Classification And Orbit Equivalence Relations

Author : Greg Hjorth
Publisher : American Mathematical Soc.
Page : 217 pages
File Size : 44,9 Mb
Release : 2000
Category : Mathematics
ISBN : 9780821820025

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Classification and Orbit Equivalence Relations by Greg Hjorth Pdf

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.

Author : Thierry Giordano,David Kerr,N. Christopher Phillips,Andrew Toms
Publisher : Springer
Page : 498 pages
File Size : 54,6 Mb
Release : 2018-08-28
Category : Mathematics
ISBN : 9783319708690

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Crossed Products of C*-Algebras, Topological Dynamics, and Classification by Thierry Giordano,David Kerr,N. Christopher Phillips,Andrew Toms Pdf

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.

Author : Vladimir Grigorʹevich Kanoveĭ
Publisher : American Mathematical Soc.
Page : 254 pages
File Size : 41,8 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821844533

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Borel Equivalence Relations by Vladimir Grigorʹevich Kanoveĭ Pdf

"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.

Author : Greg Hjorth,A. S. Kechris
Publisher : American Mathematical Soc.
Page : 109 pages
File Size : 46,5 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821837719

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Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations by Greg Hjorth,A. S. Kechris Pdf

This memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here $E$ is said to be Borel reducible to $F$ if there is a Borel function $f$ with $x E y$ if and only if $f(x) F f(y)$. Moreover, $E$ is orbit equivalent to $F$ if the respective measure spaces equipped with the extra structure provided by the equivalence relations are almost everywhere isomorphic. We consider product groups acting ergodically and by measure preserving transformations on standard Borel probability spaces.In general terms, the basic parts of the monograph show that if the groups involved have a suitable notion of 'boundary' (we make this precise with the definition of near hyperbolic), then one orbit equivalence relation can only be Borel reduced to another if there is some kind of algebraic resemblance between the product groups and coupling of the action. This also has consequence for orbit equivalence. In the case that the original equivalence relations do not have non-trivial almost invariant sets, the techniques lead to relative ergodicity results. An equivalence relation $E$ is said to be relatively ergodic to $F$ if any $f$ with $xEy \Rightarrow f(x) F f(y)$ has $[f(x)]_F$ constant almost everywhere.This underlying collection of lemmas and structural theorems is employed in a number of different ways. In the later parts of the paper, we give applications of the theory to specific cases of product groups. In particular, we catalog the actions of products of the free group and obtain additional rigidity theorems and relative ergodicity results in this context. There is a rather long series of appendices, whose primary goal is to give the reader a comprehensive account of the basic techniques. But included here are also some new results. For instance, we show that the Furstenberg-Zimmer lemma on cocycles from amenable groups fails with respect to Baire category, and use this to answer a question of Weiss. We also present a different proof that $F_2$ has the Haagerup approximation property.

Author : Su Gao,A. S. Kechris
Publisher : American Mathematical Soc.
Page : 78 pages
File Size : 53,7 Mb
Release : 2003
Category : Metric spaces
ISBN : 9780821831908

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On the Classification of Polish Metric Spaces Up to Isometry by Su Gao,A. S. Kechris Pdf

Author : James Cummings,Ernest Schimmerling
Publisher : Cambridge University Press
Page : 433 pages
File Size : 42,7 Mb
Release : 2012-11-15
Category : Mathematics
ISBN : 9781107608504

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Appalachian Set Theory by James Cummings,Ernest Schimmerling Pdf

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

Author : Su Gao
Publisher : CRC Press
Page : 392 pages
File Size : 40,7 Mb
Release : 2008-09-03
Category : Mathematics
ISBN : 158488794X

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Invariant Descriptive Set Theory by Su Gao Pdf

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

Author : Peter Gärdenfors,Jan Wolenski,K. Kijania-Placek
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 42,5 Mb
Release : 2002-12-31
Category : Computers
ISBN : 1402009291

