Groups and group theory Books

Book SynopsisThisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].Trade ReviewFrom the reviews: “The book under review has as its main goal to give an introductory overview of the construction and main properties of all finite simple groups. … This book is the first one that attempts to give a systematic treatment of all finite simple groups, using the more recent and efficient constructions … . The author succeeds in making this important but difficult area of mathematics readily accessible to a large sector of the mathematical community, and for this we should be grateful.” (Felipe Zaldivar, The Mathematical Association of America, March, 2010)“One of the great achievements of mathematics was the classification of the finite simple groups … . the book brings much information to the classroom. It contains exactly those things one would like to know if one were to meet the individual simple groups for the first time. … perfectly suitable for an advanced course or seminar. … accessible also to those who are not great experts in group theory. Anyone interested in finite groups … should have this book on his or her bookshelf.” (Gernot Stroth, Mathematical Reviews, Issue 2011 e)“The author of this book has succeeded in giving an overview of all non-abelian finite simple groups which is accessible to non-experts. … For anyone who wants to get information on finite simple groups without having to tackle massive monographs this volume will be most welcome.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)“It is the first text at this level in which all the finite simple groups are treated together, pointing out their connections. … The text is very well organised. The introduction, which forms the first chapter, contains a brief history and the statement of the classification theorem, together with sections giving remarks on the applications and the proof of the theorem. … Consequently the book may also be useful to a reader who just wants an introduction to a particular group or family of groups.” (Peter Shiu, The Mathematical Gazette, Vol. 95 (532), March, 2011)“This book is a unique introductory overview of all the finite simple groups, and thus it is suitable not only for specialists who are interested in finite simple groups but also for advanced undergraduate and graduate students in algebra. The section entitled ‘Further reading’ at the end of each chapter is a nice guide to further study of the subjects.” (Hiromichi Yamada, Zentralblatt MATH, Vol. 1203, 2011)Table of ContentsThe alternating groups.- The classical groups.- The exceptional groups.- The sporadic groups.

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Book SynopsisThis book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group. The main components are: - a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism, - the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals, - algebraic representation theory in terms of category O, and - analytic representation theory of quantized complex semisimple groups. Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.Trade Review“The book is largely self-contained. … It is highly recommended for mathematicians of all levels wishing to learn about these topics, in the algebraic setting and/or in the analytic setting.” (Huafeng Zhang, zbMATH 1514.20006, 2023)Table of Contents- Introduction. - Multiplier Hopf Algebras. - Quantized Universal Enveloping Algebras. - Complex Semisimple Quantum Groups. - Category O. - Representation Theory of Complex Semisimple Quantum Groups.

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Book SynopsisThis book focuses on the Symmetric Informationally Complete quantum measurements (SICs) in dimensions 2 and 3, along with one set of SICs in dimension 8. These objects stand out in ways that have earned them the moniker of "sporadic SICs". By some standards, they are more approachable than the other known SICs, while by others they are simply atypical. The author forays into quantum information theory using them as examples, and the author explores their connections with other exceptional objects like the Leech lattice and integral octonions. The sporadic SICs take readers from the classification of finite simple groups to Bell's theorem and the discovery that "hidden variables" cannot explain away quantum uncertainty.While no one department teaches every subject to which the sporadic SICs pertain, the topic is approachable without too much background knowledge. The book includes exercises suitable for an elective at the graduate or advanced undergraduate level.Table of ContentsEquiangular Lines.- Sporadic SICs and the Exceptional Lie Algebras.- The Hoggar-type SICs.-SICs as Equicoherent Quantum States.- SICs and Bell Inequalities.

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Book SynopsisThis book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today.The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.Table of Contents- Foreword.- Preface.- Part I Algebraic Theory: 1. Free Groups.- 2. Nilpotent Groups.- 3. Residual Finiteness and the Zassenhaus Filtration.- 4. Solvable Groups.- 5. Polycyclic Groups.- 6. The Burnside Problem.- Part II Geometric Theory: 7. Finitely Generated Groups and Their Growth Functions.- 8. Hyperbolic Plane Geometry and the Tits Alternative.- 9. Topological Groups, Lie Groups, and Hilbert Fifth Problem.- 10. Dimension Theory.- 11. Ultrafilters, Ultraproducts, Ultrapowers, and Asymptotic Cones.- 12. Gromov’s Theorem.- Part III Analytic and Probabilistic Theory: 13. The Theorems of Polya and Varopoulos.- 14. Amenability, Isoperimetric Profile, and Følner Functions.- 15. Solutions or Hints to Selected Exercises.- References.- Subject Index.- Index of Authors.

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Book SynopsisThis textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods and ideas to the attention of the public.Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.Trade Review“This book … is a nice addition to the literature. … for instructors teaching geometry courses in which these are a topic, this book should provide an excellent source of interesting examples and problems. The large number of solved problems should also make useful reading for people preparing for mathematical contests and Olympiads.” (Mark Hunacek, MAA Reviews, October 4, 2022)Table of ContentsIntroduction.- Part I: Problems - 1. Isometries.- 2. Homotheties and Spiral Similarities.- 3. Inversions.- 4. A Synthesis.- Part II: Hints - 5. Isometries.- 6. Homotheties and Spiral Similarities.- 7. Inversions.- 8. A Synthesis.- Part III: Solutions - 9. Isometries.- 10. Homotheties and Spiral Similarities.- 11. Inversions.- 12. A Synthesis.

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Book SynopsisAn Invitation to Representation Theory offers an introduction to groups and their representations, suitable for undergraduates. In this book, the ubiquitous symmetric group and its natural action on polynomials are used as a gateway to representation theory.The subject of representation theory is one of the most connected in mathematics, with applications to group theory, geometry, number theory and combinatorics, as well as physics and chemistry. It can however be daunting for beginners and inaccessible to undergraduates. The symmetric group and its natural action on polynomial spaces provide a rich yet accessible model to study, serving as a prototype for other groups and their representations. This book uses this key example to motivate the subject, developing the notions of groups and group representations concurrently.With prerequisites limited to a solid grounding in linear algebra, this book can serve as a first introduction to representation theory at the undergraduate level, for instance in a topics class or a reading course. A substantial amount of content is presented in over 250 exercises with complete solutions, making it well-suited for guided study.Trade Review“The book under review is a nice introduction to the representation theory of the symmetric group. … The book is well structured and enriched with numerous exercises, many of which are solved or with hints for the solution.” (Enrico Jabara, zbMATH 1514.20002, 2023)Table of Contents Preface Introduction Chapter 1. First Steps Chapter 2. Polynomials, Subspaces, and Subrepresentations Chapter 3. Intertwining Maps, Complete Reducibility, and Invariant Inner Products Chapter 4. The Structure of the Symmetric Group Chapter 5. Sn Decomposition of Polynomial Spaces for n= 1,2,3. Chapter 6. The Group Algebra Chapter 7. The Irreducible Representations of Sn: Characters Chapter 8. The Irreducible Representations of Sn: Young Symmetrizers Chapter 9. Cosets, Restricted and Induced Representations Chapter 10. Direct Products of Groups, Young Subgroups and Permutation Modules Chapter 11. Specht Modules Chapter 12. Decomposition of Young Permutation Modules Chapter 13. Branching Relations Bibliography Index

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Book Synopsis1 Foundations.- 2 Permutation groups and linear groups.- 3 Nilpotent groups and p-groups.- 4 Transfer and p-nilpotent groups.- 5 Representation theory.- 6 Solvable groups.

