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数学的多个领域：逻辑学、拓扑学和组合学。索莱茨基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群的某个动力学问题。到目前为止所做的分析表明，解决这个问题的方法在于描述性集合论和模型论（逻辑的分支）、拉姆齐理论（组合学的分支）和拓扑动力学（拓扑学的分支）的交叉。更确切地说，通过欧文和索莱茨基使用模型理论思想的工作，动力问题可以转化为一个关于有限物体的纯粹拉姆齐理论问题。索莱茨基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。这个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来对数学结构进行分类，这些结构涉及有向顺序，来自数学的各个领域。在这个项目中，索莱茨基将运用数理逻辑和组合学中发展起来的技术和概念来解决其他数学领域的问题。例如，他将研究一个叫做伪弧的空间的对称性。这些类型的空间在20世纪前25年首次作为复杂曲线的奇特例子引入数学。今天，我们知道它们自然地出现在许多数学环境中，例如，在流体动力学中，在欧几里得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。索莱茨基打算通过使用不同数学子领域（组合学、逻辑学）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组合和纯拓扑的新定理。不同数学领域之间这种相互刺激的互动预计将继续下去。以类似的方式，项目的其他部分在研究极端适应性和根据Tukey约简对数学对象进行分类时，以描述性集合论、模型论、拓扑动力学和组合学的相互作用为特征。