This volume presents a completely self-contained introduction to the elaborate theory of locally compact quantum groups, bringing the reader to the frontiers of present-day research. The exposition includes a substantial amount of material on functional analysis and operator algebras, subjects which in themselves have become increasingly important with the advent of quantum information theory. In particular, the rather unfamiliar modular theory of weights plays a crucial role in the theory, due to the presence of 'Haar integrals' on locally compact quantum groups, and is thus treated quite extensively The topics covered are developed independently, and each can serve either as a separate course in its own right or as part of a broader course on locally compact quantum groups. The second part of the book covers crossed products of coactions, their relation to subfactors and other types of natural products such as cocycle bicrossed products, quantum doubles and doublecrossed products. Induced corepresentations, Galois objects and deformations of coactions by cocycles are also treated. Each section is followed by a generous supply of exercises. To complete the book, an appendix is provided on topology, measure theory and complex function theory.

Today Lie group theoretical approach to differential equations has been extended to new situations and has become applicable to the majority of equations that frequently occur in applied sciences. Newly developed theoretical and computational methods are awaiting application. Students and applied scientists are expected to understand these methods. Volume 3 and the accompanying software allow readers to extend their knowledge of computational algebra.
Written by the world's leading experts in the field, this up-to-date sourcebook covers topics such as Lie-BAcklund, conditional and non-classical symmetries, approximate symmetry groups for equations with a small parameter, group analysis of differential equations with distributions, integro-differential equations, recursions, and symbolic software packages. The text provides an ideal introduction to modern group analysis and addresses issues to both beginners and experienced researchers in the application of Lie group methods.

This volume features seventeen extended conference abstracts corresponding to selected talks given by participants at the CRM research program "Automorphisms of Free Groups: Algorithms, Geometry and Dynamics", which took place at the Centre de Recerca Matematica in Barcelona in fall 2012. Most of them are short articles giving preliminary presentations of new results not yet published in regular research journals. The articles are the result from a direct collaboration among active researchers in the area after working in a dynamic and productive atmosphere. The book is intended for established researchers in the area of Group Theory, as well as for PhD and postdoc students who wish to learn more about the latest advances in this active area of research.

This book presents an introduction to the representation theory of wreath products of finite groups and harmonic analysis on the corresponding homogeneous spaces. The reader will find a detailed description of the theory of induced representations and Clifford theory, focusing on a general formulation of the little group method. This provides essential tools for the determination of all irreducible representations of wreath products of finite groups. The exposition also includes a detailed harmonic analysis of the finite lamplighter groups, the hyperoctahedral groups, and the wreath product of two symmetric groups. This relies on the generalised Johnson scheme, a new construction of finite Gelfand pairs. The exposition is completely self-contained and accessible to anyone with a basic knowledge of representation theory. Plenty of worked examples and several exercises are provided, making this volume an ideal textbook for graduate students. It also represents a useful reference for more experienced researchers.

This 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 Poincare-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.

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The Heisenberg group plays an important role in several branches of mathematics, such as representation theory, partial differential equations, number theory, several complex variables and quantum mechanics. This monograph deals with various aspects of harmonic analysis on the Heisenberg group, which is the most commutative among the non-commutative Lie groups, and hence gives the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The aim of this text is to demonstrate how the standard results of abelian harmonic analysis take shape in the non-abelian setup of the Heisenberg group. Thangavelu's exposition is clear and well developed, and leads to several problems worthy of further consideration. Any reader who is interested in pursuing research on the Heisenberg group will find this unique and self-contained text invaluable.

"These volumes collect almost all of the research and expository papers of J.-P. Serre published in mathematical journals through 1984, as well as some of his seminar reports, and a few items not previously published. .... Throughout his writings, Serre has liberally sprinkled open questions and conjectures. Most endnotes list subsequent progress made on these questions or improvements to the main results of the papers. Some make additional comments, and a few are corrections. These endnotes alone justify the publication of the collected works. Serre is one of the masters of mathematical exposition...." --James Milne, University of Michigan, in Math Reviews

This self-contained, comprehensive book tackles the principal problems and advanced questions of probability theory and random processes in 22 chapters, presented in a logical order but also suitable for dipping into. They include both classical and more recent results, such as large deviations theory, factorization identities, information theory, stochastic recursive sequences. The book is further distinguished by the inclusion of clear and illustrative proofs of the fundamental results that comprise many methodological improvements aimed at simplifying the arguments and making them more transparent. The importance of the Russian school in the development of probability theory has long been recognized. This book is the translation of the fifth edition of the highly successful Russian textbook. This edition includes a number of new sections, such as a new chapter on large deviation theory for random walks, which are of both theoretical and applied interest. The frequent references to Russian literature throughout this work lend a fresh dimension and make it an invaluable source of reference for Western researchers and advanced students in probability related subjects. Probability Theory will be of interest to both advanced undergraduate and graduate students studying probability theory and its applications. It can serve as a basis for several one-semester courses on probability theory and random processes as well as self-study.