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Book SynopsisAward-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'.Trade ReviewSubgroup Growth is an extremely well-written book and is a delight to read. It has a wealth of information making a rich and timely contribution to an emerging area in the theory of groups which has come to be known as Asymptotic Group Theory. This monograph and the challenging open problems with which it concludes are bound to play a fundamental role in the development of the subject for many years to come. —Journal Indian Inst of Science "[Subgroup growth] is one of the first books on Asymptotic Group Theory – a new, quickly developing direction in modern mathematics…The book of A. Lubotzky and D.Segal, leading specialists in group theory, answers…questions in a beautiful way .…It was natural to expect a text on the subject that would summarize the achievements in the field and we are very lucky to witness the appearance of this wonderful book. …Readers will be impressed with the encyclopedic scope of the text. It includes all, or almost all, topics related to subgroup growth ….The book also includes plenty of general information on topics that are well known to algebraic audiences and should be part of the background for every modern researcher in mathematics. …a wonderful methodological tool introduced by the authors. … The book ends [with] a section on open problems, which contains 35 problems related to subgroup growth. The list will be useful and interesting to both established mathematicians and young researchers. There is no doubt that the list includes the most important and illuminating problems in the area, and we eagerly anticipate solutions of at least some of them in the near future. The book will surely have [a] big impact on all readers interested in Group Theory, as well as in Algebra and Number Theory in general." —Bulletin of the AMS "The proofs in this book employ a remarkable variety of tools, from all branches of group theory, certainly, but also from number theory, logic, and analysis…. The authors supply surveys, and some proofs, of necessary results, in the "windows" at the end of the book. These comprise about one quarter of the full book, and they give the needy reader a handy reference, without interrupting the flow of argument in the main text…. Since the subject of this book is an active area of current research, there are many open problems in it…." —Mathematical ReviewsTable of Contents0 Introduction and Overview.- 0.1 Preliminary comments and definitions.- 0.2 Overview of the chapters.- 0.3 On CFSG.- 0.4 The windows.- 0.5 The ‘notes’.- 1 Basic Techniques of Subgroup Counting.- 1.1 Permutation representations.- 1.2 Quotients and subgroups.- 1.3 Group extensions.- 1.4 Nilpotent and soluble groups.- 1.5 Abelian groups I.- 1.6 Finite p-groups.- 1.7 Sylow’s theorem.- 1.8 Rest riet ing to soluble subgroups.- 1.9 Applications of the ‘minimal index’.- 1.10 Abelian groups II.- 1.11 Growth types.- Notes.- 2 Free Groups.- 2.1 The subgroup growth of free groups.- 2.2 Subnormal subgroups.- 2.3 Counting d-generator finite groups.- Notes.- 3 Groups with Exponential Subgroup Growth.- 3.1 Upper bounds.- 3.2 Lower bounds.- 3.3 Free pro-p groups.- 3.4 Normal subgroups in free pro-p groups.- 3.5 Relations in p-groups and Lie algebras.- Notes.- 4 Pro-p Groups.- 4.1 Pro-p groups with polynomial subgroup growth.- 4.2 Pro-p groups with slow subgroup growth.- 4.3 The groups $$SL_r^1({\mathbb{F}_p}[[t]])$$.- 4.4 A-perfect groups.- 4.5 The Nottingham group.- 4.6 Finitely presented pro-p groups.- Notes.- 5 Finitely Generated Groups with Polynomial Subgroup Growth.- 5.1 Preliminary observations.- 5.2 Linear groups with PSG.- 5.3 Upper chief factors.- 5.4 Groups of prosoluble type.- 5.5 Groups of finite upper rank.- 5.6 The degree of polynomial subgroup growth.- Notes.- 6 Congruence Subgroups.- 6.1 The characteristic 0 case.- 6.2 The positive characteristic case.- 6.3 Perfect Lie algebras.- 6.4 Normal congruence subgroups.- Notes.- 7 The Generalized Congruence Subgroup Problem.- 7.1 The congruence subgroup problem.- 7.2 Subgroup growth of lattices.- 7.3 Counting hyperbolic manifolds.- Notes.- 8 Linear Groups.- 8.1 Subgroup growth, characteristic 0.- 8.2 Residually nilpotent groups.- 8.3 Subgroup growth, characteristic p.- 8.4 Normal subgroup growth.- Notes.- 9 Soluble Groups.- 9.1 Metabelian groups.- 9.2 Residually nilpotent groups.- 9.3 Some finitely presented metabelian groups.- 9.4 Normal subgroup growth in metabelian groups.- Notes.- 10 Profinite Groups with Polynomial Subgroup Growth.- 10.1 Upper rank.- 10.2 Profinite groups with wPSG: structure.- 10.3 Quasi-semisimple groups.- 10.4 Profinite groups with wPSG: characterization.- 10.5 Weak PSG = PSG.- Notes.- 11 Probabilistic Methods.- 11.1 The probability measure.- 11.2 Generation probabilities.- 11.3 Maximal subgroups.- 11.4 Further applications.- 11.5 Pro-p groups.- Notes.- 12 Other Growth Conditions.- 12.1 Rank and bounded generation.- 12.2 Adelic groups.- 12.3 The structure of finite linear groups.- 12.4 Composition factors.- 12.5 BG, PIG and subgroup growth.- 12.6 Residually nilpotent groups.- 12.7 Arithmetic groups and the CSP.- 12.8 Examples.- Notes.- 13 The Growth Spectrum.- 13.1 Products of alternating groups.- 13.2 Some finitely generated permutation groups.- 13.3 Some profinite groups with restricted composition factors.- 13.4 Automorphisms of rooted trees.- Notes.- 14 Explicit Formulas and Asymptotics.- 14.1 Free groups and the modular group.- 14.2 Free products of finite groups.- 14.3 Modular subgroup arithmetic.- 14.4 Surface groups.- Notes.- 15 Zeta Functions I: Nilpotent Groups.- 15.1 Local zeta functions as p-adic integrals.- 15.2 Alternative methods.- 15.3 The zeta function of a nilpotent group.- Notes.- 16 Zeta Functions II: p-adic Analytic Groups.- 16.1 Integration on pro-p groups.- 16.2 Counting subgroups in a p-adic analytic group.- 16.3 Counting orbits.- 16.4 Counting p-groups.- Notes.- Windows.- 1 Finite Group Theory.- 1 Hall subgroups and Sylow bases.- 2 Carter subgroups.- 3 The Fitting subgroup.- 4 The generalized Fitting subgroup.- 5 Tate’s theorem.- 6 Rank and p-rank.- 7 Schur multiplier.- 8 Powerful p-groups.- 9 GLn and Sym(n).- 2 Finite Simple Groups.- 1 The list.- 2 Generators.- 3 Subgroups.- 4 Representations.- 5 Automorphisms.- 6 Schur multipliers.- 7 An elementary proof.- 3 Permutation Groups.- 1 Primitive groups.- 2 Groups with restricted sections.- 3 Subgroups of alternating groups.- 4 Profinite Groups.- 1 Completions.- 2 Free profinite groups.- 3 Profinite presentations.- 5 Pro-p Groups.- 1 Generators and relations.- 2 Pro-p groups of finite rank.- 3 Linear pro-p groups over local fields.- 4 Automorphisms of finite p-groups.- 5 Hall’s enumeration principle.- 6 Soluble Groups.- 1 Nilpotent groups.- 2 Soluble groups of finite rank.- 3 Finitely generated metabelian groups.- 7 Linear Groups.- 1 Soluble groups.- 2 Jordan’s theorem.- 3 Monomial groups.- 4 Finitely generated groups.- 5 Lang’s theorem.- 8 Linearity Conditions for Infinite Groups.- 1 Variations on Mal’cev’s local theorem.- 2 Groups that are residually of bounded rank.- 3 Applications of Ado’s theorem.- 9 Strong Approximation for Linear Groups.- 1 A variant of the Strong Approximation Theorem.- 2 Subgroups of SLn(Fp).- 3 The ‘Lubotzky alternative’.- 4 Strong approximation in positive characteristic.- 10 Primes.- 1 The Prime Number Theorem.- 2 Arithmetic progressions and the Bombieri-Vinogradov theorem.- 3 Global fields and Chebotarev’s theorem.- 11 Probability.- 12 p-adic Integrals and Logic.- 1 Results.- 2 A peek inside the black box.- Open Problems.- 1 ‘Growth spectrum’.- 2 Normal subgroup growth in pro-p groups and metabelian groups.- 3 The degree of f.g. nilpotent groups.- 4 Finite extensions.- 5 Soluble groups.- 6 Isospectral groups.- 7 Congruence subgroups, lattices in Lie groups.- 8 Other growth conditions.- 9 Zeta functions.

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Book SynopsisThe aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Aix-Marseille Université, France Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

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Table of ContentsI.- II.- III.- IV.- V.

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Table of ContentsLocally compact groupoids.- The C*-algebra of a groupoid.- Some examples.

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Book SynopsisThis introduction to the representation theory of compact Lie groups follows Herman Weyl’s original approach. It discusses all aspects of finite-dimensional Lie theory, consistently emphasizing the groups themselves. Thus, the presentation is more geometric and analytic than algebraic. It is a useful reference and a source of explicit computations. Each section contains a range of exercises, and 24 figures help illustrate geometric concepts.Table of ContentsI Lie Groups and Lie Algebras.- II Elementary Representation Theory.- III Representative Functions.- IV The Maximal Torus of a Compact Lie Group.- V Root Systems.- VI Irreducible Characters and Weights.- Symbol Index.

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Book SynopsisThis (post) graduate text gives a broad introduction to Lie groups and algebras with an emphasis on differential geometrical methods. It analyzes the structure of compact Lie groups in terms of the action of the group on itself by conjugation, culminating in the classification of the representations of compact Lie groups and their realization as sections of holomorphic line bundles over flag manifolds. Appendices provide background reviews.Trade ReviewFrom the reviews: "This one is worth to read and to keep on your shelf! It presents the theory of Lie groups not only in the usual way of Lie algebraic treatment, but also from the global point of view. … Every chapter ends with very useful notes on the origins and connections of the chapter’s subject. References are given separately in each chapter. ... It is highly recommended to advanced undergraduate and graduated students in mathematics and physics." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 75, 2009)Table of ContentsPreface 1 Lie Groups and Lie Algebras 1.1 Lie Groups and their Lie Algebras 1.2 Examples 1.3 The Exponential Map 1.4 The Exponential Map for a Vector Space 1.5 The Tangent Map of Exp 1.6 The Product in Logarithmic Coordinates 1.7 Dynkin's Formula 1.8 Lie's Fundamental Theorems 1.9 The Component of the Identity 1.10 Lie Subgroups and Homomorphisms 1.11 Quotients 1.12 Connected Commutative Lie Groups 1.13 Simply Connected Lie Groups 1.14 Lie's Third Fundamental Theorem in Global Form 1.15 Exercises 1.16 Notes References for Chapter One 2 Proper Actions 2.1 Review 2.2 Bochner's Linearization Theorem 2.3 Slices 2.4 Associated Fiber Bundles 2.5 Smooth Functions on the Orbit Space 2.6 Orbit Types and Local Action Types 2.7 The Stratification by Orbit Types 2.8 Principal and Regular Orbits 2.9 Blowing Up 2.10 Exercises 2.11 Notes References for Chapter Two 3 Compact Lie Groups 3.0 Introduction 3.1 Centralizers 3.2 The Adjoint Action 3.3 Connectedness of Centralizers 3.4 The Group of Rotations and its Covering Group 3.5 Roots and Root Spaces 3.6 Compact Lie Algebras 3.7 Maximal Tori 3.8 Orbit Structure in the Lie Algebra 3.9 The Fundamental Group 3.10 The Weyl Group as a Reflection Group 3.11 The Stiefel Diagram 3.12 Unitary Groups 3.13 Integration 3.14 The Weyl Integration Theorem 3.15 Nonconnected Groups 3.16 Exercises 3.17 Notes References for Chapter Three 4 Representations of Compact Groups 4.0 Introduction 4.1 Schur's Lemma 4.2 Averaging 4.3 Matrix Coefficients and Characters 4.4 G-types 4.5 Finite Groups 4.6 The Peter-Weyl Theorem 4.7 Induced Representations 4.8 Reality 4.9 Weyl's Character Formula 4.10 Weight Exercises 4.11 Highest Weight Vectors 4.12 The Borel-Weil Theorem 4.13 The Nonconnected Case 4.14 Exercises 4.15 Notes References for Chapter Four Appendix A Appendix B Appendix

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Book SynopsisIn the early 70's and 80's the field of integrable systems was in its prime youth: results and ideas were mushrooming all over the world. It was during the roaring 70's and 80's that a first version of the book was born, based on our research and on lectures which each of us had given. We owe many ideas to our colleagues Teruhisa Matsusaka and David Mumford, and to our inspiring graduate students (Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first version lacked rigor and precision, was rough, dis- connected and incomplete...In the early 90's new problems appeared on the horizon and the project came to a complete standstill, ultimately con- fined to a floppy. A few years ago, under the impulse of Pol Vanhaecke, the project was revived and gained real momentum due to his insight, vision and determination. The leap from the old to the new version is gigantic. The book is designed as a teaching textbook and is aimed at a wide read- ership of mathematicians and physicists, graduate students and professionals.Trade ReviewFrom the reviews of the first edition: "The aim of this book is to explain ‘how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations’. … One of the main advantages of this book is that the authors … succeeded to present the material in a self-contained manner with numerous examples. As a result it can be also used as a reference book for many subjects in mathematics. In summary … a very good book which covers many interesting subjects in modern mathematical physics." (Vladimir Mangazeev, The Australian Mathematical Society Gazette, Vol. 33 (4), 2006) "This is an extensive volume devoted to the integrability of nonlinear Hamiltonian differential equations. The book is designed as a teaching textbook and aims at a wide readership of mathematicians and physicists, graduate students and professionals. … The book provides many useful tools and techniques in the field of completely integrable systems. It is a valuable source for graduate students and researchers who like to enter the integrability theory or to learn fascinating aspects of integrable geometry of nonlinear differential equations." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1083, 2006)Table of Contents1 Introduction.- 2 Lie Algebras.- 3 Poisson Manifolds.- 4 Integrable Systems on Poisson Manifolds.- 5 The Geometry of Abelian Varieties.- 6 A.c.i. Systems.- 7 Weight Homogeneous A.c.i. Systems.- 8 Integrable Geodesic Flow on SO(4).- 9 Periodic Toda Lattices Associated to Cartan Matrices.- 10 Integrable Spinning Tops.- References.

