Actions of Polish Groups and Classification Problems

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.

[1]  Alain Louveau,et al.  Countable Borel Equivalence Relations , 2002, J. Math. Log..

[2]  A. Kechris,et al.  A strong generic ergodicity property of unitary and self-adjoint operators , 2001, Ergodic Theory and Dynamical Systems.

[3]  G. Hjorth,et al.  The complexity of the classification of Riemann surfaces and complex manifolds , 2000 .

[4]  Greg Hjorth,et al.  Classification and Orbit Equivalence Relations , 1999 .

[5]  A. Kechris.  Rigidity properties of Borel ideals on the integers , 1998 .

[6]  Greg Hjorth,et al.  Borel Equivalence Relations Induced by Actions of the Symmetric Group , 1998, Ann. Pure Appl. Log..

[7]  Greg Hjorth,et al.  Borel Equivalence Relations and Classifications of Countable Models , 1996, Ann. Pure Appl. Log..

[8]  A. Kechris,et al.  The Descriptive Set Theory of Polish Group Actions: BETTER TOPOLOGIES , 1996 .

[9]  T. Giordano,et al.  Topological orbit equivalence and C*-crossed products. , 1995 .

[10]  Greg Hjorth,et al.  Analytic equivalence relations and Ulm-type classifications , 1995, Journal of Symbolic Logic.

[11]  R. Dougherty,et al.  The structure of hy-per nite Borel equivalence relations , 1994 .

[12]  Alexander S. Kechris.  Amenable Equivalence Relations and Turing Degrees , 1991, J. Symb. Log..

[13]  Klaus Schmidt,et al.  Algebraic ideas in ergodic theory , 1990 .

[14]  Alain Louveau,et al.  A Glimm-Effros dichotomy for Borel equivalence relations , 1990 .

[15]  George W. Mackey,et al.  Unitary group representations - in physis, probability, and number theory , 1989, Advanced book classics.

[16]  Indecomposability of treed equivalence relations , 1988 .

[17]  V. Wagh.  A descriptive version of Ambrose’s representation theorem for flows , 1988 .

[18]  Theodore A. Slaman,et al.  Definable functions on degrees , 1988 .

[19]  S. Ulam,et al.  Mathematical problems and games , 1987 .

[20]  D. Sullivan,et al.  Generic dynamics and monotone complete C*-algebras , 1986 .

[21]  D. Newton.  AN INTRODUCTION TO ERGODIC THEORY (Graduate Texts in Mathematics, 79) , 1982 .

[22]  Benjamin Weiss,et al.  An amenable equivalence relation is generated by a single transformation , 1981, Ergodic Theory and Dynamical Systems.

[23]  Lee A. Rubel,et al.  First-Order Conformal Invariants , 1980 .

[24]  John P. Burgess.  A selection theorem for group actions. , 1979 .

[25]  J. Feldman,et al.  Orbit structure and countable sections for actions of continuous groups , 1978 .

[26]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[27]  Calvin C. Moore,et al.  Ergodic equivalence relations, cohomology, and von Neumann algebras. II , 1977 .

[28]  J. Feldman,et al.  Ergodic equivalence relations, cohomology, and von Neumann algebras , 1975 .

[29]  Edwin Hewitt,et al.  Real And Abstract Analysis , 1967 .

[30]  W. Ambrose.  Representation of Ergodic Flows , 1941 .