Alexander Lubotzky's Books

Varieties of Representations of Finitely Generated Groups

By: Alexander Lubotzky,Andy R. Magid,Andy Roy Magid

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Varieties of Representations of Finitely Generated Groups

By: Alexander Lubotzky,Andy R. Magid,Andy Roy Magid

The n-dimensional representations, over an algebraically closed characteristic zero field k, of a finitely generated group are parameterized by an affine algebraic variety over k. The tangent spaces of this variety are subspaces of spaces of one-cocycles and thus the geometry of the variety is locally related to the cohomology of the group. The cohomology is also related to the prounipotent radical of the proalgebraic hull of the group. This paper exploits these two relations to compute dimensions of representation varieties, especially for nilpotent groups and their generalizations. It also presents the foundations of the theory of representation varieties in an expository, self-contained manner.

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Page Count

134

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082182337X

ISBN 13

9780821823378

Subgroup Growth

By: Alexander Lubotzky,Dan Segal

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Subgroup Growth

By: Alexander Lubotzky,Dan Segal

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2001. Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index. In the last two decades this topic has developed into one of the most active areas of research in infinite group theory; this book is a systematic and comprehensive account of the substantial theory which has emerged. As well as determining the range of possible 'growth types', for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure. A wide range of mathematical disciplines play a significant role in this work: as well as various aspects of infinite group theory, these include finite simple groups and permutation groups, profinite groups, arithmetic groups and Strong Approximation, algebraic and analytic number theory, probability, and p-adic model theory. Relevant aspects of such topics are explained in self-contained 'windows'.

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Page Count

463

Publisher

Birkhäuser

Published Date

ISBN 10

3034889658

ISBN 13

9783034889650

Tree Lattices

By: Hyman Bass,Alexander Lubotzky

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Tree Lattices

By: Hyman Bass,Alexander Lubotzky

This monograph extends this approach to the more general investigation of X-lattices, and these "tree lattices" are the main object of study. The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems. Tree Lattices should be a helpful resource to researchers in the field, and may also be used for a graduate course on geometric methods in group theory.

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Page Count

239

Publisher

Springer Science & Business Media

Published Date

ISBN 10

146122098X

ISBN 13

9781461220985

Complexity and Randomness in Group Theory

GAGTA BOOK 1

By: Frédérique Bassino,Ilya Kapovich,Markus Lohrey,Alexei Miasnikov,Cyril Nicaud,Andrey Nikolaev,Igor Rivin,Vladimir Shpilrain,Alexander Ushakov,Pascal Weil

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Complexity and Randomness in Group Theory

GAGTA BOOK 1

By: Frédérique Bassino,Ilya Kapovich,Markus Lohrey,Alexei Miasnikov,Cyril Nicaud,Andrey Nikolaev,Igor Rivin,Vladimir Shpilrain,Alexander Ushakov,Pascal Weil

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Page Count

386

Publisher

Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110667029

ISBN 13

9783110667028

Cooperative Task-Oriented Computing

Algorithms and Complexity

By: Chryssis Georgiou,Alexander Shvartsman

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Cooperative Task-Oriented Computing

Algorithms and Complexity

By: Chryssis Georgiou,Alexander Shvartsman

Cooperative network supercomputing is becoming increasingly popular for harnessing the power of the global Internet computing platform. A typical Internet supercomputer consists of a master computer or server and a large number of computers called workers, performing computation on behalf of the master. Despite the simplicity and benefits of a single master approach, as the scale of such computing environments grows, it becomes unrealistic to assume the existence of the infallible master that is able to coordinate the activities of multitudes of workers. Large-scale distributed systems are inherently dynamic and are subject to perturbations, such as failures of computers and network links, thus it is also necessary to consider fully distributed peer-to-peer solutions. We present a study of cooperative computing with the focus on modeling distributed computing settings, algorithmic techniques enabling one to combine efficiency and fault-tolerance in distributed systems, and the exposition of trade-offs between efficiency and fault-tolerance for robust cooperative computing. The focus of the exposition is on the abstract problem, called Do-All, and formulated in terms of a system of cooperating processors that together need to perform a collection of tasks in the presence of adversity. Our presentation deals with models, algorithmic techniques, and analysis. Our goal is to present the most interesting approaches to algorithm design and analysis leading to many fundamental results in cooperative distributed computing. The algorithms selected for inclusion are among the most efficient that additionally serve as good pedagogical examples. Each chapter concludes with exercises and bibliographic notes that include a wealth of references to related work and relevant advanced results. Table of Contents: Introduction / Distributed Cooperation and Adversity / Paradigms and Techniques / Shared-Memory Algorithms / Message-Passing Algorithms / The Do-All Problem in Other Settings / Bibliography / Authors' Biographies