The theory of R-trees is a well-established and important area of geometric group theory and in this book the authors introduce a construction that provides a new perspective on group actions on R-trees. They construct a group RF(G), equipped with an action on an R-tree, whose elements are certain functions from a compact real interval to the group G. They also study the structure of RF(G), including a detailed description of centralizers of elements and an investigation of its subgroups and quotients. Any group acting freely on an R-tree embeds in RF(G) for some choice of G. Much remains to be done to understand RF(G), and the extensive list of open problems included in an appendix could potentially lead to new methods for investigating group actions on R-trees, particularly free actions. This book will interest all geometric group theorists and model theorists whose research involves R-trees.

This 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.

The dual space of a locally compact group G consists of the equivalence classes of irreducible unitary representations of G. This book provides a comprehensive guide to the theory of induced representations and explains its use in describing the dual spaces for important classes of groups. It introduces various induction constructions and proves the core theorems on induced representations, including the fundamental imprimitivity theorem of Mackey and Blattner. An extensive introduction to Mackey analysis is applied to compute dual spaces for a wide variety of examples. Fell's contributions to understanding the natural topology on the dual are also presented. In the final two chapters, the theory is applied in a variety of settings including topological Frobenius properties and continuous wavelet transforms. This book will be useful to graduate students seeking to enter the area as well as experts who need the theory of unitary group representations in their research.

This volume, dedicated to Bertram Kostant on the occasion of his 65th birthday, is a collection of 22 invited papers by leading mathematicians working in Lie theory, geometry, algebra, and mathematical physics. Kostant's fundamental work in all these areas has provided deep new insights and connections, and has created new fields of research. The papers gathered here present original research articles as well as expository papers, broadly reflecting the range of Kostant's work.

This is the first volume of a revised edition of P.M. Cohn's classic three-volume text Algebra, widely regarded as one of the most outstanding introductory algebra textbooks. This volume covers the important results of algebra. Readers should have some knowledge of linear algebra, groups and fields, although all the essential facts and definitions are recalled.

This standard reference on applications of invariant theory to the construction of moduli spaces is a systematic exposition of the geometric aspects of classical theory of polynomial invariants. This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by Professor Frances Kirwan. It includes a fully updated bibliography of work in this area.

The aim of this book is the classification of symplectic amalgams - structures which are intimately related to the finite simple groups. In all there sixteen infinite families of symplectic amalgams together with 62 more exotic examples. The classification touches on many important aspects of modern group theory: * p-local analysis * the amalgam method * representation theory over finite fields; and * properties of the finite simple groups. The account is for the most part self-contained and the wealth of detail makes this book an excellent introduction to these recent developments for graduate students, as well as a valuable resource and reference for specialists in the area.

This book is intended for graduate students in Physics. It starts with a discussion of angular momentum and rotations in terms of the orthogonal group in three dimensions and the unitary group in two dimensions and goes on to deal with these groups in any dimensions. All representations of su(2) are obtained and the Wigner-Eckart theorem is discussed. Casimir operators for the orthogonal and unitary groups are discussed. The exceptional group G2 is introduced as the group of automorphisms of octonions. The symmetric group is used to deal with representations of the unitary groups and the reduction of their Kronecker products. Following the presentation of Cartan's classification of semisimple algebras Dynkin diagrams are described. The book concludes with space-time groups - the Lorentz, Poincare and Liouville groups - and a derivation of the energy levels of the non-relativistic hydrogen atom in n space dimensions.

A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.

