Ramsey theory, dynamics of Polish groups, and Tukey functions

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In the project, Solecki will explore several problems whose solutions will involve interactions of a number of areas ofmathematics: logic, topology, and combinatorics. Solecki has been investigating properties of the pseudo-arc, which can be thought of as the generic curve, with emphasis on a certain dynamical problem concerning the group of symmetries of this mathematical object. The analysis done so far indicates that the solution to the problem lies on the intersection of descriptive set theory and model theory (branches of logic), Ramsey theory (a branch of combinatorics), and topological dynamics (a branch of topology).To be more precise, by a work of Irwin and Solecki that uses model theoretic ideas, the dynamical problem can be translated into a purely Ramsey theoretic question concerning finite objects. A more recent work of Solecki shows that structural Ramsey theoretic theorems of the appropriate type do hold in certain situations.One of the aims of this project is to extend these combinatorial results. The desired extension is in a very close agreement with the expected internal developments of structural Ramsey theory. In another part of the project, Solecki will apply methods coming from descriptive set theory and topology to classifying mathematical structures that involve directed orders and come from various areas of mathematics.In the project, Solecki will apply techniques and notions developed in mathematical logic and in combinatorics to problems in other areas of mathematics. For example, he will investigate the symmetries of a space called the pseudo-arc. These types of spaces were first introduced in mathematics in the first quarter of the twentieth century as curious examples of complicated curves. Today, we know that they appear naturally in many mathematical contexts, for example, in fluid dynamics, in smooth dynamical systems living in Euclidean spaces, among topological groups, in the study of continuous functions, etc. Solecki intends to gain a better understanding of the pseudo-arc by using methods from diverse subfields of mathematics (combinatorics, logic).Problems that have arisen in these investigations have already lead to new theorems that are of purely combinatorial and of purely topological interest. Such mutually stimulating interactions between distinct areas of mathematics are expected to continue. In a similar manner, other parts of the project feature interactions of descriptive set theory, model theory, topological dynamics, and combinatorics in the study of extreme amenability and in classification of mathematical objects according to Tukey reductions.