Ramsey theory, dynamics of Polish groups, and Tukey functions

在这个项目中，Solecki将探索几个问题，这些问题的解决方案将涉及到许多数学领域的相互作用：逻辑、拓扑学和组合学。Solecki一直在研究伪弧的性质，它可以被认为是一般曲线，重点是关于这个数学对象的对称性群的某个动力学问题。到目前为止的分析表明，问题的解决依赖于描述集合论和模型理论(逻辑的分支)、Ramsey理论(组合学的一个分支)和拓扑动力学(拓扑学的一个分支)的交叉。更准确地说，通过Irwin和Solecki的工作，利用模型理论的思想，动力学问题可以转化为关于有限对象的纯Ramsey理论问题。Solecki最近的一项工作表明，适当类型的结构Ramsey理论在某些情况下确实成立。这个项目的目的之一是推广这些组合结果。所希望的扩展与结构Ramsey理论预期的内部发展非常一致。在该项目的另一部分中，Solecki将应用来自描述集合论和拓扑学的方法对涉及有向序的数学结构进行分类，这些结构来自不同的数学领域。在该项目中，Solecki将把数理逻辑和组合学中发展起来的技术和概念应用于其他数学领域的问题。例如，他将研究一种称为伪弧的空间的对称性。这些类型的空间在20世纪第一季度作为复杂曲线的奇特例子首次被引入数学。今天，我们知道它们自然地出现在许多数学背景中，例如，在流体力学中，在欧几里德空间中的光滑动力系统中，在拓扑群中，在连续函数的研究中，等等。Solecki打算通过使用数学的不同子领域(组合学，逻辑)的方法来更好地理解伪弧。在这些研究中出现的问题已经导致了纯粹具有组合和纯粹拓扑意义的新定理。不同数学领域之间的这种相互刺激的互动预计将继续下去。以类似的方式，项目的其他部分以描述集合论、模型理论、拓扑动力学和组合学的交互作用为特色，在研究极端舒适性和根据Tukey约化对数学对象进行分类时。

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.