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Book SynopsisCe deuxième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, comprend les chapitres: 2. Algèbres de Lie libres; 3. Groupes de Lie. Le chapitre 2 poursuit la présentation des notions fondamentales des algèbres de Lie avec l’introduction des algèbres de Lie libres et de la série de Hausdorff. Le chapitre 3 est consacré aux concepts de base pour les groupes de Lies sur un corps archimédien ou ultramétrique.Table of ContentsAlgèbres de Lie libres.- Groupes de Lie.

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Book SynopsisLes Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce neuvième chapitre du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, comprend les paragraphes: §1 Algèbres de Lie compactes; §2 Tores maximaux des groupes de Lie compacts; §3 Fromes compactes des algèbres de Lie semi-simples complexes; §4 Système de raciness associé à un groupe compact; §5 Classes de conjugaison; §6 Intégration dans les groupes de Lie compacts; §7 Représentations irréductibles des groupes de Lie compacts connexes; §8 Transformation de Fourier; §9 Opération des groupes de Lie compacts sur les variétés. Ce volume a été publié en 1982.Table of ContentsGroupes de Lie réels compacts.

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Book SynopsisCe huitième chapitre du Livre d'Algèbre, deuxième Livre des Éléments de mathématique, est consacré à l'étude de certaines classes d'anneaux et des modules sur ces anneaux.Il couvre les notions de module et d'anneau noethérien et artinien, ainsi que celle de radical. Ce chapitre décrit également la structure des anneaux semi-simples. Nous y donnons aussi la définition de divers groupes de Grothendieck qui jouent un rôle universel pour les invariants de modules et plusieurs descriptions du groupe de Brauer qui intervient dans la classification des anneaux simples.Une note historique en fin de volume, reprise de l'édition précédente, retrace l'émergence d'une grande partie des notions développées.Ce volume est une deuxième édition entièrement refondue de l'édition de 1958.Trade ReviewFrom the reviews of the second edition:“This book is intended as a comprehensive exposition of the theory of semi-simple rings and modules, with special emphasis on the Noetherian and Artinian cases. … Each section ends with a large collection of related exercises in the typical Bourbaki-style … . Certainly, it has been both a splendid idea and a great undertaking to rewrite N. Bourbaki’s classic Chapter 8 of Book II of the ‘Elements of Mathematics’ in such excellent a manner, very much so to the benefit of further generations of mathematicians.” (Werner Kleinert, Zentralblatt MATH, Vol. 1245, 2012)Table of ContentsIntroduction.- Chapitre VIII. Modules et anneaux semi-simples.- 1. Modules artiniens et modules noethériens.- 2. Structure des modules de longueur finie.- 3. Modules simples.- 4. Modules semi-simples.- 5. Commutation.- 6. Équivalence de Morita des modules et des algèbres.- 7. Anneaux simples.- 8. Anneaux semi-simples.- 9. Radical.- 10. Modules sur un anneau artinien.- 11. Groupes de Grothendieck.- 12. Produit tensoriel de modules semi-simples.- 13. Algèbres absolument semi-simples.- 14. Algèbres centrales et simples.- 15. Groupes de Brauer.- 16. Autres descriptions du groupe de Brauer.- 17. Normes et traces réduites.- 18. Algèbres simples sur un corps fini.- 20. Représentations linéaires des algèbres.- 21. Représentations linéaires des groupes finis.- Appendice 1. Algèbres sans élément unité.- Appendice 2. Déterminants sur un corps non commutatif.- Appendice 3. Le théorème des zéros de Hilbert.- Appendice 4. Trace d’un endomorphisme de rang fini.- Note Historique.- Bibliographie.- Index des notations.- Index terminologique

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Book SynopsisCe volume du Livre d’Intégration, sixième Livre du traité, traite de l’intégration sur les groupes localement compacts et de ses applications. Les notions introduites, telles que les mesures de Haar et le produit de convolution, sont à la base de l’analyse harmonique. Il comprend les chapitres : -1. Mesure de Haar ; -2. Convolution et représentations.Table of ContentsMesure de Haar.- Convolution et Représentation.

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Book SynopsisThe 1963 Göttingen notes of T. A. Springer are well known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. Trade ReviewFrom the reviews of the first edition: "This book is an updated and revised English version of the German notes on octaves, Jordan algebras and exceptional groups which appeared as mimeographed lecture notes of Göttingen University in 1963. It is still an excellent reference on the subject … ." (Huberta Lausch, Zentralblatt MATH, Vol. 1087, 2006)Table of Contents1. Composition Algebras.- 2. The Automorphism Group of an Octonion Algebra.- 3. Triality.- 4. Twisted Composition Algebras.- 5. J-algebras and Albert Algebras.- 6. Proper J-algebras and Twisted Composition Algebras.- 7. Exceptional Groups.- 8. Cohomological Invariants.

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Book SynopsisFrom the reviews of the French edition: "This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a (B,N)-pair structure, or ‘Tits systems’". --G.B. Seligman in MathReviews.Table of ContentsCoxeter Groups and Tits Systems.- Groups Generated by Reflections.- Root Systems.

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Book SynopsisThe Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.Trade ReviewFrom the reviews:"In this book the authors present a complete proof of the Langlands conjecture for GL (2) over a non-archimedean local field, which uses local methods and is accessible to students. … The book is very well written and easy to read." (J. G. M. Mars, Zentralblatt MATH, Vol. 1100 (2), 2007)"The book under review gives a complete and self-contained insight into the theory of representations of G. … We highly recommend this book to Ph.D. students as well as to specialists. The book contains a huge amount of information, definition and facts … . The book has a Bibliography containing 91 references … ." (Alexandru Ioan Badulescu, Mathematical Reviews, Issue 2007 m)“The aim of this monograph is to present a complete and self-contained proof of the Langlands conjecture for GL(2) over a non-archimedean local field. … This volume presents a large amount of difficult material in a clear and readable manner. It can be recommended to anyone interested in representations of linear algebraic groups.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 154 (4), August, 2008)Table of ContentsSmooth Representations.- Finite Fields.- Induced Representations of Linear Groups.- Cuspidal Representations.- Parametrization of Tame Cuspidals.- Functional Equation.- Representations of Weil Groups.- The Langlands Correspondence.- The Weil Representation.- Arithmetic of Dyadic Fields.- Ordinary Representations.- The Dyadic Langlands Correspondence.- The Jacquet-Langlands Correspondence.

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Book SynopsisThis concise, class-tested book was refined over the authors’ 30 years as instructors at MIT and the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory along with applications helps students to learn, understand and use it for their own needs. Thus, the theoretical background is confined to introductory chapters. Subsequent chapters develop new theory alongside applications so that students can retain new concepts, build on concepts already learned, and see interrelations between topics. Essential problem sets between chapters aid retention of new material and consolidate material learned in previous chapters.Trade ReviewFrom the reviews:"It was developed for a graduate course taught mostly by Millie Dresselhaus at MIT for more than 30 years, with many revisions of lecture notes. Very much a graduate text or specialist monograph, the book covers a wealth of applications across solid-state physics. … The book can be warmly recommended to students and researchers in solid-state physics, either to serve as a text for an advanced lecture course or for individual study … ." (Volker Heine, Physics Today, November, 2008)"This textbook is based on the authors’ pedagogical experience during their 30 years at MIT. … the book develops all of the relevant mathematics (linear algebra) and the necessary physics (quantum mechanics), it is eminently suitable to a wide audience in physics, chemistry and materials science." (Barry R. Masters, Optics and Photonics News, July/August, 2009)“This is an excellent text … . originates from lectures by Charles Kittel and J. H. van Vleck in the 1950s and much of the material was presented in courses by the authors over the last three decades. The material is meant for Electrical Engineering and Physics students at the graduate level … . has exercises at the end of each chapter and an extensive set of appendices. The exposition is clear and detailed. This is a very good book for its target audience.” (W. Miller Jr., Zentralblatt MATH, Vol. 1175, 2010)“The goal of the book under review is to teach group theory in close connection to applications. … Every chapter of the book finishes with several selected problems. Specific to this book is the feature that every abstract theoretical group concept is introduced and applied in a concrete physical way. This is why the book is very useful for anyone interested in applications of group theory to the wide range of condensed matter phenomena.”­­­ (Oktay K. Pashaev, Mathematical Reviews, Issue 2010 i)“It is highly welcomed because of its well-thought structuring and the plenty of non-trivial examples. The authors develop those parts of the theory of groups which are interesting for physicists, from chapter to chapter offering nearly at any step one or more informative application.” (G. Kowol, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)Table of ContentsBasic Mathematics.- Basic Mathematical Background: Introduction.- Representation Theory and Basic Theorems.- Character of a Representation.- Basis Functions.- Introductory Application to Quantum Systems.- Splitting of Atomic Orbitals in a Crystal Potential.- Application to Selection Rules and Direct Products.- Molecular Systems.- Electronic States of Molecules and Directed Valence.- Molecular Vibrations, Infrared, and Raman Activity.- Application to Periodic Lattices.- Space Groups in Real Space.- Space Groups in Reciprocal Space and Representations.- Electron and Phonon Dispersion Relation.- Applications to Lattice Vibrations.- Electronic Energy Levels in a Cubic Crystals.- Energy Band Models Based on Symmetry.- Spin–Orbit Interaction in Solids and Double Groups.- Application of Double Groups to Energy Bands with Spin.- Other Symmetries.- Time Reversal Symmetry.- Permutation Groups and Many-Electron States.- Symmetry Properties of Tensors.

