Invariant Descriptive Set Theory and Its Applications

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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.