Ramsey theory, dynamics of Polish groups, and Tukey functions

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，拓扑学和组合学。索莱茨基一直在研究伪弧的性质，伪弧可以被认为是通用曲线，重点是关于这个数学对象的对称群的某些动力学问题。分析表明，解决这一问题的关键在于描述集合论和模型论的交叉（逻辑分支），拉姆齐理论（组合数学的一个分支）和拓扑动力学（拓扑学的一个分支）.更确切地说，通过欧文和索莱茨基使用模型论思想的工作，动力学问题可以转化为关于有限对象的纯粹拉姆齐理论问题.最近的工作Solecki表明，结构拉姆齐理论定理的适当类型不举行在某些situations.One的目的，这个项目是扩大这些组合的结果。所需的扩展是在一个非常密切的协议与预期的内部发展的结构拉姆齐理论。在该项目的另一部分，Solecki将应用来自描述集合论和拓扑学的方法对涉及有向序和来自数学各个领域的数学结构进行分类。在该项目中，Solecki将应用数理逻辑和组合学中发展的技术和概念来解决其他数学领域的问题。例如，他将研究称为伪弧的空间的对称性。这些类型的空间在世纪的前25年首次被引入数学，作为复杂曲线的奇怪例子。今天，我们知道，他们自然地出现在许多数学背景下，例如，在流体动力学，在光滑的动力系统生活在欧几里德空间，之间的拓扑群，在连续函数的研究等Solecki打算获得更好的理解伪弧通过使用方法从不同的数学子领域在这些研究中出现的问题已经导致了纯组合和纯拓扑的新定理。这种不同数学领域之间相互促进的互动预计将继续下去。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和分类的数学对象根据图基减少。

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.