Borel Equivalence Relations
Borel Equivalence Relations

Author: Vladimir Grigorʹevich Kanoveĭ

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 254

ISBN-13: 0821844539

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"Over the last 20 years, the theory of Borel equivalence relations and related topics have been very active areas of research in set theory and have important interactions with other fields of mathematics, like ergodic theory and topological dynamics, group theory, combinatorics, functional analysis, and model theory. The book presents, for the first time in mathematical literature, all major aspects of this theory and its applications."--BOOK JACKET.

Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations

Author: Greg Hjorth

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 126

ISBN-13: 0821837710

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Contributes to the theory of Borel equivalence relations, considered up to Borel reducibility, and measures preserving group actions considered up to orbit equivalence. This title catalogs the actions of products of the free group and obtains additional rigidity theorems and relative ergodicity results in this context.

Classification and Orbit Equivalence Relations
Classification and Orbit Equivalence Relations

Author: Greg Hjorth

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 217

ISBN-13: 0821820028

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Actions of Polish groups are ubiquitous in mathematics. In certain branches of ergodic theory and functional analysis, one finds a systematic study of the group of measure-preserving transformations and the unitary group. In logic, the analysis of countable models intertwines with results concerning the actions of the infinite symmetric group. This text develops the theory of Polish group actions entirely from scratch, ultimately presenting a coherent theory of the resulting orbit equivalence classes that may allow complete classification by invariants of an indicated form. The book concludes with a criterion for an orbit equivalence relation classifiable by countable structures considered up to isomorphism. This self-contained volume offers a complete treatment of this active area of current research and develops a difficult general theory classifying a class of mathematical objects up to some relevant notion of isomorphism or equivalence.

Topics in Orbit Equivalence
Topics in Orbit Equivalence

Author: Alexander S. Kechris

Publisher: Springer Science & Business Media

Published: 2004-08-26

Total Pages: 148

ISBN-13: 9783540226031

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This volume provides a self-contained introduction to some topics in orbit equivalence theory, a branch of ergodic theory. The first two chapters focus on hyperfiniteness and amenability. Included here are proofs of Dye's theorem that probability measure-preserving, ergodic actions of the integers are orbit equivalent and of the theorem of Connes-Feldman-Weiss identifying amenability and hyperfiniteness for non-singular equivalence relations. The presentation here is often influenced by descriptive set theory, and Borel and generic analogs of various results are discussed. The final chapter is a detailed account of Gaboriau's recent results on the theory of costs for equivalence relations and groups and its applications to proving rigidity theorems for actions of free groups.

Invariant Descriptive Set Theory
Invariant Descriptive Set Theory

Author: Su Gao

Publisher: CRC Press

Published: 2008-09-03

Total Pages: 392

ISBN-13: 9781584887942

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Presents Results from a Very Active Area of ResearchExploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathem

Descriptive Set Theory and Dynamical Systems
Descriptive Set Theory and Dynamical Systems

Author: M. Foreman

Publisher: Cambridge University Press

Published: 2000-05-25

Total Pages: 304

ISBN-13: 9780521786447

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In recent years there has been a growing interest in the interactions between descriptive set theory and various aspects of the theory of dynamical systems, including ergodic theory and topological dynamics. This volume, first published in 2000, contains a collection of survey papers by leading researchers covering a wide variety of recent developments in these subjects and their interconnections. This book provides researchers and graduate students interested in either of these areas with a guide to work done in the other, as well as with an introduction to problems and research directions arising from their interconnections.

Appalachian Set Theory
Appalachian Set Theory

Author: James Cummings

Publisher: Cambridge University Press

Published: 2012-11-15

Total Pages: 433

ISBN-13: 1107608503

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Papers based on a series of workshops where prominent researchers present exciting developments in set theory to a broad audience.

Classical and New Paradigms of Computation and their Complexity Hierarchies
Classical and New Paradigms of Computation and their Complexity Hierarchies

Author: Benedikt Löwe

Publisher: Springer Science & Business Media

Published: 2007-11-04

Total Pages: 266

ISBN-13: 1402027761

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The notion of complexity is an important contribution of logic to theoretical computer science and mathematics. This volume attempts to approach complexity in a holistic way, investigating mathematical properties of complexity hierarchies at the same time as discussing algorithms and computational properties. A main focus of the volume is on some of the new paradigms of computation, among them Quantum Computing and Infinitary Computation. The papers in the volume are tied together by an introductory article describing abstract properties of complexity hierarchies. This volume will be of great interest to both mathematical logicians and theoretical computer scientists, providing them with new insights into the various views of complexity and thus shedding new light on their own research.