Invariant Descriptive Set Theory and Its Applications

这个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。Gao研究了大型波兰群的结构，如Graev度量群和泛Urysohn空间的等距群。在这个项目中，既要研究超级普遍的波兰群体问题，也要研究群体参与的概念。该项目的另一个目标是关于可能产生超有限等价关系的可数群行动。为了这个目标，高晓松和杰克逊将通力合作。重点将放在可数可解群体的普遍行动上。对于不变描述集合论的应用，PI建议作为当前项目的一部分来研究可分Banach空间的一致分类问题。这涉及到与其他Banach空间理论专家的进一步合作。不变描述集论是一种针对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，从而允许应用不变描述集合论的框架。近年来，不变描述集理论得到了很大的发展，并成功地应用于理解许多有意义的数学分类问题的复杂性。本课题致力于该理论及其应用的进一步发展。这个项目研究的主题是跨学科的，汇集了来自数学和逻辑不同领域的概念、方法和技术。

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.