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.
