In the project, Solecki will explore several problems whose solutions will involve interactions of a number of areas of mathematics: logic (descriptive set theory and elements of model theory), topological dynamics, and combinatorics. For example, Solecki intends to study questions concerning dynamical properties of the homeomorphism group of the pseudo-arc. By a recent work of Irwin and Solecki, these dynamical questions can be approached using projective Fra""iss'e limits, which are dual versions of the classical construction of Fra""iss'e limits from model theory. Now, the corresponding questions about projective Fra""iss'e limits can be translated to purely combinatorial problems concerning finite objects. To solve these problems Solecki will attempt to develop new combinatorial techniques with the guidance of analogies with the Ramsey theory for finite structures, a classical branch of combinatorics.
This research will not only involve applying results from one part of mathematics to another. The expectation is that the interactions will be mutually beneficial: the topological dynamics question suggests a new model theoretic construction, which in turn leads to an interesting combinatorial problem, whose solution yields, on the one hand, a new combinatorial theorem and, on the other hand, an answer to the original dynamical question. 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, Borel equivalence relations, and algebraic properties of isometry groups. It may be hoped that crossing boundaries of subfields of mathematics will lead to new and substantial insights.
