Download or read book The Theory of Countable Borel Equivalence Relations written by Alexander S. Kechris and published by Cambridge University Press. This book was released on 2024-11-30 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of definable equivalence relations has been a vibrant area of research in descriptive set theory for the past three decades. It serves as a foundation of a theory of complexity of classification problems in mathematics and is further motivated by the study of group actions in a descriptive, topological, or measure-theoretic context. A key part of this theory is concerned with the structure of countable Borel equivalence relations. These are exactly the equivalence relations generated by Borel actions of countable discrete groups and this introduces important connections with group theory, dynamical systems, and operator algebras. This text surveys the state of the art in the theory of countable Borel equivalence relations and delineates its future directions and challenges. It gives beginning graduate students and researchers a bird's-eye view of the subject, with detailed references to the extensive literature provided for further study.
Download or read book Recursion Theory and Countable Borel Equivalence Relations written by Andrew Marks and published by . This book was released on 2012 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: We investigate the problem of what equivalence relations from recursion theory are universal countable Borel equivalence relations. While this question is interesting in its own right, it has also been a particularly rich source of connections between recursion theory, countable Borel equivalence relations, and Borel combinatorics. Tools developed by this investigation have proved very applicable to other problems in these fields. In Chapter 2, we prove a model universality theorem, and introduce several themes of the thesis. A corollary of this first theorem is that polynomial time Turing equivalence is a universal countable Borel equivalence relation. Slaman and Steel have shown that arithmetic equivalence is a universal countable Borel equivalence relation. In Chapter 3, we combine this fact with the existence of a cone measure for arithmetic equivalence to prove several structural results about universal countable Borel equivalence relations in general. We show that universality for Borel reductions coincides with universality for Borel embeddings, and a universal countable Borel equivalence relation is always universal on some nullset with respect to any Borel probability measure. We also settle questions of Thomas, and Jackson, Kechris, and Louveau by showing that a smooth disjoint union of non-universal countable Borel equivalence relations is non-universal. This result can be significantly strengthened by assuming a conjecture of Martin which states that every Turing invariant function is equivalent to a uniformly Turing invariant function on a Turing cone. In Chapter 4, we investigate uniformity of homomorphisms among equivalence relations from recursion theory. We pose several open questions in this context, and investigate the implications of the uniformity that they imply. We introduce the concept of a Borel metric on a countable Borel equivalence relation, and show that this concept is closely connected to a weakening of the notion of a uniform homomorphism. Using this language of metrics and the machinery of Slaman and Steel for proving the universality of arithmetic equivalence, we construct an example of a homomorphism between equivalence relations coarser than Turing equivalence which is not uniform on any pointed perfect set. This is the first example of a nonuniform homomorphism in this sort of recursion-theoretic context, and it places some limits on how abstract a proof of Martin's conjecture could be. In Chapter 5, we turn to the question of whether recursive isomorphism is a universal countable Borel equivalence relation. Improving prior results of Dougherty and Kechris and Andretta, Camerlo, and Hjorth, we show that recursive isomorphism on $3\̂omega$ is a universal countable Borel equivalence relation. We isolate a question of Borel combinatorics for which a positive answer would imply that recursive isomorphism on $2\̂omega$ is universal. We show that this question is equivalent to the problem of whether $\omega$ many 2-regular Borel graphs on the same space can be simultaneously Borel 3-colored so that there are no monochromatic points. We then show that this question has an affirmative answer if and only if many-one equivalence on $2\̂omega$ is a uniformly universal countable Borel equivalence relation. Thus, we have an exact combinatorial calibration of the difficulty of this universality problem. In Chapter 6, we consider the question of whether there exist disjoint Borel complete sections for every pair of aperiodic countable Borel equivalence relations. We show that this question is very robust, and has many equivalent formulations. A positive answer to this question would positively answer the combinatorial question of the previous paragraph, while a negative answer would settle several open questions of Borel combinatorics. We also show that this question is true in both the measure and category context, in all its equivalent forms. One application of this fact is that every Borel bipartite 3-regular graph has measurable and Baire measurable edge colorings with 4 colors. This is a descriptive analogue of a special case of Vizing's theorem on edge colorings from classical combinatorics. Finally, we see that recursive isomorphism on $2\̂omega$ is measure universal. Thus, purely measure-theoretic tools cannot be used to prove that it is not universal.