The last two decades have produced tremendous developments in the mathematical theory of wavelets and their great variety of applications in science and engineering. Wavelets allow complex information, such as music, speech, images and patterns to be decomposed into an elementary form called 'building blocks' at different positions and scales. The information is reconstructed with high precision. In an effort to acquaint researchers in applied mathematics, physics, statistics, computer science, and engineering and to stimulate further research, a regional research convergence was organized at the University of Central Florida in May 1998. Many distinguished applied mathematicians and engineering scientists participated in the conference and presented a digest of recent developments, open questions and unsolved problems in this rapidly growing and important field. As a follow-up project, this book has developed from chapters written by renowned mathematicians and engineering scientists who have important contribution to the subject of wavelets, wavelet transforms, and time-frequency signal analysis.The carefully selected chapters in this new text will appeal to the reader interested in a broad perspective of wavelet analysis and time- frequency signal analysis. Wavelet Transforms and Time-Frequency Signal Analysis brings together recent developments in theory and applications of wavelet transforms that are likely to determine fruitful directions for future advanced study and research. The book is designed as a new source for modern topics dealing with wavelets, wavelet transforms, time-frequency signal analysis and other applications for future development of this new, important and useful subject for mathematics, science and engineering.

"An excellent up-to-date introduction to the theory of groups. It is general yet comprehensive, covering various branches of group theory. The 15 chapters contain the following main topics: free groups and presentations, free products, decompositions, Abelian groups, finite permutation groups, representations of groups, finite and infinite soluble groups, group extensions, generalizations of nilpotent and soluble groups, finiteness properties." --ACTA SCIENTIARUM MATHEMATICARUM

The British mathematician William Burnside (1852 1927) and Ferdinand Georg Frobenius (1849 1917), Professor at Zurich and Berlin universities, are considered to be the founders of the modern theory of finite groups. Not only did Burnside prove many important theorems, but he also laid down lines of research for the next hundred years: two Fields Medals have been awarded for work on problems suggested by him. The Theory of Groups of Finite Order, originally published in 1897, was the first major textbook on the subject. The 1911 second edition (reissued here) contains an account of Frobenius's character theory, and remained the standard reference for many years.

The central theme of this monograph is the relation between the structure of a group and the structure of its lattice of subgroups. Since the first papers on this topic have appeared, notably those of BAER and ORE, a large body of literature has grown up around this theory, and it is our aim to give a picture of the present state of this theory. To obtain a systematic treatment of the subject quite a few unpublished results of the author had to be included. On the other hand, it is natural that we could not reproduce every detail and had to treat some parts some wh at sketchily. We have tried to make this report as self-contained as possible. Accordingly we have given some proofs in considerable detail, though of course it is in the nature of such areport that many proofs have to be omitted or can only be given in outline. Similarly references to the concepts and theorems used are almost exclusively references to standard works like BIRKHOFF [lJ and ZASSENHAUS [lJ. The author would like to express his sincere gratitude to Professors REINHOLD BAER and DONALD G. HIGMAN for their kindness in giving hirn many valuable suggestions. His thanks are also due to Dr. NOBORU ITo who, during stimulating conversations, contributed many useful ideas. Urbana, May, 1956. M. Suzuki. Contents.

Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science. The three parts of this first volume of a two-volume set deal with the stability problem for smooth mappings, critical points of smooth functions, and caustics and wave front singularities. The second volume describes the topological and algebro-geometrical aspects of the theory: monodromy, intersection forms, oscillatory integrals, asymptotics, and mixed Hodge structures of singularities.

The first volume has been adapted for the needs of non-mathematicians, presupposing a limited mathematical background and beginning at an elementary level. With this foundation, the book's sophisticated development permits readers to explore more applications than previous books on singularities."

Number theory currently has at least three different perspectives on non-abelian phenomena: the Langlands programme, non-commutative Iwasawa theory and anabelian geometry. In the second half of 2009, experts from each of these three areas gathered at the Isaac Newton Institute in Cambridge to explain the latest advances in their research and to investigate possible avenues of future investigation and collaboration. For those in attendance, the overwhelming impression was that number theory is going through a tumultuous period of theory-building and experimentation analogous to the late 19th century, when many different special reciprocity laws of abelian class field theory were formulated before knowledge of the Artin-Takagi theory. Non-abelian Fundamental Groups and Iwasawa Theory presents the state of the art in theorems, conjectures and speculations that point the way towards a new synthesis, an as-yet-undiscovered unified theory of non-abelian arithmetic geometry.

Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory.

The present volume is the second in a two-volume set entitled "Singularities of Differentiable Maps." While the first volume, subtitled "Classification of Critical Points" and originallypublishedas Volume82 in the Monographs in Mathematics series, contained the zoology of differentiable maps, that is, it was devoted to a description of what, where, and how singularities could be encountered, this second volume concentrates on elements of theanatomy and physiology of singularities of differentiable functions. The questions considered are about the structure of singularities and how they function."