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Book SynopsisThis monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem.This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.Table of ContentsPreface.- Conventions.- List of Symbols.- 1 Symmetric bilinear forms.- 2 Clifford algebras.- 3 The spin representation.- 4 Covariant and contravariant spinors.- 5 Enveloping algebras.- 6 Weil algebras.- 7 Quantum Weil algebras.- 8 Applications to reductive Lie algebras.- 9 D(g; k) as a geometric Dirac operator.- 10 The Hopf–Koszul–Samelson Theorem.- 11 The Clifford algebra of a reductive Lie algebra.- A Graded and filtered super spaces.- B Reductive Lie algebras.- C Background on Lie groups.- References.- Index.

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Book SynopsisThe German edition of this book appeared in 1932 under the title "Die gruppentheoretische Methode in der Quantenmechanik". Its aim was, to explain the fundamental notions of the Theory of Groups and their Representations, and the application of this theory to the Quantum Mechanics of Atoms and Molecules. The book was mainly written for the benefit of physicists who were supposed to be familiar with Quantum Mechanics. However, it turned out that it was also used by. mathematicians who wanted to learn Quantum Mechanics from it. Naturally, the physical parts were too difficult for mathematicians, whereas the mathematical parts were sometimes too difficult for physicists. The German language created an additional difficulty for many readers. In order to make the book more readable for physicists and mathe­ maticians alike, I have rewritten the whole volume. The changes are most notable in Chapters 1 and 6. In Chapter t, I have tried to give a mathematically rigorous exposition of the principles of Quantum Mechanics. This was possible because recent investigations in the theory of self-adjoint linear operators have made the mathematical foundation of Quantum Mechanics much clearer than it was in t 932. Chapter 6, on Molecule Spectra, was too much condensed in the German edition. I hope it is now easier to understand. In Chapter 2-5 too, numerous changes were made in order to make the book more readable and more useful.Table of Contents1. Fundamental Notions of Quantum Mechanics.- § 1. Wave Functions.- § 2. Hilbert Spaces.- § 3. Linear Operators.- § 4. Hypermaximal Operators.- § 5. Separation of Variables.- § 6. One Electron in a Central Field.- § 7. Perturbation Theory.- § 8. Angular Momentum and Infinitesimal Rotations.- 2. Groups and Their Representations.- § 9. Linear Transformations.- § 10. Groups.- § 11. Equivalence and Reducibility of Representations.- § 12. Representations of Abelian Groups. Examples.- § 13. Uniqueness Theorems.- § 14. Kronecker’s Product Transformation.- § 15. The Operators Commuting with all Operators of a Given Representation.- § 16. Representations of Finite Groups.- § 17. Group Characters.- 3. Translations, Rotations and Lorentz Transformations.- § 18. Lie Groups and their Infinitesimal Transformations.- A. Lie Groups.- B. One-dimensional Lie Groups and Semi-Groups.- C. Causality and Translations in Time.- D. The Lie Algebra of a Lie Group.- E. Representations of Lie Groups.- § 19. The Unitary Groups SU(2) and the Rotation Group O3.- § 20. Representations of the Rotation Group O3.- § 21. Examples and Applications.- A. The Product Representation ?j × ?j’.- B. The Clebsch-Gordan Series.- C. Applications of (21.1).- D. The Reflection Character.- § 22. Selection and Intensity Rules.- § 23. The Representations of the Lorentz Group.- A. The Group SL(2) and the Restricted Lorentz Group.- B. Infinitesimal Transformations.- C. The Relation between World Vectors and Spinors.- IV. The Spinning Electron.- § 24. The Spin.- § 25. The Wave Function of the Spinning Electron.- A. Pauli’s Pair of Functions (?1, ?2).- B. Transformation of the Pair (?1, ?2).- C. Infinitesimal Rotations.- D. The Angular Momenta.- E. The Doublet Splitting of the Alkali Terms.- G. The Inversion s.- § 26. Dirac’s Wave Equation.- § 27. Two-Component Spinors.- A. Dirac’s Equation Rewritten.- B. Weyl’s Equation.- § 28. The Several Electron Problem. Multiplet Structure. Zeeman Effect.- V. The Group of Permutations and the Exclusion Principle.- § 29. The Resonance of Equal Particles.- § 30. The Exclusion Principle and the Periodical System.- § 31. The Eigenfunctions of the Atom.- § 32. The Calculation of the Energy Values.- § 33. Pure Spin Functions and their Transformation under Rotations and Permutations.- § 34. Representations of the Symmetric Group Sn.- VI. Molecule Spectra.- § 35. The Quantum Numbers of the Molecule.- § 36. The Rotation Levels.- § 37. The Case of Two Equal Nuclei.- Author and Subject Index.

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Table of Contents1. Einhüllende Algebren.- 2. Halbeinfache Lie-Algebren.- 3. Zentralisatoren in Einhüllenden halbeinfacher Lie-Algebren.- 4. Moduln mit einem höchsten Gewicht.- 5. Annullatoren einfacher Moduln mit einem höchsten Gewicht.- 6. Harish-Chandra-Moduln.- 7. Primitive Ideale und Harish-Chandra-Moduln.- 8. Gel’fand-Kirillov-Dimension und Multiplizität.- 9. Die Multiplizität von Moduln in der Kategorie O.- 10. Gel’fand-Kirillov-Dimension von Harish-Chandra-Moduln.- 11. Lokalisierungen von Harish-Chandra-Moduln.- 12. Goldie-Rang und Kostants Problem.- 13. Schiefpolynomringe und der Übergang zu den m-Invarianten.- 14. Goldie-Rang-Polynome und Darstellungen der Weylgruppe.- 15. Induzierte Ideale und eine Vermutung von Gel’fand und Kirillov.- 16. Kazhdan-Lusztig-Polynome und spezielle Darstellungen der Weylgruppe.- 17. Assoziierte Varietäten.- Literatur.- Verzeichnis der Notationen.

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Book Synopsisby a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat­ egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap­ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.Table of ContentsI. Hopf Algebras.- 1.1 Axioms of a Hopf Algebra.- 1.2 Group Algebras and Enveloping Algebras.- 1.3 Adjoint Action.- 1.4 The Hopf Dual.- 1.5 Comments and Complements.- 2. Excerpts from the Classical Theory.- 2.1 Lie Algebras.- 2.2 Algebraic Lie Algebras.- 2.3 Algebraic Groups.- 2.4 Lie Algebras of Algebraic Groups.- 2.5 Comments and Complements.- 3. Encoding the Cartan Matrix.- 3.1 Quantum Weyl Algebras.- 3.2 The Drinfeld Double.- 3.3 The Rosso Form and the Casimir Invariant.- 3.4 The Classical Limit and the Shapovalev Form.- 3.5 Comments and Complements.- 4. Highest Weight Modules.- 4.1 The Jantzen Filtration and Sum Formula.- 4.2 Kac-Moody Lie Algebras.- 4.3 Integrable Modules for Uq(gc).- 4.4 Demazure Modules and Product Formulae.- 4.5 Comments and Complements.- 5. The Crystal Basis.- 5.1 Operators in the Crystal Limit.- 5.2 Crystals.- 5.3 Ad-invariant Filtrations, Twisted Actions and the Crystal Basis for Uq(n-).- 5.4 The Grand Loop.- 5.5 Comments and Complements.- 6. The Global Bases.- 6.1 The ? Operation and the Embedding Theorem.- 6.2 Globalization.- 6.3 The Demazure Property.- 6.4 Littelmann’s Path Crystals.- 6.5 Comments and Complements.- 7. Structure Theorems for Uq(g).- 7.1 Local Finiteness for the Adjoint Action.- 7.2 Positivity of the Rosso Form.- 7.3 The Separation Theorem.- 7.4 Noetherianity.- 7.5 Comments and Complements.- 8. The Primitive Spectrum of Uq(g).- 8.1 The Poincaré Series of the Harmonic Space.- 8.2 Factorization of the Quantum PRV Determinants.- 8.3 Verma Module Annihilators.- 8.4 Equivalence of Categories.- 8.5 Comments and Complements.- 9. Structure Theorems for Rq[G].- 9.1 Commutativity Relations.- 9.2 Surjectivity and Injectivity Theorems.- 9.3 The Adjoint Action.- 9.4 The R-Matrix.- 9.5 Comments and Complements.- 10. The Prime Spectrum of Rq[G].- 10.1 Highest Weight Modules.- 10.2 The Quantum Weyl Group.- 10.3 Prime and Primitive Ideals of Rq[G].- 10.4 Hopf Algebra Automorphisms.- 10.5 Comments and Complements.- A.2 Excerpts from Ring Theory.- A.3 Combinatorial Identities and Dimension Theory.- A.4 Remarks on Constructions of Quantum Groups.- A.5 Comments and Complements.- Index of Notation.

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Book SynopsisCe livre des Éléments de mathématique est consacré à la Topologie algébrique. Les quatre premiers chapitres présentent la théorie des revêtements d'un espace topologique et du groupe de Poincaré. On construit le revêtement universel d'un espace connexe pointé délaçable et on établit l'équivalence de catégories entre revêtements de cet espace et actions du groupe de Poincaré. On démontre une version générale du théorème de van Kampen exprimant le groupoïde de Poincaré d'un espace topologique comme un coégalisateur de diagrammes de groupoïdes. Dans de nombreuses situations géométriques, on en déduit une présentation explicite du groupe de Poincaré.Table of ContentsMode d'Emploi.- Introduction.- Chapitre I. Revêtements.- 1. Produits fibrés et carrés cartésiens.- 2. Applications étales.- 3. Faisceaux.- 4. Revêtements.- 5. Revêtements principaux.- 6. Espaces simplement connexes.- Exercices.- Chapitre II. Groupoïdes.- 1. Carquois.- 2. Graphes.- 3. Groupoïdes.- 4. Homotopies.- 5. Coégalisateur.- Exercices.- Chapitre III. Homotopie et Groupoïdes de Poincaré.- 1. Homotopies, homéotopies.- 2. Homotopie et chemins.- 3. Groupoïde de Poincaré.- 4. Homotopie et revêtements.- 5. Homotopie et revêtements (cas des espaces localement connexes par arcs).- Exercices.- Chapitre IV. Espaces Delaçables.- 1. Espaces délaçables.- 2. Groupes de Poincaré des espaces délaçables.- 3. Groupes de Poincaré des groupes topologiques.- 4. Théorie de la descente.- 5. Théorème de van Kampen.- 6. Espaces classifiants.- Exercices.- Index des notations.- Index terminologique.

