Measurable dynamics of Polish groups and Ramsey theory

Solecki将研究某些大型波兰团体的措施保持行动。希望这项研究将揭示可测量的动态，这样的群体和一个单一的措施保持变换之间的类比。这项研究也有望揭示一般的测度保持变换的结构。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）。基于他对拉姆齐理论的抽象方法，他将尝试证明过去提出的某些具体的拉姆齐陈述。他还将尝试分类的具体拉姆齐定理在某些有限的情况下。最后，在项目的第三部分，他将研究非常一般的集合论方法（Tukey约简），用于比较数学各个领域中出现的偏序。Solecki将研究涉及数学各个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。三个相关的主题作为该项目资助的研究的动机和应用领域：研究广泛理解的数学对象对称性的群体;研究通用，即表现出所有可能的随机行为，对称性;以及研究表现出强烈形式的同质性的对象空间。

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.