Complex definable equivalence relations and applications

这个项目涉及可定义的等价关系和分类问题，这些问题比可数结构的同构更复杂。其中一个主要的焦点是泛轨道等价关系，即由波兰群的Borel作用量引起的最复杂的等价关系。研究主题包括：(1)Urysohn泛度量空间，等距群及其作用，(2)酉群的作用，以及(3)各种拓扑空间的分类问题。这些主题与分析、几何和拓扑学的明显联系也将描述集合论和其他数学领域的发展结合在一起。可定义等价关系的描述集合论为衡量数学中各种分类问题的复杂性提供了一个标准尺度。这一规模的低端通过之前在该领域的工作得到了很好的理解。但有关高端市场的信息相对较少。研究者试图提高对复杂的可定义等价关系的认识。通过这项研究，不仅可以更好地理解所涉及的数学问题的复杂性或难度，而且可以获得处理复杂和大规模系统的方法，并将其应用于数学和科学的其他领域。

This project concerns definable equivalence relations and classification problems that are more complex than the isomorphism of countable structures. One of the main focuses is the universal orbit equivalence relation, namely, the most complex equivalence relation induced by a Borel action of a Polish group. The investigator studies topics such as (1) the Urysohn universal metric space, the isometry group and its actions, (2) actions of the unitary group, and (3) classification problems for various topological spaces. The obvious connections of these topics to analysis, geometry and topology also bring together the development of descriptive set theory and other fields of mathematics.The descriptive set theory of definable equivalence relations has provided a standard scale on which complexity of various classification problems in mathematics can be measured. The low end of this scale is very well understood through the previous work in the field. But there has been relatively little information known for the high end. The investigator seeks to advance the knowledge of complex definable equivalence relations. Through this research it is hopeful that not only the complexity or difficulty of the mathematical problems involved can be better understood but also methods to deal with complicated and large-scale systems can be obtained and applied to other fields of mathematics and sciences.