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.
Normal 0 false false false EN-US X-NONE X-NONE The research results obtained in this funded project significantly advanced the theory and applications of the invariant descriptive set theory. The invariant set theory studies Polish groups, their actions, and the orbit equivalence relations induced by these actions. It can be viewed as a complexity theory for classification problems in mathematics. Enough techniques have been developed in this funded project and in previous, related projects, which led to the solution of a twenty-year-old open problem on the existence of surjectively universal Polish groups. In addition, a majority of open problems on the topic of universality of Polish groups have been resolved. The study of 2-colorings and marker structures on countable groups, initiated by the PI and his collaborators as a tool to understand the actions of countable groups, has been developed greatly in this funded project. It has been proven a powerful tool to give sweeping answers to many open questions in previous studies. In addition to the known connections with symbolic dynamics, this study also established new connections of the subject with forcing and graph theory. The invariant descriptive set theory provides a natural framework to study the relative complexity of classification problems in mathematics. These classification problems are often central problems in the respective fields. In this funded project, the PI and his collaborators investigated the uniform homeomorphism problem for separable Banach spaces. A complete determination of the relative complexity of this problem has not been obtained, but there is a better understanding of the interaction with the nonlinear theory of Banach space which is yet to be fully developed. Alternatively, the PI and his collaborators were able to make progress on the understanding of the classification problems for symbolic dynamical systems. The complexity for several important classification problems for symbolic dynamical systems was completely classified. Due to the interdisciplinary nature of the subject, research results in this funded project have had significant influence in other areas of mathematics such as topological groups, dynamical systems, ergodic theory, and Banach space theory. In each area the problems addressed in this funded project are at the center of the relevant field and the research results generated a lot of interest from the experts in those fields. The project has also had significant broader impact on the development of human resources. During the project period the PI graduated 5 PhDs and 1 Master’s student, and worked closely with 1 postdoctoral fellow. Most of the trainees have published as coauthors during the project period. The group contains women and minorities. The research topics have also attracted other graduate students into the field, with whom the PI is currently working.