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Book SynopsisA clear explanation of the mathematics that underlies comprehension of Rubik's Cube, and how the cube itself can be used to grasp the fundamentals of group theory.Trade Review"The best book yet written - or likely to be written - on the wonders and the dark, unsolved mysteries of the cube" Martin Gardiner "Frey and Singmaster give a nicely blended discussion of specific algorithms for the cube together with some underlying theoretical concepts from group theory. Thus, cubists will be able to cure their insomnia by reading Chapter 3, and then perhaps satisfy their aroused curiosity about what is really going on by reading further" David M. Goldschmidt, Professor of Mathematics, University of California, Berkeley "Frey and Singmaster masterfully use the Rubik's cube to give a very tangible and tactile feel to the abstract concept of a group ... the book remains the definitive source for cube mathematics and presents an interesting and accessible introduction to group theory." David A. Reimann, in Mathematical Reviews, Issue 2012dTable of ContentsPreface 1. Introduction 2. A Cubik Orientation 3. Restoring the Cube 4. The What, Why, and How of Cube Movements 5. Improved Restoration Processes 6. The Cube Group and Subgroups 7. Permutation Structures and the Order of Groups 8. Advanced Restoration Methods 9. Epilogue A. A Small Catalogue of Processes B. Solutions to Exercises Index

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Book SynopsisThe second of two volumes addressing theoretical problems associated with consensual action of task-orientated small groups. Less formal than the first volume, this work tends to be more general in approach focusing on interpersonal relations and social processes within such groups.Trade Review"The field has long been preoccupied by certain topics--decision making, leadership, and performance--and the contents reflect that continuing emphasis. Witte and Davis have done the field a great service in bringing these papers together for publication."—Contemporary PsychologyTable of ContentsVolume 1.Contents: Preface. Part I: Introduction.J.H. Davis, Small-Group Research and the Steiner Questions: The Once and Future Thing. Part II: Social Aggregation and Combination Models.H.W. Crott, J. Werner, C. Hoffmann, A Probabilistic Model of Opinion Change Considering Distance Between Alternatives: An Application to Mock Jury Data. J.H. Davis, Group Decision Making and Quantitative Judgments: A Consensus Model. P.R. Laughlin, Group Decision Making and Collective Induction. R.S. Tindale, C.M. Smith, L.S. Thomas, J. Filkins, S. Sheffey, Shared Representations and Asymmetric Social Influence Processes in Small Groups. N.L. Kerr, R.J. MacCoun, G.P. Kramer, "When are N Heads Better (or Worse) Than One?": Biased Judgment in Individuals Versus Groups. T. Kameda, Procedural Influence in Consensus Formation: Evaluating Group Decision Making From a Social Choice Perspective. Part III: Social Information-Processing Models.G. Stasser, S.I. Vaughan, Models of Participation During Fact-to-Face Unstructured Discussion. B. Latané, Strength From Weakness: The Fate of Opinion Minorities in Spatially Distributed Groups. D. Gigone, R. Hastie, The Impact of Information on Group Judgment: A Model and Computer Simulation. E.H. Witte, The Extended Group Situation Theory (EGST): Explaining the Amount of Change. S. Galam, When Humans Interact Like Atoms. J.H. Davis, Concluding Remarks. Volume 2.Contents: Preface. Part I: Introduction.E.H. Witte, Small-Group Research and the Crisis of Social Psychology: An Introduction. Part IIGroup Structure and Composition.R.L. Moreland, J.M. Levine, M.L. Weingert, Creating the Ideal Group: Composition Effects at Work. W. Stroebe, M. Diehl, G. Abakoumkin, Social Compensation and the Köhler Effect: Toward a Theoretical Explanation of Motivation Gains in Group Productivity. H.A.M. Wilke, Status Congruence in Small Groups. J.C. Jesuino, Leadership: Micro-Macro Links. W. Scholl, Effective Teamwork -- A Theoretical Model and a Test in the Field. M. von Cranach, Toward a Theory of the Acting Group. Part III: Interpersonal Influence, Conflict, and Resolution.J.A. Pérez, G. Mugny, The Conflict Elaboration Theory of Social Influence. D. Frey, S. Schulz-Hardt, D. Stahlberg, Information Seeking Among Individuals and Groups and Possible Consequences for Decision Making in Business and Politics. M.A. Hogg, Social Identity, Self-Categorization, and the Small Group. J.M. Rabbie, H.F.M. Lodewijkx, A Behavioral Interaction Model: Toward an Integrative Theoretical Framework for Studying Intra- and Intergroup Dynamics. E.H. Witte, Similarities Among Various Conceptual Positions and Theoretical Points of View.

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Book SynopsisThere is a mysterious connection between our experiences of intimacy--of love, the longing to feel connected, and sexual embrace--and the human sense of time--eternity, impermanence, and rhythm. In this critical analysis of the time-intimacy equation, Bennett shows how the scientific study of personal relationships can address this mystery. As a study of transpersonal science, this book points to the possible evolution of intimacy and of our consciousness of time, and how the two evolutionary paths weave together. Dr. Bennett draws from a wide array of resources to advance and marry two compelling themes: first, the social and clinical science of personal relationships should integrate the spiritual or transpersonal dimension of intimacy, and second, science can contribute to lay understandings by describing the richly temporal aspects of relationships. In blending popular literature, transpersonal psychology, and scientific research and theory, this work also attempts to addrTrade Review"Joel Bennett's Time and Intimacy talks about intimacy in a way that recreates the feeling of intimacy, and recapitulates his vision of an intimate interaction through the structure of the book itself. Just as intimate interactions provide a sense of transcending time even as they can only be understood within a temporary trajectory, Bennett's prose, by refusing linearity and traditional scientific discourse, recapitulates the felt sense of an intimate interaction through its poetic style. Time and Intimacy is a book that can be opened anywhere and enjoyed, as it is composed of many loosely woven strands pulled together graphically and sometimes metaphorically. When it is opened spontaneously and absorbed in bursts, with full attention given to the language itself, it can be appreciated like poetry. Unlike poetry, however, Bennett's work has woven within it references to the latest research on close relationships. For readers who believe that the structure of the text should speak to the subject matter of that text, there is much experimentation with textual structure to be discovered in Time and Intimacy. Language alternates in unpredictable ways between the poetic and the philosophical, between the scientific and the transpersonal, between the objective and the subjective. Much as an intimate conversation itself might, the book invites the reader to transcend the ordinary boundaries of academic disciplines and to think about intimate experience in language that goes beyond ordinary discourse."—Karen PragerUniversity of Texas, Dallas"This book delves deeply and reveals to the reader a rich and new way to look at the realm of relationships. Simultaneously exploring time and intimacy creates a profound shift in our understanding of how people relate. In a modern era that views time as a commodity, intimacy is often lost as relationship suffers. Joel Bennett provides a brilliant approach to a very important subject--well worth any reader's time."—Stephan Rechtschaffen, MDAuthor, Time Shifting: Creating More Time to Enjoy Your Life; Cofounder and CEO,"Bennett has done an extraordinary job of combining three forms of reflection to treat the topic of intimacy. The integration of three themes--quantitative psychology research, transpersonal depth psychology, and temporality/transcendence--constitutes the unique contribution of this book. Time and Intimacy: A New Science of Personal Relationships presents a hypothesis for further reflection in psychology, and it organizes a whole field of research for study. Most important, it deals sensitively with a very important topic."—Robert Cummings NevilleDean, School of Theology, Boston UniversityTable of ContentsContents: S. Duck, Series Editor's Foreword. Preface. Introduction. Intimacy as a Journey Through This Text: Time and Change From Transcription to Transcendence. Unfolding in Time: Intimacy Across Situations. The Forces of Time: The Process of Intimacy. Time, Addiction, and Intimacy: Scripts, Archescripts, & Holoscripts. Temporal Context in Love and Science: The Weave of Temporal Sensitivity. In Search of the Temporal in Personal Relationship Research: From Temporal Initiative to Temporal Scale. Theories of Personal Relationships and the Contextual Model. Summary and Integration: Toward a Transpersonal Science. Appendix.

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Book SynopsisRelational maintenance provides a rallying point for those seeking to discover the behaviors that individuals utilize to sustain their personal relationships. Theoretical models, research programs, and specific studies have examined how people in a variety of close relationships choose to define and maintain those relationships. In addition, relational maintenance turns our attention to communicative processes that help people sustain their close relationships. In this collection, editors Daniel J. Canary and Marianne Dainton focus on the communicative processes critical to the maintenance and enhancement of personal relationships. The volume considers variations in maintaining different types of personal relationships; structural constraints on relationship maintenance; and cultural variations in relational maintenance. Contributions to the volume cover a broad range of relational types, including romantic relationships, family relationships, long-distance relationships, workplace relationships, and Gay and Lesbian relationships, among others. Maintaining Relationships Through Communication: Relational, Contextual, and Cultural Variations synthesizes current research in relationship maintenance, emphasizes the ways that behaviors vary in their maintenance functions across relational contexts, discusses alternative explanations for maintaining relationships, and presents avenues for future research. As such, it is intended for students and scholars studying interpersonal communication and personal relationships.Table of ContentsChapter 1 Definitions and Perspectives on Relational Maintenance Communication, Kathryn Dindia; Part 1 Maintaining Different Types of Relationships; Chapter 2 Maintaining Family Relationships, Sally Vogl-Bauer; Chapter 3 Maintaining Romantic Relationships: A Summary and Analysis of One Research Program, Laura Stafford; Chapter 4 Maintaining Friendships Throughout the Lifespan, Marianne Dainton, Elaine Zelley, Emily Langan; Chapter 5 Maintaining Undesired Relationships, Jon A. Hess; Part 2 Contextual Variations in Maintaining Relationships; Chapter 6 Maintaining Long-Distance Relationships, Brooks A. Aylor; Chapter 7 Computer-Mediated Communication Effects on Relationship Formation and Maintenance, Michael K. Rabby, Joseph B. Walther; Chapter 8 Relationship Maintenance in Organizational Settings, Vincent R. Waldron; Chapter 9 When Part ners Falter: Repair After a Transgression, Tara M. Emmers-Sommer; Part 3 Cultural Variations in Maintaining Relationships; Chapter 10 Relationship Maintenance in Same-Sex Couples, Stephen M. Haas; Chapter 11 Relationship Maintenance in Intercultural Couples: An Interdependence Analysis, Stanley O. Gaines, Jr., Christopher R. Agnew; Chapter 12 Maintaining Marriages in Russia: Managing Societal Influences and Communication Dynamics, Deborah S. Ballard-Reisch, Marat Zaguidoulline, Daniel J. Weigel; Chapter 13 Maintaining Relationships in Korea and the United States: Features of Korean Culture that Affect Relational Maintenance Beliefs and Behaviors, Young-Ok Yum, Daniel J. Canary; Part 4 Epilogue; Chapter 14 Framing the Maintenance of Relationships Through Communication: An Epilogue, Marianne Dainton;