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In the Scope of Logic, Methodology and Philosophy of Science by Peter Gärdenfors,Jan Wolenski,K. Kijania-Placek Pdf

This is the first of two volumes containing papers submitted by the invited speakers to the 11th International Congress of Logic, Methodology and Philosophy of Science, held in Cracow in 1999, under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science. The invited speakers are the leading researchers and accordingly the book presents the current state of the intellectual discourse in the respective fields. The papers delivered at the congress were divided into 17 sections. Thus the structure of the volume corresponds to the very schedule of the congress. Volume one contains the opening lecture by Andrzej K. Wróblewski as well as invited papers in sections of Proof Theory, Model Theory, Recursion Theory, Axiomatic Set Theory, Logic and Computation, Logic, Language and Cognition, Methodology, Probability, Induction, and Decision Theory, Philosophy of Logic, Mathematics, and Computer Science, and Philosophy of the Physical Sciences.

Author : M. Husek,J. van Mill
Publisher : Elsevier
Page : 652 pages
File Size : 43,7 Mb
Release : 2002-11-13
Category : Mathematics
ISBN : 9780444509802

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Recent Progress in General Topology II by M. Husek,J. van Mill Pdf

The book presents surveys describing recent developments in most of the primary subfields of General Topology and its applications to Algebra and Analysis during the last decade. It follows freely the previous edition (North Holland, 1992), Open Problems in Topology (North Holland, 1990) and Handbook of Set-Theoretic Topology (North Holland, 1984). The book was prepared in connection with the Prague Topological Symposium, held in 2001. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs slightly from those chosen in 1992. The following areas experienced significant developments: Topological Groups, Function Spaces, Dimension Theory, Hyperspaces, Selections, Geometric Topology (including Infinite-Dimensional Topology and the Geometry of Banach Spaces). Of course, not every important topic could be included in this book. Except surveys, the book contains several historical essays written by such eminent topologists as: R.D. Anderson, W.W. Comfort, M. Henriksen, S. Mardeŝić, J. Nagata, M.E. Rudin, J.M. Smirnov (several reminiscences of L. Vietoris are added). In addition to extensive author and subject indexes, a list of all problems and questions posed in this book are added. List of all authors of surveys: A. Arhangel'skii, J. Baker and K. Kunen, H. Bennett and D. Lutzer, J. Dijkstra and J. van Mill, A. Dow, E. Glasner, G. Godefroy, G. Gruenhage, N. Hindman and D. Strauss, L. Hola and J. Pelant, K. Kawamura, H.-P. Kuenzi, W. Marciszewski, K. Martin and M. Mislove and M. Reed, R. Pol and H. Torunczyk, D. Repovs and P. Semenov, D. Shakhmatov, S. Solecki, M. Tkachenko.

Author : Matthew Foreman,Akihiro Kanamori
Publisher : Springer Science & Business Media
Page : 2200 pages
File Size : 47,7 Mb
Release : 2009-12-10
Category : Mathematics
ISBN : 9781402057649

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Handbook of Set Theory by Matthew Foreman,Akihiro Kanamori Pdf

Numbers imitate space, which is of such a di?erent nature —Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.

Author : Chitat Chong,Qi Feng,Theodore A Slaman,W Hugh Woodin,Yue Yang
Publisher : World Scientific
Page : 228 pages
File Size : 46,8 Mb
Release : 2014-05-28
Category : Mathematics
ISBN : 9789814602655

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E-Recursion, Forcing and C*-Algebras by Chitat Chong,Qi Feng,Theodore A Slaman,W Hugh Woodin,Yue Yang Pdf

This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians. Contents:Selected Applications of Logic to Classification Problem for C*-Algebras (Ilijas Farah)Subcomplete Forcing and L-Forcing (Ronald Jensen)E-Recursion (Gerald E Sacks) Readership: Mathematics graduate students, researchers in logic, set theory and related areas. Key Features:These are notes based on short courses given by three leading experts in set theory, recursion theory and their applicationsKeywords:Logic;Set Theory;Forcing;E-recursion;C*-Algebra;Recursion Theory;Computability Theory