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Page Count

155

Publisher

Springer Nature

Published Date

ISBN 10

3031020057

ISBN 13

9783031020056

The Classical Groups and K-Theory

By: Alexander J. Hahn,O.Timothy O'Meara

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The Classical Groups and K-Theory

By: Alexander J. Hahn,O.Timothy O'Meara

It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).

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Page Count

589

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3662131528

ISBN 13

9783662131527

Introduction to Analysis on Graphs

By: Alexander Grigor’yan

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Introduction to Analysis on Graphs

By: Alexander Grigor’yan

A central object of this book is the discrete Laplace operator on finite and infinite graphs. The eigenvalues of the discrete Laplace operator have long been used in graph theory as a convenient tool for understanding the structure of complex graphs. They can also be used in order to estimate the rate of convergence to equilibrium of a random walk (Markov chain) on finite graphs. For infinite graphs, a study of the heat kernel allows to solve the type problem—a problem of deciding whether the random walk is recurrent or transient. This book starts with elementary properties of the eigenvalues on finite graphs, continues with their estimates and applications, and concludes with heat kernel estimates on infinite graphs and their application to the type problem. The book is suitable for beginners in the subject and accessible to undergraduate and graduate students with a background in linear algebra I and analysis I. It is based on a lecture course taught by the author and includes a wide variety of exercises. The book will help the reader to reach a level of understanding sufficient to start pursuing research in this exciting area.

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Page Count

160

Publisher

American Mathematical Soc.

Published Date

ISBN 10

147044397X

ISBN 13

9781470443979

Cohomology of Number Fields

By: Jürgen Neukirch,Alexander Schmidt,Kay Wingberg

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Cohomology of Number Fields

By: Jürgen Neukirch,Alexander Schmidt,Kay Wingberg

This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

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Page Count

831

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3540378898

ISBN 13

9783540378891

Transformation Groups And Invariant Measures: Set-theoretical Aspects

By: Alexander B Kharazishvili

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Transformation Groups And Invariant Measures: Set-theoretical Aspects

By: Alexander B Kharazishvili

This book is devoted to some topics of the general theory of invariant and quasi-invariant measures. Such measures are usually defined on various σ-algebras of subsets of spaces equipped with transformation groups, and there are close relationships between purely algebraic properties of these groups and the corresponding properties of invariant (quasi-invariant) measures. The main goal of the book is to investigate several aspects of those relationships (primarily from the set-theoretical point of view). Also of interest are the properties of some natural classes of sets, important from the viewpoint of the theory of invariant (quasi-invariant) measures.

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Page Count

270

Publisher

World Scientific

Published Date

ISBN 10

9814518220

ISBN 13

9789814518222

Nonmeasurable Sets and Functions

By: Alexander Kharazishvili

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Nonmeasurable Sets and Functions

By: Alexander Kharazishvili

The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense. This theorem stimulated the development of the following interesting topics in mathematics:1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces;2. The theory of non-real-valued-measurable cardinals;3. The theory of invariant (quasi-invariant)extensions of invariant (quasi-invariant) measures.These topics are under consideration in the book. The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated . Among them there are: Vitali sets, Bernstein sets, Sierpinski sets, nontrivial solutions of the Cauchy functional equation, absolutely nonmeasurable sets in uncountable groups, absolutely nonmeasurable additive functions, thick uniform subsets of the plane, small nonmeasurable sets, absolutely negligible sets, etc. The importance of properties of nonmeasurable sets for various aspects of the measure extension problem is shown. It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups. Many open attractive problems are formulated concerning nonmeasurable sets and functions.· highlights the importance of nonmeasurable sets (functions) for general measure extension problem.· Deep connections of the topic with set theory, real analysis, infinite combinatorics, group theory and geometry of Euclidean spaces shown and underlined.· self-contained and accessible for a wide audience of potential readers.· Each chapter ends with exercises which provide valuable additional information about nonmeasurable sets and functions.· Numerous open problems and questions.

