Dynamics, descriptive set theory, and Ramsey theory

在这个项目中，Solecki将探索几个问题，这些问题的解决将涉及数学的许多领域：逻辑学（描述性集合论和模型论的要素）、拓扑动力学和组合学。例如，Solecki打算研究伪弧的同胚群的动力学性质问题。根据Irwin和solecki最近的一项工作，这些动力学问题可以使用射光Fra\“iss\”e极限来解决，这是模型理论中经典Fra\“iss\”e极限构造的双重版本。现在，关于射影极限的相应问题可以转化为关于有限对象的纯组合问题。为了解决这些问题，索莱茨基将尝试在与有限结构的拉姆齐理论（组合学的一个经典分支）的类比指导下开发新的组合技术。这项研究不仅涉及将数学的一个部分的结果应用到另一个部分。期望相互作用将是互利的：拓扑动力学问题提出了一种新的模型理论结构，这反过来又导致了一个有趣的组合问题，其解一方面产生了一个新的组合定理，另一方面又得到了原始动力学问题的答案。以类似的方式，项目的其他部分以描述集合论、模型论、拓扑动力学和组合学的相互作用为特征，研究了极端适应性、Borel等价关系和等距群的代数性质。跨越数学子领域的界限，可能会带来新的、实质性的见解。

In the project, Solecki will explore several problems whosesolutions will involve interactions of a number of areas ofmathematics: logic (descriptive set theory and elements of modeltheory), topological dynamics, and combinatorics. For example,Solecki intends to study questions concerning dynamical propertiesof the homeomorphism group of the pseudo-arc. By a recent work of Irwin andSolecki, these dynamical questions can be approached usingprojective Fra\"iss\'e limits, which are dual versions of theclassical construction of Fra\"iss\'e limits from model theory.Now, the corresponding questions about projective Fra\"iss\'elimits can be translated to purely combinatorial problems concerningfinite objects. To solve these problems Solecki will attempt todevelop new combinatorial techniques with the guidance of analogieswith the Ramsey theory for finite structures, a classical branch ofcombinatorics. This research will not only involve applying resultsfrom one part of mathematics to another. The expectation is that theinteractions will be mutually beneficial: the topological dynamicsquestion suggests a new model theoretic construction, which in turnleads to an interesting combinatorial problem, whose solutionyields, on the one hand, a new combinatorial theorem and, on theother hand, an answer to the original dynamical question. In asimilar manner, other parts of the project feature interactions ofdescriptive set theory, model theory, topological dynamics, andcombinatorics in the study of extreme amenability, Borel equivalencerelations, and algebraic properties of isometry groups. It may be hoped that crossing boundaries of subfields of mathematics will lead to new and substantial insights.