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Book SynopsisThis volume addresses the role of communication in stereotype dynamics, while placing the phenomenon of social stereotypes appropriately in the socio-cultural context.Trade Review"I think it is critically important for social psychologists to focus more on the influence of language and communication across the areas within the discipline, and I believe this volume will stand as a milestone in the movement of social psychology back to studying human interaction." - David Roskos-Ewoldsen, Ph.D., University of Alabama"The authors are all outstanding ... The chapters are timely and important." - Charles Stangor, Ph.D., University of Maryland"… Kashima et al. achieve their goals for this volume… The reader is left with a new appreciation for the social nature of stereotyping, as well as an awareness of the limitations of a strict social cognition perspective… I would recommend this volume for the editors’ intended audience of advanced scholars and students in social psychology, linguistics, and communication." – Matthew L. Newman, PsycCRITIQUES“I think it is critically important for social psychologists to focus more on the influence of language and communication across the areas within the discipline, and I believe this volume will stand as a milestone in the movement of social psychology back to studying human interaction.” —David Roskos-Ewoldsen, Ph.D . University of Alabama“The authors are all outstanding...The chapters are timely and important.” —Charles Stangor, Ph.D. University of MarylandTable of ContentsY. Kashima, K. Fiedler, P. Freytag, Stereotype Dynamics: An Introduction and Overview. Part 1. Stereotype Dynamics.G. Semin, Stereotypes in the Wild. V. Yzerbyt, A. Carnaghi, Stereotype Change in the Social Context. A. Lyons, A. Clark, Y. Kashima, T. Kurz, Cultural Dynamics of Stereotyping. Part 2. Symbolic Mediation and Stereotyping.K. Fiedler, M. Blümke, P. Freytag, S. Koch, C. Unkelbach, A Semiotic Approach to Understanding the Role of Communication in Stereotyping. A. Carnaghi, A. Maass, Derogatory Language in Intergroup Context: Are “Gay” and “Fag” Synonymous? S. Sczesny, J. Bosak, A.B. Diekman, J. Twenge, Dynamics of Sex Role Stereotypes. Part 3. Stereotype and Language Use.C. Wenneker, D. Wigboldus, Interpersonal Consequences and Intrapersonal Underpinnings of the Linguistic Expectancy Bias. K.M. Douglas, R.M. Sutton, C. McGarty, Strategic Language Use in Interpersonal and Intergroup Communication. P. Freytag, Sender-Receiver-Constellations as a Moderator of Linguistic Abstraction Biases. Part 4. Stereotype Sharedness and Distinctiveness.M. Karasawa, S. Suga, Retention and Transmission of Socially Shared Beliefs: The Role of Linguistic Abstraction in Stereotypic Communication. O. Klein, S. Tindale, M. Brauer, The Consensualization of Stereotypes in Small Groups. F. Pratto, P.J. Hegarty, J.D. Korchmaros, How Communication Practices and Category Norms Lead People to Stereotype Particular People and Groups. Part 5. Identity, Self-Regulation, and Stereotyping.M. Hornsey, Intergroup Sensitivity Effect: Responses to Criticisms of Groups. R.M. Sutton, K.M. Douglas, T.J. Elder, M. Tarrant, Social Identity and Social Convention in Responses to Criticisms of Groups. J. Keller, H. Bless, Communicating Stereotype Expectancies: The Interplay of Stereotype Threat and Regulatory Focus.

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Book SynopsisThe Handbook of Relationship Initiation is the first volume to focus specifically on the very beginning stage of relationships â their origin. In this Handbook, leading scholars on relationships review the literature on various processes related to the initiation of relationships: how people meet, communicate for the first time, and begin to define themselves as being in a relationship. Topics include attraction, mate selection, influence of social networks on relationship initiation, initiation over the internet, hook-ups among young adults, and flirting and opening gambits. In addition, the dark side of relationship initiation is considered, including unwanted relationship pursuit and barriers to relationship initiation including social anxiety.This volume provides an overdue synthesis of the literature on this topic. It is especially timely in view of the growing prevalence on relationship initiation online, through matchmaking and other social networking sites, which has increased awareness that science can be used to understand, create, and facilitate compatible matching.This Handbook provides an essential resource for an interdisciplinary range of researchers and students who study relationships, including social psychologists, communication scientists and scholars of marriage and the family.Trade Review"I am somewhat chagrined to find that after 20 years in the field, I found a great deal that I did not know regarding relationship initiation. ... I strongly suspect that this latest work will be an instant classic. ... I highly recommend the handbook for relationship scholars from a variety of disciplines. It is also appropriate for graduate students and upper level undergraduates." - Gregory D. Morrow, Edinboro University of Pennsylvania, in PsycCRITIQUESTable of ContentsPart 1. Introduction. S. Sprecher, A. Wenzel, J. Harvey, Preface. C. A. Bredow, R. M. Cate, T. L. Huston, Have We Met Before? A Conceptual Model of First Romantic Encounters. B. Fehr, Friendship Formation. D. P. Schmitt, An Evolutionary Perspective on Mate Choice and Relationship Initiation. G. Creasey, P. Jarvis, Attachment Theory and Research: A Special Focus on Relationship Initiation. Part 2. The Process of Relationship Initiation. M. R. Cunningham, A. P. Barbee, Prelude to a Kiss: Nonverbal Flirting, Opening Gambits, and Other Communication Dynamics in the Initiation of Romantic Relationships. L. K. Knoblock, L. E. Miller, Uncertainty and Relationship Initiation. W. A. Afifi, A. A. Lucas, Information Seeking in Initial Stages of Relational Development. V. J. Derlega, B. A. Winstead, K. Greene, Self-Disclosure and Starting a Close Relationship. L. K. Guerrero, P. A. Mongeau, On Becoming "More than Friends": The Transition from Friendship to Romantic Relationship. Part 3. Diverse Contexts of Relationship Initiation. X. B. Arriaga, C. R. Agnew, N. M. Capezza, J. J. Lehmiller, The Social and Physical Environment of Relationship Initiation: An Interdependence Analysis. P. W. Eastwick, E. J. Finkel, Speed-Dating: A Powerful and Flexible Paradigm for Studying Relationship Initiation. K. Y. A. McKenna, MySpace or Your Place: Relationship Initiation and Development in the Wired and Wireless World. S. Sprecher, P. Schwartz, J. Harvey, E. Hatfield, Thebusinessoflove.com: Relationship Initiation at Internet Matchmatching Services. Part 4. Attraction and Other Emotions in Relationship Initiation. J. W. Bruce, W. G. Graziano, Attraction and the Initiation of Relationships: A Review of the Empirical Literature. S. Sprecher, D. Felmlee, Insider Perspectives to Attraction. A. Aron, H. E. Fisher, G. Strong, B. Acevedo, S. Riela, I. Tsapelas, Falling in Love. S. S. Hencrick, C. Hendrick, Satisfaction, Love, and Respect in the Initiation of Romantic Relationships. S. Metts, S. L. Mikucki, The Emotional Landscape of Relationship Initiation. Part 5. Challenges and Problematic Relationship Initiation. E. L. Paul, A. Wenzel, J. H. Harvey, Hookups: A Facilitator or a Barrier to Relationship Initiation and Intimacy Development? M. A. Fine, T. A. Coffelt, L. N. Olson, Romantic Relationship Initiation following Relationship Dissolution. W. R. Cupach, B. H. Spitzberg, "Thanks, but No Thanks…." The Occurrence and Management of Unwanted Relationship Pursuit. A. Wenzel, T. B. Kashdan, Emotional Disturbances and the Initial Stages of Relationship Development: Processes and Consequences of Social Anxiety and Depression. Part 6. Cognitions, Beliefs, and Memories About Relationship Initiation. L. Custer, D. Holmberg, K. Blair, T. L. Orbuch, "So How Did You Two Meet?": Narratives of Relationship Initiation. C. R. Knee, A. L. Bush, Relationship Beliefs and Their Role in Relationship Initiation. S. Tran, J. A. Simpson, G. J. O. Fletcher, The Role of Ideal Standards in Relationship Initiation Processes. G. D. Reeder, Perceptions of Goals and Motives in Romantic Relationships. Part 7. Commentary. D. Perlman, Ending the Beginning of Relationships.

