Henson, C. Ward and Iovino, José and Kechris, Alexander S. and Odell, Edward
(2003)
*Introduction.*
In:
Analysis and Logic.
London Mathematical Society Lecture Note Series.
No.262.
Cambridge University Press
, Cambridge, pp. 117-119.
ISBN 9781107360006.
https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145363

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## 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. Classically, in various branches of dynamics one studies actions of the groups of integers ℤ, reals ℝ, Lie groups, or even more generally (second countable) locally compact groups. One of the goals of the theory is to expand this scope by considering the more comprehensive class of Polish groups (separable completely metrizable topological groups), which seems to be the widest class of well-behaved (for our purposes) groups and which includes practically every type of topological group we are interested in.

Item Type: | Book Section | ||||||
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Additional Information: | © 2003 Cambridge University Press. | ||||||

Series Name: | London Mathematical Society Lecture Note Series | ||||||

Issue or Number: | 262 | ||||||

DOI: | 10.1017/cbo9781107360006.017 | ||||||

Record Number: | CaltechAUTHORS:20180816-160145363 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145363 | ||||||

Official Citation: | Henson, C., Iovino, J., Kechris, A., & Odell, E. (2003). Introduction. In C. Finet & C. Michaux (Eds.), Analysis and Logic (London Mathematical Society Lecture Note Series, pp. 117-119). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107360006.017 | ||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 88881 | ||||||

Collection: | CaltechAUTHORS | ||||||

Deposited By: | George Porter | ||||||

Deposited On: | 17 Aug 2018 14:35 | ||||||

Last Modified: | 16 Nov 2021 00:30 |

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