Measurable dynamics of Polish groups and Ramsey theory

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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.