Author : Alessandro Andretta,Luca Motto Ros
Publisher : American Mathematical Society
Page : 189 pages
File Size : 44,7 Mb
Release : 2022-05-24
Category : Mathematics
ISBN : 9781470452735

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Souslin Quasi-Orders and Bi-Embeddability of Uncountable Structures by Alessandro Andretta,Luca Motto Ros Pdf

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Author : Cesar E. Silva,Alexandre I. Danilenko
Publisher : Springer Nature
Page : 707 pages
File Size : 55,7 Mb
Release : 2023-07-31
Category : Mathematics
ISBN : 9781071623886

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Ergodic Theory by Cesar E. Silva,Alexandre I. Danilenko Pdf

This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, covers recent developments in classical areas of ergodic theory, including the asymptotic properties of measurable dynamical systems, spectral theory, entropy, ergodic theorems, joinings, isomorphism theory, recurrence, nonsingular systems. It enlightens connections of ergodic theory with symbolic dynamics, topological dynamics, smooth dynamics, combinatorics, number theory, pressure and equilibrium states, fractal geometry, chaos. In addition, the new edition includes dynamical systems of probabilistic origin, ergodic aspects of Sarnak's conjecture, translation flows on translation surfaces, complexity and classification of measurable systems, operator approach to asymptotic properties, interplay with operator algebras

Author : Rodrick Wallace,Deborah Wallace,Robert G. Wallace
Publisher : Springer Science & Business Media
Page : 220 pages
File Size : 47,8 Mb
Release : 2009-06-12
Category : Computers
ISBN : 9780387922133

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Farming Human Pathogens by Rodrick Wallace,Deborah Wallace,Robert G. Wallace Pdf

Farming Human Pathogens: Ecological Resilience and Evolutionary Process introduces a cutting-edge mathematical formalism based on the asymptotic limit theorems of information theory to describe how punctuated shifts in mesoscale ecosystems can entrain patterns of gene expression and organismal evolution. The authors apply the new formalism toward characterizing a number of infectious diseases that have evolved in response to the world as humans have made it. Many of the human pathogens that are emerging out from underneath epidemiological control are 'farmed' in the metaphorical sense, as the evolution of drug-resistant HIV makes clear, but also quite literally, as demonstrated by avian influenza's emergence from poultry farms in southern China. The most successful pathogens appear able to integrate selection pressures humans have imposed upon them from a variety of socioecological scales. The book also presents a related treatment of Eigen's Paradox and the RNA 'error catastrophe' that bedevils models of the origins of viruses and of biological life itself.

Author : Daniel Parrochia,Pierre Neuville
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 44,8 Mb
Release : 2013-05-14
Category : Mathematics
ISBN : 9783034806091

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Towards a General Theory of Classifications by Daniel Parrochia,Pierre Neuville Pdf

This book is an essay on the epistemology of classifications. Its main purpose is not to provide an exposition of an actual mathematical theory of classifications, that is, a general theory which would be available to any kind of them: hierarchical or non-hierarchical, ordinary or fuzzy, overlapping or non-overlapping, finite or infinite, and so on, establishing a basis for all possible divisions of the real world. For the moment, such a theory remains nothing but a dream. Instead, the authors essentially put forward a number of key questions. Their aim is rather to reveal the “state of art” of this dynamic field and the philosophy one may eventually adopt to go further. To this end they present some advances made in the course of the last century, discuss a few tricky problems that remain to be solved, and show the avenues open to those who no longer wish to stay on the wrong track. Researchers and professionals interested in the epistemology and philosophy of science, library science, logic and set theory, order theory or cluster analysis will find this book a comprehensive, original and progressive introduction to the main questions in this field.