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Book Synopsis0 Introduction and Foundations.- 0.1 The Fundamental Concepts and Problems of Topology.- 0.2 Simplicial Complexes.- 0.3 The Jordan Curve Theorem.- 0.4 Algorithms.- 0.5 Combinatorial Group Theory.- 1 Complex Analysis and Surface Topology.- 1.1 Riemann Surfaces.- 1.2 Nonorientable Surfaces.- 1.3 The Classification Theorem for Surfaces.- 1.4 Covering Surfaces.- 2 Graphs and Free Groups.- 2.1 Realization of Free Groups by Graphs.- 2.2 Realization of Subgroups.- 3 Foundations for the Fundamental Group.- 3.1 The Fundamental Group.- 3.2 The Fundamental Group of the Circle.- 3.3 Deformation Retracts.- 3.4 The SeifertVan Kampen Theorem.- 3.5 Direct Products.- 4 Fundamental Groups of Complexes.- 4.1 Poincaré's Method for Computing Presentations.- 4.2 Examples.- 4.3 Surface Complexes and Subgroup Theorems.- 5 Homology Theory and Abelianization.- 5.1 Homology Theory.- 5.2 The Structure Theorem for Finitely Generated Abelian Groups.- 5.3 Abelianization.- 6 Curves on Surfaces.- 6.1 Dehn's Algorithm.Table of Contents0 Introduction and Foundations.- 0.1 The Fundamental Concepts and Problems of Topology.- 0.2 Simplicial Complexes.- 0.3 The Jordan Curve Theorem.- 0.4 Algorithms.- 0.5 Combinatorial Group Theory.- 1 Complex Analysis and Surface Topology.- 1.1 Riemann Surfaces.- 1.2 Nonorientable Surfaces.- 1.3 The Classification Theorem for Surfaces.- 1.4 Covering Surfaces.- 2 Graphs and Free Groups.- 2.1 Realization of Free Groups by Graphs.- 2.2 Realization of Subgroups.- 3 Foundations for the Fundamental Group.- 3.1 The Fundamental Group.- 3.2 The Fundamental Group of the Circle.- 3.3 Deformation Retracts.- 3.4 The Seifert—Van Kampen Theorem.- 3.5 Direct Products.- 4 Fundamental Groups of Complexes.- 4.1 Poincaré’s Method for Computing Presentations.- 4.2 Examples.- 4.3 Surface Complexes and Subgroup Theorems.- 5 Homology Theory and Abelianization.- 5.1 Homology Theory.- 5.2 The Structure Theorem for Finitely Generated Abelian Groups.- 5.3 Abelianization.- 6 Curves on Surfaces.- 6.1 Dehn’s Algorithm.- 6.2 Simple Curves on Surfaces.- 6.3 Simplification of Simple Curves by Homeomorphisms.- 6.4 The Mapping Class Group of the Torus.- 7 Knots and Braids.- 7.1 Dehn and Schreier’s Analysis of the Torus Knot Groups.- 7.2 Cyclic Coverings.- 7.3 Braids.- 8 Three-Dimensional Manifolds.- 8.1 Open Problems in Three-Dimensional Topology.- 8.2 Polyhedral Schemata.- 8.3 Heegaard Splittings.- 8.4 Surgery.- 8.5 Branched Coverings.- 9 Unsolvable Problems.- 9.1 Computation.- 9.2 HNN Extensions.- 9.3 Unsolvable Problems in Group Theory.- 9.4 The Homeomorphism Problem.- Bibliography and Chronology.

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Book SynopsisManifolds over complete nonarchimedean fields together with notions like tangent spaces and vector fields form a convenient geometric language to express the basic formalism of p-adic analysis. The volume starts with a self-contained and detailed introduction to this language. This includes the discussion of spaces of locally analytic functions as topological vector spaces, important for applications in representation theory. The author then sets up the analytic foundations of the theory of p-adic Lie groups and develops the relation between p-adic Lie groups and their Lie algebras. The second part of the book contains, for the first time in a textbook, a detailed exposition of Lazard's algebraic approach to compact p-adic Lie groups, via his notion of a p-valuation, together with its application to the structure of completed group rings.Trade ReviewFrom the reviews:“The book is divided into two parts … . The author’s style of writing is elegant … . this is a demanding book, but a rewarding one … . any person who intends to work in this area will want to have it close at hand.” (Mark Hunacek, The Mathematical Gazette, Vol. 98 (541), March, 2014)“This book presents the foundations of the theory of p-adic Lie groups in a systematic and self-contained way. … Schneider’s book on p-adic Lie groups systematically develops the analytic theory of p-adic Lie groups and also Lazard’s algebraic approach to p-adic Lie groups. It is highly recommended.” (Dubravka Ban, Mathematical Reviews, Issue 2012 h)“The notion of a p-adic Lie group has been around for a while, but they have recently become more prominent in number theory and representation theory. … Schneider’s Grundlehren volume is an attempt to fill that gap by giving a systematic treatment of the subject. … this is a book to be welcomed and studied carefully by anyone who wants to learn about p-adic Lie theory.” (Fernando Q. Gouvêa, The Mathematical Association of America, August, 2011)“The book thoroughly discusses the analytic aspects of p-adic manifolds and p-adic lie groups. … this clearly written book by Schneider will be very useful … to all those interested learning the basic theory of p-adic groups or about the completed group ring of a p-adic group with number theoretical applications in mind.” (Bala Sury, Zentralblatt MATH, Vol. 1223, 2011)Table of ContentsIntroduction.- Part A: p-Adic Analysis and Lie Groups.- I.Foundations.- I.1.Ultrametric Spaces.- I.2.Nonarchimedean Fields.- I.3.Convergent Series.- I.4.Differentiability.- I.5.Power Series.- I.6.Locally Analytic Functions.- II.Manifolds.- II.7.Charts and Atlases.- II.8.Manifolds.- II.9.The Tangent Space.- II.10.The Topological Vector Space C^an(M,E), part 1.- II.11 Locally Convex K-Vector Spaces.- II.12 The Topological Vector Space C^an(M,E), part 2.- III.Lie Groups.- III.13.Definitions and Foundations.- III.14.The Universal Enveloping Algebra.- III.15.The Concept of Free Algebras.- III.16.The Campbell-Hausdorff Formula.- III.17.The Convergence of the Hausdorff Series.- III.18.Formal Group Laws.- Part B:The Algebraic Theory of p-Adic Lie Groups.- IV.Preliminaries.- IV.19.Completed Group Rings.- IV.20.The Example of the Group Z^d_p.- IV.21.Continuous Distributions.- IV.22.Appendix: Pseudocompact Rings.- V.p-Valued Pro-p-Groups.- V.23.p-Valuations.- V.24.The free Group on two Generators.- V.25.The Operator P.- V.26.Finite Rank Pro-p-Groups.- V.27.Compact p-Adic Lie Groups.- VI.Completed Group Rings of p-Valued Groups.- VI.28.The Ring Filtration.- VI.29.Analyticity.- VI.30.Saturation.- VII.The Lie Algebra.- VII.31.A Normed Lie Algebra.- VII.32.The Hausdorff Series.- VII.33.Rational p-Valuations and Applications.- VII.34.Coordinates of the First and of the Second Kind.- References.- Index.

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Book SynopsisThis work seeks to offer a concise introduction to geometric group theory - a method for studying infinite groups via their intrinsic geometry. Basic combinatorial and geometric group theory is presented, along with research on the growth of groups, and exercises and problems.

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Book SynopsisThis volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, Lie Algebras, the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, The Structure of Locally Compact Groups, deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952.

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Book SynopsisSpecific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering.Trade Review"The book is self-contained.It remains a good and solid introduction to this subject."Nieuw Archief voor Wiskunde, March 2001 "... This book takes the reader on one of the greatest journeys in modern mathematics that has as its roots a subject that is more than 300 years old. Armed with this knowledge a reader is ready to pursue numerous topics of active mathematical research, from the more pure domains of symplectic geometry and topology to the geometric analysis of the limitless supply of examples from mechanics."Newsletter of the Newzealand Mathematical Society, No. 81, April 2001 Second Edition J.E. Marsden and T.S. Ratiu Introduction to Mechanics and Symmetry A Basic Exposition of Classical Mechanical Systems "As the name of the book implies, a consistent theme running through the book is that of symmetry. Indeed the latter half of the book focuses on Poisson manifolds, momentum maps, Lie-Poisson reduction, co-adjoint orbits and the integrability of the rigid body. The discussion of reduction must be the most comprehensive yet given. A pleasant feature of the book is that most of the theory that relates to finite-dimensional mechanical systems is illustrated concretely in terms of local coordinates, thereby making the book accessible even to beginners in the field."—MATHEMATICAL REVIEWSTable of ContentsPreface * About the Authors * 1 Introduction and Overview * 2 Hamiltonian Systems on Linear Symplectic Spaces * 3 An Introduction to Infinite-Dimensional Systems * 4 Manifolds, Vector Fields, and Differential Forms * 5 Hamiltonian Systems on Symplectic Manifolds * 6 Cotangent Bundles * 7 Lagrangian Mechanics * 8 Variational Principles, Constraints, and Rotating Systems * 9 An Introduction to Lie Groups * 10 Poisson Manifolds * 11 Momentum Maps *12 Computation and Properties of Momentum Maps * 13 Lie-Poisson and Euler-Poincare Reduction * 14 Coadjoint Orbits * 15 The Free Rigid Body * References

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Book SynopsisThis book presents ordinary differential equations based on Lie group analysis and related invariance principles. The author provides students and teachers with a text for one-semester undergraduate and graduate courses that spans a variety of topics, from the basic theory through to applications.Trade Review"…this is the first self-contained university text on ordinary differential equations…" (Zentralblatt Math, Vol.1047, No.22, 2004)Table of ContentsIntroduction to Differential Equations. Transformation Groups. Lie Group Analysis of Ordinary Differential Equations. Brief on Lie Algebras. First Order Differential Equations. Integration of Second Order Equations. Basic Theory of Linear Equations. Nonlinear Second Order Equations. Integration of Third Order Equations. Nonlinear Superposition Principle. Index.

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Book SynopsisAs algebra becomes more widely used in a variety of applications and computers are developed to allow efficient calculations in the field, so there becomes a need for new techniques to further this area of research. Grobner Bases is one topic which has recently become a very popular and important area of modern algebra.Table of ContentsRings. Fields, and Ideals. Monomial Ideals. Gröbner Bases. Algebraic Sets. Primary Decomposition of Ideals. Solving Systems of Polynomial Equations. Applications of Gröbner Bases. Homogeneous Algebras. Projective Varieties. The Associated Graded Ring. Hilbert Series. Variations of Gröbner Bases. Improvements to Buchberger's Algorithm. Software. Hints to Some Exercises. Answers to Exercises. Bibliography. Index.