Download or read book Borel Equivalence Relations written by Vladimir Grigorʹevich Kanoveĭ and published by American Mathematical Soc.. This book was released on 2008 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.
Download or read book Classification and Orbit Equivalence Relations written by Greg Hjorth and published by American Mathematical Soc.. This book was released on 2000 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations written by Greg Hjorth and published by American Mathematical Soc.. This book was released on 2005 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Topics in Orbit Equivalence written by Alexander S. Kechris and published by Springer Science & Business Media. This book was released on 2004-08-26 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Countable Borel Quasi orders written by Jay Williams and published by . This book was released on 2012 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal in cite{LR05} for the study of analytic equivalence relations, we study countable Borel quasi-orders. We are largely concerned in this thesis with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We first establish that there is a universal countable Borel quasi-order, using a Feldman-Moore-type result for countable Borel quasi-orders and an argument similar to that of Dougherty, Jackson, and Kechris in cite{DJK94}. We then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees. This is used in many of the other proofs of universality. Our main result is Theorem 5.5.2, which states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal in cite{LR05}. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. Although it may have been possible to prove this only using results on countable Borel equivalence relations, the use of quasi-orders seems to be the most direct route to this result. The proof uses small cancellation theory. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order. Finally, we discuss the structure of countable Borel quasi-orders under Borel reducibility, and we present some open problems.
Download or read book Invariant Descriptive Set Theory written by Su Gao and published by CRC Press. This book was released on 2008-09-03 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book Full Groups Classification and Equivalence Relations written by Benjamin David Miller and published by . This book was released on 2004 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Global Aspects of Ergodic Group Actions written by A. S. Kechris and published by American Mathematical Soc.. This book was released on 2010 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Descriptive Set Theory Equivalence Relations and Classification Problems in Analysis written by John Daniel Clemens and published by . This book was released on 2001 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Linear Algebraic Groups written by Armand Borel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This revised, enlarged edition of Linear Algebraic Groups (1969) starts by presenting foundational material on algebraic groups, Lie algebras, transformation spaces, and quotient spaces. It then turns to solvable groups, general properties of linear algebraic groups, and Chevally’s structure theory of reductive groups over algebraically closed groundfields. It closes with a focus on rationality questions over non-algebraically closed fields.
Download or read book Ordinal Definability and Recursion Theory written by Alexander S. Kechris and published by Cambridge University Press. This book was released on 2016-01-11 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: The third in a series of four books presenting the seminal papers from the Caltech-UCLA 'Cabal Seminar'.
Download or read book Appalachian Set Theory written by James Cummings and published by Cambridge University Press. This book was released on 2012-11-15 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume takes its name from a popular series of intensive mathematics workshops hosted at institutions in Appalachia and surrounding areas. At these meetings, internationally prominent set theorists give one-day lectures that focus on important new directions, methods, tools and results so that non-experts can begin to master these and incorporate them into their own research. Each chapter in this volume was written by the workshop leaders in collaboration with select student participants, and together they represent most of the meetings from the period 2006–2012. Topics covered include forcing and large cardinals, descriptive set theory, and applications of set theoretic ideas in group theory and analysis, making this volume essential reading for a wide range of researchers and graduate students.
Download or read book Classical Descriptive Set Theory written by Alexander Kechris and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text presents a largely balanced approach to the subject, which combines many elements of the different traditions. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.
Download or read book On the Computational Content of the Theory of Borel Equivalence Relations written by Nikolay Bazhenov and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Ergodic Theory written by Cesar E. Silva and published by Springer Nature. This book was released on 2023-07-31 with total page 707 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
