Dynamics, descriptive set theory, and Ramsey theory

在该项目中，Solecki将探讨几个问题，其解决方案将涉及一些数学领域的相互作用：逻辑（描述性集合论和模型论元素），拓扑动力学和组合学。例如，Solecki打算研究有关伪弧的同胚群的动力学性质的问题。通过欧文和索莱茨基最近的工作，这些动力学问题可以使用投影Fra\“iss\'e极限来解决，这是模型论中Fra\“iss\' e极限经典构造的双重版本。现在，有关投影Fra\“iss\'e极限的相应问题可以转化为有关有限对象的纯粹组合问题。为了解决这些问题，Solecki将尝试开发新的组合技术的指导下analogies与拉姆齐理论的有限结构，一个经典的分支组合。这项研究将不仅涉及应用结果从一个部分的数学到另一个。期望的是，相互作用将是互利的：拓扑动力学问题提出了一个新的模型理论建设，这反过来又导致一个有趣的组合问题，其解决方案产量，一方面，一个新的组合定理，另一方面，一个答案，原来的动力学问题。以类似的方式，该项目的其他部分功能的相互作用的描述集理论，模型理论，拓扑动力学，和组合学的研究极端的顺从性，Borel equivalencerrelations，代数性质的等距群。人们希望，跨越数学子领域的界限将导致新的和实质性的见解。

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.