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Page Count

350

Publisher

Elsevier

Published Date

ISBN 10

0080479766

ISBN 13

9780080479767

The Mathematical Coloring Book

Mathematics of Coloring and the Colorful Life of its Creators

By: Alexander Soifer

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The Mathematical Coloring Book

Mathematics of Coloring and the Colorful Life of its Creators

By: Alexander Soifer

This book provides an exciting history of the discovery of Ramsey Theory, and contains new research along with rare photographs of the mathematicians who developed this theory, including Paul Erdös, B.L. van der Waerden, and Henry Baudet.

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Page Count

619

Publisher

Springer Science & Business Media

Published Date

ISBN 10

0387746420

ISBN 13

9780387746425

Codes and Association Schemes

DIMACS Workshop Codes and Association Schemes, November 9-12, 1999, DIMACS Center

By: Alexander Barg

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Codes and Association Schemes

DIMACS Workshop Codes and Association Schemes, November 9-12, 1999, DIMACS Center

By: Alexander Barg

This volume presents papers related to the DIMACS workshop, "Codes and Association Schemes". The articles are devoted to the following topics: applications of association schemes and of the polynomial method to properties of codes, structural results for codes, structural results for association schemes, and properties of orthogonal polynomials and their applications in combinatorics. Papers on coding theory are related to classical topics, such as perfect codes, bounds on codes, codes and combinatorial arrays, weight enumerators, and spherical designs. Papers on orthogonal polynomials provide new results on zeros and symptotic properties of standard families of polynomials encountered in coding theory. The theme of association schemes is represented by new classification results and new classes of schemes related to posets. This volume collects up-to-date applications of the theory of association schemes to coding and presents new properties of both polynomial and general association schemes. It offers a solid representation of results in problems in areas of current interest.

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Page Count

317

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821820745

ISBN 13

9780821820742

Subgroup Growth

By: Alexander Lubotzky,Dan Segal

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Subgroup Growth

By: Alexander Lubotzky,Dan Segal

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Page Count

463

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Birkhäuser

Published Date

ISBN 10

3034889658

ISBN 13

9783034889650

Varieties of Representations of Finitely Generated Groups

By: Alexander Lubotzky,Andy R. Magid,Andy Roy Magid

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Varieties of Representations of Finitely Generated Groups

By: Alexander Lubotzky,Andy R. Magid,Andy Roy Magid

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134

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082182337X

ISBN 13

9780821823378

Tree Lattices

By: Hyman Bass,Alexander Lubotzky

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Tree Lattices

By: Hyman Bass,Alexander Lubotzky

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Page Count

239

Publisher

Springer Science & Business Media

Published Date

ISBN 10

146122098X

ISBN 13

9781461220985

Book's by Alexander Lubotzky

Varieties of Representations of Finitely Generated Groups

Varieties of Representations of Finitely Generated Groups

Subgroup Growth

Subgroup Growth

Tree Lattices

Tree Lattices

Complexity and Randomness in Group Theory

Complexity and Randomness in Group Theory

Cooperative Task-Oriented Computing

Cooperative Task-Oriented Computing

The Classical Groups and K-Theory

The Classical Groups and K-Theory

Introduction to Analysis on Graphs

Introduction to Analysis on Graphs

Cohomology of Number Fields

Cohomology of Number Fields

Transformation Groups And Invariant Measures: Set-theoretical Aspects

Transformation Groups And Invariant Measures: Set-theoretical Aspects

Nonmeasurable Sets and Functions

Nonmeasurable Sets and Functions

The Mathematical Coloring Book

The Mathematical Coloring Book

Codes and Association Schemes

Codes and Association Schemes

Subgroup Growth

Subgroup Growth

Varieties of Representations of Finitely Generated Groups

Varieties of Representations of Finitely Generated Groups

Tree Lattices

Tree Lattices

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