki将研究与拓扑动力学有关的某些组合现象(拉姆齐现象)。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声明。他还将尝试在某些有限的上下文中对具体的Ramsey定理进行分类。最后，在项目的第三部分，他将致力于非常一般的集合论方法(图基约化)，这些方法用于比较不同数学领域中出现的偏序。Solecki将研究涉及不同数学领域的相互作用的问题：逻辑、遍历理论、拓扑动力学和组合学。三个相互关联的主题是该项目资助的研究的动机和应用领域：对数学对象的一组被广泛理解的对称性的研究；对类属的研究，即展示所有可能的随机行为、对称性；以及对表现出强烈形式的同质性的对象的空间的研究。

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.