Investigations into Set Theory and Descriptive Dynamics

集合论和描述动力学的研究

• 批准号: 9803126
• 负责人: Matthew Foreman
• 金额: $ 13.12万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803126&HistoricalAwards=false
• 关键词: Investigations; into; Set; Theory; Descriptive

This project consists of two parts. The first involves continuing investigations into combinatorial set theory. This focuses on the use of strong ideals and generic embeddings to prove reflection properties and other combinatorial consequences. These are wide ranging, from stationary set reflection at singular cardinals to infinitary Ramsey theory. The tools include methods of forcing, the PCF theory and large cardinals. The second is the use of descriptive set theory to provide new invariants for studying objects of interest to ergodic theory. The main thrust is to be able to classify the complexity of various classes of transformations with the hope of being able to distinguish between previously indistinguishable classes (e.g one may be Borel and the other true analytic.) An example of an important problem that may yield to such a technique is to show that there is an ergodic, finite entropy, measure preserving transformation that is not isomorphic to a smooth measure preserving transformation on a compact manifold. This work is in the Foundations of Mathematics: how mathematics fits together and why it works. These studies often involve consideration of problems that are not solvable by the usual assumptions of mathematics: The Zermelo-Frankel Axioms with the Axiom of Choice. Methods to be used imclude generic large cardinals, or symmetries of the mathematical universe that reveal powerful regularities that often solve intractable problems. Considerations arising from studies of the foundations of mathematics have led to classifications of problems based on their inherent complexity. These measures of complexity, in turn, can be applied to natural problems in dynamical systems and ergodic theory, as will be pursued in this project.

该项目由两个部分组成。 第一个涉及对组合集理论的持续调查。这重点是使用强大的理想和通用嵌入来证明反射特性和其他组合后果。 这些都是广泛的范围，从奇异的红衣主教到无限的拉姆西理论。工具包括强迫方法，PCF理论和大型红衣主教。 第二个是使用描述性集理论来提供新的不变性，以研究厄运理论感兴趣的对象。 主要目的是能够对各种转换的各种类别的复杂性进行分类，以期使以前的不可分辨性类别（例如，一个可能是Borel和另一个真实的分析。）一个重要问题的一个例子。这可能会给这种技术带来的一个示例，即表明，有限的，有限的，有限的，有限的胚胎术语是一种构造的构图，即在构成一个构造的构造中，以衡量量的构造，以衡量一个平稳的构造。这项工作是在数学的基础上：数学如何融合在一起以及它为什么起作用。 这些研究通常涉及考虑数学通常假设无法解决的问题：具有所选公理的zermelo-frankel公理。 用于使用的通用大型红衣主教或数学宇宙的对称性的方法，这些宇宙揭示了强大的规律性，通常可以解决棘手的问题。 对数学基础的研究产生的考虑因素导致问题基于固有的复杂性进行了分类。 这些复杂性的衡量标准又可以应用于动态系统和厄贡理论中的自然问题，正如该项目所追求的那样。