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Published January 2003 | Published
Book Section - Chapter Open

Actions of Polish Groups and Classification Problems

Abstract

We will discuss in this paper some aspects of a general program whose goal is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory, which has been growing rapidly over the last few years, is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions of it with other areas of mathematics, such as the theory of topological groups, topological dynamics, ergodic theory and its relationships with the theory of operator algebras, model theory, and recursion theory.

Additional Information

© 2002 Cambridge University Press. Publication date: January 2003. Research and preparation of this paper have been partially supported by NSF Grants DMS 96-19880 and DMS 99-87437.

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August 19, 2023
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