Invariant Descriptive Set Theory and Its Applications

这个项目涉及不变描述集理论及其在数学分类问题中的应用的几个方面。高研究结构的大型波兰团体，如Graev度量团体和等距组的普遍Urysohn空间。在这个项目中，我们将研究满射普遍波兰群体的问题和群体参与的概念。该项目的另一个目标是关注可数的群体行动，可能会产生超有限等价关系。为了这个目标，高和杰克逊将通力合作。重点将放在可数可解群的普遍作用上。对于不变描述集理论的应用，PI建议研究可分Banach空间的统一分类问题，作为当前项目的一部分。不变描述集理论（Invariant Descriptive Set Theory）是一种结构复杂性理论，用于研究逻辑和数学中的等价关系。数学中的许多重要问题都要求对数学对象进行满意的分类。这些分类问题可以被视为等价关系，允许不变描述集理论的框架被应用。近年来，不变描述集理论得到了极大的发展，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。该项目旨在进一步发展该理论及其应用。在这个项目中调查的主题是跨学科的，并汇集了来自数学和逻辑的不同领域的概念，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.