Classification and Orbit Equivalence Relations

Classification and Orbit Equivalence Relations
Author: Greg Hjorth
Publisher: American Mathematical Soc.
Total Pages: 217
Release: 2000
Genre: Mathematics
ISBN: 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.

Borel Equivalence Relations

Borel Equivalence Relations
Author: Vladimir Grigorʹevich Kanoveĭ
Publisher: American Mathematical Soc.
Total Pages: 254
Release: 2008
Genre: Mathematics
ISBN: 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.

Crossed Products of C*-Algebras, Topological Dynamics, and Classification

Crossed Products of C*-Algebras, Topological Dynamics, and Classification
Author: Thierry Giordano
Publisher: Springer
Total Pages: 498
Release: 2018-08-28
Genre: Mathematics
ISBN: 3319708694

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This book collects the notes of the lectures given at an Advanced Course on Dynamical Systems at the Centre de Recerca Matemàtica (CRM) in Barcelona. The notes consist of four series of lectures. The first one, given by Andrew Toms, presents the basic properties of the Cuntz semigroup and its role in the classification program of simple, nuclear, separable C*-algebras. The second series of lectures, delivered by N. Christopher Phillips, serves as an introduction to group actions on C*-algebras and their crossed products, with emphasis on the simple case and when the crossed products are classifiable. The third one, given by David Kerr, treats various developments related to measure-theoretic and topological aspects of crossed products, focusing on internal and external approximation concepts, both for groups and C*-algebras. Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.

Restricted Orbit Equivalence

Restricted Orbit Equivalence
Author: Daniel J. Rudolph
Publisher: American Mathematical Soc.
Total Pages: 159
Release: 1985
Genre: Mathematics
ISBN: 0821823248

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These notes develop a theory of restricted orbit equivalence which has as particular examples Ornstein's isomorphism theorem for Bernoulli processes, Dye's orbit equivalence theorem for ergodic processes and the theory of Kakutani equivalence developed by Feldman, Ornstein, Weiss and Katok. A number of results from the Bernoulli theory are shown to be true for any restricted orbit equivalence.

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.
Total Pages: 126
Release: 2005
Genre: Mathematics
ISBN: 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.

Global Aspects of Ergodic Group Actions

Global Aspects of Ergodic Group Actions
Author: A. S. Kechris
Publisher: American Mathematical Soc.
Total Pages: 258
Release: 2010
Genre: Mathematics
ISBN: 0821848941

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A study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces.

Topics in Orbit Equivalence

Topics in Orbit Equivalence
Author: Alexander Kechris
Publisher: Springer
Total Pages: 144
Release: 2004-09-08
Genre: Mathematics
ISBN: 3540445080

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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.

Sets and Proofs

Sets and Proofs
Author: S. Barry Cooper
Publisher: Cambridge University Press
Total Pages: 450
Release: 1999-06-17
Genre: Mathematics
ISBN: 9780521635493

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First of two volumes providing a comprehensive guide to mathematical logic.

Mathematical Approach To Multilevel, Multiscale Health Interventions, A: Pharmaceutical Industry Decline And Policy Response

Mathematical Approach To Multilevel, Multiscale Health Interventions, A: Pharmaceutical Industry Decline And Policy Response
Author: Rodrick Wallace
Publisher: World Scientific
Total Pages: 238
Release: 2013-01-28
Genre: Mathematics
ISBN: 1848169981

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This book, based on published studies, takes a unique perspective on the 30-year collapse of pharmaceutical industry productivity in the search for small molecule “magic bullet” interventions. The relentless escalation of inflation-adjusted cost per approved medicine in the United States — from $200 million in 1950 to $1.2 billion in 2010 — has driven industry giants to, at best, slavish imitation in drug design, and at worst, abandonment of research and embracing of widespread fraud in consumer marketing.The book adapts formalism across a number of disciplines to the strategy for design of mutilevel interventions, focusing first on molecular, cellular, and larger scale examples, and then extending the argument to the simplifications provided by the dominant role of social and cultural structures and processes in individual and population patterns of health and illness.In place of “magic bullets”, we must now apply “magic strategies” that act across both the scale and level of organization. This book provides an introductory roadmap to the new tools that will be needed for the design of such strategies./a