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Book SynopsisDevelops a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. This book formulates simple general conditions on the spectral theory of the group and the regularity of the averaging sets, which suffice to guarantee convergence to the ergodic mean.Table of ContentsPreface vii 0.1 Main objectives vii 0.2 Ergodic theory and amenable groups viii 0.3 Ergodic theory and nonamenable groups x Chapter 1. Main results: Semisimple Lie groups case 1 1.1 Admissible sets 1 1.2 Ergodic theorems on semisimple Lie groups 2 1.3 The lattice point-counting problem in admissible domains 4 1.4 Ergodic theorems for lattice subgroups 6 1.5 Scope of the method 8 Chapter 2. Examples and applications 11 2.1 Hyperbolic lattice points problem 11 2.2 Counting integral unimodular matrices 12 2.3 Integral equivalence of general forms 13 2.4 Lattice points in S-algebraic groups 15 2.5 Examples of ergodic theorems for lattice actions 16 Chapter 3. Definitions, preliminaries, and basic tools 19 3.1 Maximal and exponential-maximal inequalities 19 3.2 S-algebraic groups and upper local dimension 21 3.3 Admissible and coarsely admissible sets 21 3.4 Absolute continuity and examples of admissible averages 23 3.5 Balanced and well-balanced families on product groups 26 3.6 Roughly radial and quasi-uniform sets 27 3.7 Spectral gap and strong spectral gap 29 3.8 Finite-dimensional subrepresentations 30 Chapter 4. Main results and an overview of the proofs 33 4.1 Statement of ergodic theorems for S-algebraic groups 33 4.2 Ergodic theorems in the absence of a spectral gap: overview 35 4.3 Ergodic theorems in the presence of a spectral gap: overview 38 4.4 Statement of ergodic theorems for lattice subgroups 40 4.5 Ergodic theorems for lattice subgroups: overview 42 4.6 Volume regularity and volume asymptotics: overview 44 Chapter 5. Proof of ergodic theorems for S-algebraic groups 47 5.1 Iwasawa groups and spectral estimates 47 5.2 Ergodic theorems in the presence of a spectral gap 50 5.3 Ergodic theorems in the absence of a spectral gap, I 56 5.4 Ergodic theorems in the absence of a spectral gap, II 57 5.5 Ergodic theorems in the absence of a spectral gap, III 60 5.6 The invariance principle and stability of admissible averages 67 Chapter 6. Proof of ergodic theorems for lattice subgroups 71 6.1 Induced action 71 6.2 Reduction theorems 74 6.3 Strong maximal inequality 75 6.4 Mean ergodic theorem 78 6.5 Pointwise ergodic theorem 83 6.6 Exponential mean ergodic theorem 84 6.7 Exponential strong maximal inequality 87 6.8 Completion of the proofs 90 6.9 Equidistribution in isometric actions 91 Chapter 7. Volume estimates and volume regularity 93 7.1 Admissibility of standard averages 93 7.2 Convolution arguments 98 7.3 Admissible, well-balanced, and boundary-regular families 101 7.4 Admissible sets on principal homogeneous spaces 105 7.5 Tauberian arguments and Holder continuity 107 Chapter 8. Comments and complements 113 8.1 Lattice point-counting with explicit error term 113 8.2 Exponentially fast convergence versus equidistribution 115 8.3 Remark about balanced sets 116 Bibliography 117 Index 121

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Book SynopsisMumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book provides a comprehensive exploration of Mumford-Tate groups and domains.Trade Review"The brilliance of the results and their broad spectrum of their applications makes this book an outstanding piece. Yet, there is more to write and to develop: the authors suggest the existence of future lines of research for a next book."--Jonathan Sanchez Hernandez, European Mathematical SocietyTable of ContentsIntroduction 1 I Mumford-Tate Groups 28 I.A Hodge structures 28 I.B Mumford-Tate groups 32 I.C Mixed Hodge structures and their Mumford-Tate groups 38 II Period Domains and Mumford-Tate Domains 45 II.A Period domains and their compact duals 45 II.B Mumford-Tate domains and their compact duals 55 II.C Noether-Lefschetz loci in period domains 61 III The Mumford-Tate Group of a Variation of Hodge Structure 67 III.A The structure theorem for variations of Hodge structures 69 III.B An application of Mumford-Tate groups 78 III.C Noether-Lefschetz loci and variations of Hodge structure .81 IV Hodge Representations and Hodge Domains 85 IV.A Part I: Hodge representations 86 IV.B The adjoint representation and characterization of which weights give faithful Hodge representations 109 IV.C Examples: The classical groups 117 IV.D Examples: The exceptional groups 126 IV.E Characterization of Mumford-Tate groups 132 IV.F Hodge domains 149 IV.G Mumford-Tate domains as particular homogeneous complex manifolds 168 Appendix: Notation from the structure theory of semisimple Lie algebras 179 V Hodge Structures with Complex Multiplication 187 V.A Oriented number fields 189 V.B Hodge structures with special endomorphisms 193 V.C A categorical equivalence 196 V.D Polarization and Mumford-Tate groups . 198 V.E An extended example 202 V.F Proofs of Propositions V.D.4 and V.D.5 in the Galois case 209 VI Arithmetic Aspects of Mumford-Tate Domains 213 VI.A Groups stabilizing subsets of D 215 VI.B Decomposition of Noether-Lefschetz into Hodge orientations 219 VI.C Weyl groups and permutations of Hodge orientations 231 VI.D Galois groups and fields of definition 234 Appendix: CM points in unitary Mumford-Tate domains 239 VII Classification of Mumford-Tate Subdomains 240 VII.A A general algorithm 240 VII.B Classification of some CM-Hodge structures 243 VII.C Determination of sub-Hodge-Lie-algebras 246 VII.D Existence of domains of type IV(f) 251 VII.E Characterization of domains of type IV(a) and IV(f) 253 VII.F Completion of the classification for weight 3 256 VII.G The weight 1 case 260 VII.H Algebro-geometric examples for the Noether-Lefschetzlocus types 265 VIII Arithmetic of Period Maps of Geometric Origin 269 VIII.A Behavior of fields of definition under the period Map -- image and preimage 270 VIII.B Existence and density of CM points in motivic VHS 275 Bibliography 277 Index 287

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Book SynopsisMumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book provides a comprehensive exploration of Mumford-Tate groups and domains.Trade Review"The brilliance of the results and their broad spectrum of their applications makes this book an outstanding piece. Yet, there is more to write and to develop: the authors suggest the existence of future lines of research for a next book."--Jonathan Sanchez Hernandez, European Mathematical SocietyTable of ContentsIntroduction 1 I Mumford-Tate Groups 28 I.A Hodge structures 28 I.B Mumford-Tate groups 32 I.C Mixed Hodge structures and their Mumford-Tate groups 38 II Period Domains and Mumford-Tate Domains 45 II.A Period domains and their compact duals 45 II.B Mumford-Tate domains and their compact duals 55 II.C Noether-Lefschetz loci in period domains 61 III The Mumford-Tate Group of a Variation of Hodge Structure 67 III.A The structure theorem for variations of Hodge structures 69 III.B An application of Mumford-Tate groups 78 III.C Noether-Lefschetz loci and variations of Hodge structure .81 IV Hodge Representations and Hodge Domains 85 IV.A Part I: Hodge representations 86 IV.B The adjoint representation and characterization of which weights give faithful Hodge representations 109 IV.C Examples: The classical groups 117 IV.D Examples: The exceptional groups 126 IV.E Characterization of Mumford-Tate groups 132 IV.F Hodge domains 149 IV.G Mumford-Tate domains as particular homogeneous complex manifolds 168 Appendix: Notation from the structure theory of semisimple Lie algebras 179 V Hodge Structures with Complex Multiplication 187 V.A Oriented number fields 189 V.B Hodge structures with special endomorphisms 193 V.C A categorical equivalence 196 V.D Polarization and Mumford-Tate groups . 198 V.E An extended example 202 V.F Proofs of Propositions V.D.4 and V.D.5 in the Galois case 209 VI Arithmetic Aspects of Mumford-Tate Domains 213 VI.A Groups stabilizing subsets of D 215 VI.B Decomposition of Noether-Lefschetz into Hodge orientations 219 VI.C Weyl groups and permutations of Hodge orientations 231 VI.D Galois groups and fields of definition 234 Appendix: CM points in unitary Mumford-Tate domains 239 VII Classification of Mumford-Tate Subdomains 240 VII.A A general algorithm 240 VII.B Classification of some CM-Hodge structures 243 VII.C Determination of sub-Hodge-Lie-algebras 246 VII.D Existence of domains of type IV(f) 251 VII.E Characterization of domains of type IV(a) and IV(f) 253 VII.F Completion of the classification for weight 3 256 VII.G The weight 1 case 260 VII.H Algebro-geometric examples for the Noether-Lefschetzlocus types 265 VIII Arithmetic of Period Maps of Geometric Origin 269 VIII.A Behavior of fields of definition under the period Map -- image and preimage 270 VIII.B Existence and density of CM points in motivic VHS 275 Bibliography 277 Index 287

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Book SynopsisProvides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. This title delves into arguments originating in Nori's work that have been further developed by others.Trade Review"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH "[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews ClippingsTable of ContentsPreface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163

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Book SynopsisProvides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. This title delves into arguments originating in Nori's work that have been further developed by others.Trade Review"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH "[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews ClippingsTable of ContentsPreface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163

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Book SynopsisProvides an introduction to Hodge theory - one of the central and most vibrant areas of contemporary mathematics - from leading specialists on the subject. This book includes topics that range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps.Trade Review"Charles and Schnell's chapter beautifully surveys the theory of absolute Hodge classes, giving in particular a complete proof of Deligne's theorem on absolute Hodge classes on abelian varieties... A welcome addition to the literature and should be useful to both graduate students and researchers working in Hodge theory."--Dan Petersen, MathSciNetTable of Contents*FrontMatter, pg. i*Contributors, pg. v*Contents, pg. vii*Preface, pg. xv*Chapter One. Introduction to Kahler Manifolds, pg. 1*Chapter Two. From Sheaf Cohomology to the Algebraic de Rham Theorem, pg. 70*Chapter Three. Mixed Hodge Structures, pg. 123*Chapter Four. Period Domains and Period Mappings, pg. 217*Chapter Five. The Hodge Theory of Maps, pg. 257*Chapter Six The Hodge Theory of Maps, pg. 273*Chapter Seven. Introduction to Variations of Hodge Structure, pg. 297*Chapter Eight. Variations of Mixed Hodge Structure, pg. 333*Chapter Nine. Lectures on Algebraic Cycles and Chow Groups, pg. 410*Chapter Ten. The Spread Philosophy in the Study of Algebraic Cycles, pg. 449*Chapter Eleven. Notes on Absolute Hodge Classes, pg. 469*Chapter Twelve. Shimura Varieties: A Hodge-Theoretic Perspective, pg. 531*Bibliography, pg. 574*Index, pg. 577

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Book SynopsisTrade Review"The authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts."--Gerd Faltings, Zentralblatt MATHTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Foreword, pg. ix*Chapter I. Representations of the fundamental group and the torsor of deformations. An overview, pg. 1*Chapter II. Representations of the fundamental group and the torsor of deformations. Local study, pg. 27*Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects, pg. 179*Chapter IV. Cohomology of Higgs isocrystals, pg. 307*Chapter V. Almost etale coverings, pg. 449*Chapter VI. Covanishing topos and generalizations, pg. 485*Facsimile : A p-adic Simpson correspondence, pg. 577*Bibliography, pg. 595*Indexes, pg. 599

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