Invariant Descriptive Set Theory and Its Applications

不变描述集合论及其应用

• 批准号: 0901853
• 负责人: Su Gao
• 金额: $ 25.67万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901853&HistoricalAwards=false
• 关键词: Invariant; Descriptive; Set; Theory; Its

This project concerns several aspects of invariant descriptive set theory and its applications to classification problems in mathematics. Gao studies the structures of large Polish groups such as Graev metric groups and the isometry group of the universal Urysohn space. In this project both the problem of surjectively universal Polish groups and the notion of group involvement will be studied. Another objective of the project concerns countable group actions that are likely to generate hyperfinite equivalence relations. For this objective Gao and Jackson will work collaboratively. The focus will be on universal actions of countable solvable groups. For applications of invariant descriptive set theory the PI proposes to study the uniform classification problem for separable Banach spaces as a part of the current project. This involves further collaborations with other experts in Banach space theory.Invariant descriptive set theory is a structural complexity theory for equivalence relations arising in logic and mathematics. Many significant problems in mathematics ask for satisfactory classification of mathematical objects. These classification problems can be viewed as equivalence relations, allowing the framework of invariant descriptive set theory to be applied. In the recent years invariant descriptive set theory has been greatly advanced and successfully applied to obtain an understanding of the complexity of many meaningful mathematical classification problems. This project seeks further development of the theory and its applications. The topics investigated in this project are interdisciplinary and bring together concepts, methods, and techniques from different areas of mathematics and logic.

该项目涉及不变描述集合论的几个方面及其在数学分类问题中的应用。高研究了大型波兰群的结构，例如格雷夫度量群和通用 Urysohn 空间的等距群。在这个项目中，波兰群体的普遍普遍性问题和群体参与的概念都将被研究。该项目的另一个目标涉及可能产生超有限等价关系的可数群体行动。为了这个目标，高和杰克逊将共同努力。重点将放在可数可解群的普遍行动上。对于不变描述集合论的应用，PI 建议研究可分离 Banach 空间的统一分类问题，作为当前项目的一部分。这涉及与巴拿赫空间理论的其他专家的进一步合作。不变描述集合论是逻辑和数学中出现的等价关系的结构复杂性理论。数学中的许多重要问题都要求对数学对象进行令人满意的分类。这些分类问题可以被视为等价关系，允许应用不变描述集合论的框架。近年来，不变描述集合论得到了极大的发展并成功应用于理解许多有意义的数学分类问题的复杂性。该项目寻求理论及其应用的进一步发展。该项目研究的主题是跨学科的，汇集了数学和逻辑不同领域的概念、方法和技术。