Ramsey theory, dynamics of Polish groups, and Tukey functions

拉姆齐理论、波兰群动力学和图基函数

• 批准号: 1001623
• 负责人: Slawomir Solecki
• 金额: $ 29.01万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-15 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001623&HistoricalAwards=false
• 关键词: Ramsey; theory; dynamics; Polish; groups

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.

在该项目中，索莱茨基将探索几个问题，其解决方案将涉及多个数学领域的相互作用：逻辑、拓扑和组合数学。索莱茨基一直在研究伪弧的性质，可以将其视为通用曲线，重点是与该数学对象的对称性群有关的某些动力学问题。到目前为止所做的分析表明，问题的解决在于描述集合论和模型论（逻辑的分支）、拉姆齐理论（组合学的分支）和拓扑动力学（拓扑的分支）的交叉。更准确地说，通过欧文和索莱茨基使用模型理论思想的工作，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。 Solecki 最近的一项工作表明，适当类型的结构拉姆齐理论定理在某些情况下确实成立。该项目的目标之一是扩展这些组合结果。期望的扩展与结构拉姆齐理论的预期内部发展非常接近。在该项目的另一部分中，Solecki 将应用来自描述性集合论和拓扑的方法对涉及有向顺序且来自数学各个领域的数学结构进行分类。在该项目中，Solecki 将应用在数理逻辑和数学中开发的技术和概念。组合数学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。这些类型的空间在二十世纪上半叶首次被引入数学中，作为复杂曲线的奇怪例子。今天，我们知道它们自然地出现在许多数学背景中，例如流体动力学、欧几里得空间中的光滑动力系统、拓扑群、连续函数的研究等。索莱茨基打算更好地理解通过使用数学不同子领域（组合学、逻辑）的方法来绘制伪弧。这些研究中出现的问题已经导致了纯粹组合和纯粹拓扑兴趣的新定理。数学不同领域之间的这种相互促进的相互作用预计将继续下去。以类似的方式，该项目的其他部分以描述性集合论、模型论、拓扑动力学和组合学在极端顺应性研究和根据图基约简的数学对象分类中的相互作用为特色。