Complex definable equivalence relations and applications

点击翻译按钮获取中文摘要

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.