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，Solecki将探讨几个问题，其解决方案将涉及多个教学领域的相互作用：逻辑，拓扑和组合学。 Solecki一直在研究伪弧的属性，可以将其视为通用曲线，重点是关于该数学对象的对称组的某些动态问题。到目前为止所做的分析表明，解决问题的解决方案在于描述性集理论与模型理论（逻辑的分支），Ramsey理论（组合学的分支）和拓扑动态（拓扑结构的一个分支）（拓扑结构）的相互作用。要通过Irwin和Solecki进行模型的理论问题，可以将动态问题转换为对象。 Solecki的最新工作表明，在某些情况下，适当类型的结构性Ramsey理论理论确实存在。该项目的目的之一是扩展这些组合结果。所需的扩展与结构性拉姆西理论的预期内部发展非常密切。在项目的另一部分中，Solecki将应用来自描述性集理论和拓扑的方法来对涉及定向订单并来自数学领域的数学结构进行分类。在该项目中，Solecki将应用在数学逻辑和组合中，将技术和概念应用于数学其他领域的问题中。例如，他将研究一个称为伪弧的空间的对称性。这些类型的空间首先是在20世纪第一季度在数学中引入的，这是复杂曲线的奇怪例子。今天，我们知道它们在许多数学环境中自然而然地出现在欧几里得空间中的平稳动态系统中，在拓扑组中，在连续功能等方面的流畅动力学系统，等等。Solecki打算通过使用来自这些数学的多样化的数学来实现这些研究的方法来更好地理解伪库的方法。纯粹是组合和纯粹拓扑兴趣的新定理。预计数学不同领域之间的这种相互刺激的相互作用将继续。以类似的方式，该项目的其他部分具有描述性集理论，模型理论，拓扑动力学和组合学的相互作用，并根据Tukey减少的数学对象进行了研究。