Orbit Equivalence Relations and Classification Problems

这个项目涉及可定义等价关系的描述集合论的两个重要方面。在理论方面，PI建议研究波兰群体的结构及其行动的动力学。群的范围从可数幂零群到完全酉群。在应用方面的PI建议研究的主要开放问题有关分类的数学结构所产生的分析，拓扑和几何。然而，主要的焦点将是分类问题，这些问题与酉群的作用引起的轨道等价关系相当。可分复Hilbert空间上的有界线性算子的分类就是这样一个问题，在数学的各个领域中，许多重要的开放性问题都要求对该领域研究的数学对象进行满意的分类。这些分类问题通常可以形式化为可定义的等价关系，有时甚至是由波兰群作用引起的轨道等价关系。在过去的15年左右的时间里，一个引人注目的可定义等价关系的理论已经发展起来，从而对许多分类问题的复杂性有了完整的理解。这个项目旨在进一步发展可定义等价关系的描述集理论。预计不同领域的数学将汇集在这个基础框架。对于许多旧的分类问题，描述集理论的观点是新的，值得研究。

This project concerns two important aspects of the descriptive set theory of definable equivalence relations. On the theoretical side the PI proposes to study the structures of Polish groups and the dynamics of their actions. The scope of the groups ranges from countable nilpotent groups to the full unitary group. On the application side the PI proposes to study the major open problems concerning classification of mathematical structures arising in analysis, topology and geometry. The main focus, though, will be classification problems that are comparable to the orbit equivalence relations induced by actions of the unitary group. One example of such a problem is the classification of bounded linear operators on a separable complex Hilbert space.Many important open problems in various fields of mathematics ask for satisfactory classification of mathematical objects studied in the fields. These classification problems can often be formalized as definable equivalence relations, sometimes even orbit equivalence relations induced by Polish group actions. A striking theory of definable equivalence relations has been developed in the past 15 years or so and complete understanding of the complexity of many classification problems has thus been obtained. This project seeks further development of the descriptive set theory of definable equivalence relations. It is anticipated that mathematics of different fields be brought together in this foundational framework. For many of the old classification problems the descriptive set theoretic perspective is new and worth investigating.