The 12 lectures presented in Representation Theories and Algebraic Geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, Hecke algebras, restricted Lie algebras, and their companions. This interplay has been extensively exploited during recent years, resulting in great progress in these representation theories. Conversely, a great stimulus has been given to the development of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology. The range of topics covered is wide, from equivariant Chow groups, decomposition classes and Schubert varieties, multiplicity free actions, convolution algebras, standard monomial theory, and canonical bases, to annihilators of quantum Verma modules, modular representation theory of Lie algebras and combinatorics of representation categories of Harish-Chandra modules.

Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title Lie Theory, feature survey work and original results by well-established researchers in key areas of semisimple Lie theory.

Harmonic Analysis on Symmetric Spacesa "General Plancherel Theorems presents extensive surveys by E.P. van den Ban, H. Schlichtkrull, and P. Delorme of the spectacular progress over the past decade in deriving the Plancherel theorem on reductive symmetric spaces.

Van den Bana (TM)s introductory chapter explains the basic setup of a reductive symmetric space along with a careful study of the structure theory, particularly for the ring of invariant differential operators for the relevant class of parabolic subgroups. Advanced topics for the formulation and understanding of the proof are covered, including Eisenstein integrals, regularity theorems, Maassa "Selberg relations, and residue calculus for root systems. Schlichtkrull provides a cogent account of the basic ingredients in the harmonic analysis on a symmetric space through the explanation and definition of the Paleya "Wiener theorem. Approaching the Plancherel theorem through an alternative viewpoint, the Schwartz space, Delorme bases his discussion and proof on asymptotic expansions of eigenfunctions and the theory of intertwining integrals.

Well suited for both graduate students and researchers in semisimple Lie theory and neighboring fields, possibly even mathematical cosmology, Harmonic Analysis on Symmetric Spacesa "General Plancherel Theorems provides abroad, clearly focused examination of semisimple Lie groups and their integral importance and applications to research in many branches of mathematics and physics. Knowledge of basic representation theory of Lie groups as well as familiarity with semisimple Lie groups, symmetric spaces, and parabolic subgroups is required.

A consistent and near complete survey of the important progress made in the field over the last few years, with the main emphasis on the rigidity method and its applications. Among others, this monograph presents the most successful existence theorems known and construction methods for Galois extensions as well as solutions for embedding problems combined with a collection of the existing Galois realizations.

This text offers an introduction to error-correcting linear codes for researchers and graduate students in mathematics, computer science and engineering. The book differs from other standard texts in its emphasis on the classification of codes by means of isometry classes. The relevant algebraic are developed rigorously. Cyclic codes are discussed in great detail. In the last four chapters these isometry classes are enumerated, and representatives are constructed algorithmically.

The second volume of this work contains Parts 2 and 3 of the "Handbook of Coding Theory". Part 2, "Connections", is devoted to connections between coding theory and other branches of mathematics and computer science. Part 3, "Applications", deals with a variety of applications for coding.

It became more and more usual, from, say, the 1970s, for each book on Module Theory, to point out and prove some (but in no more than 15 to 20 pages) generalizations to (mostly modular) lattices. This was justified by the nowadays widely accepted perception that the structure of a module over a ring is best understood in terms of the lattice struc ture of its submodule lattice. Citing Louis H. Rowen "this important example (the lattice of all the submodules of a module) is the raison d'etre for the study of lattice theory by ring theorists". Indeed, many module-theoretic results can be proved by using lattice theory alone. The purpose of this book is to collect and present all and only the results of this kind, although for this purpose one must develop some significant lattice theory. The results in this book are of the following categories: the folklore of Lattice Theory (to be found in each Lattice Theory book), module theoretic results generalized in (modular, and possibly compactly gen erated) lattices (to be found in some 6 to 7 books published in the last 20 years), very special module-theoretic results generalized in lattices (e. g. , purity in Chapter 9 and several dimensions in Chapter 13, to be found mostly in [27], respectively, [34] and [18]) and some new con cepts (e. g.

Symmetry is one of the most important organising principles in the natural sciences. The mathematical theory of symmetry has long been associated with group theory, but it is a basic premise of this book that there are aspects of symmetry which are more faithfully represented by a generalization of groups called inverse semigroups. The theory of inverse semigroups is described from its origins in the foundations of differential geometry through to its most recent applications in combinatorial group theory, and the theory tilings.

This book is concerned with the structure of linear semigroups, that is, subsemigroups of the multiplicative semigroup Mn(K) of n x n matrices over a field K (or, more generally, skew linear semigroups - if K is allowed to be a division ring) and its applications to certain problems on associative algebras, semigroups and linear representations. It is motivated by several recent developments in the area of linear semigroups and their applications. It summarizes the state of knowledge in this area, presenting the results for the first time in a unified form. The book's point of departure is a structure theorem, which allows the use of powerful techniques of linear groups. Certain aspects of a combinatorial nature, connections with the theory of linear representations and applications to various problems on associative algebras are also discussed.

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature - from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world. With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Grobner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics."

Orthogonal designs have proved fundamental to constructing code division multiple antenna systems for more efficient mobile communications. Starting with basic theory, this book develops the algebra and combinatorics to create new communications modes. Intended primarily for researchers, it is also useful for graduate students wanting to understand some of the current communications coding theories.

For courses in Prealgebra. Trusted author content. Thoughtful innovation. In this revision of the Bittinger Paperback Worktext Series, the Bittinger author team brings their extensive experience to developmental math courses, paired with thoughtful integration of technology and content. The Bittinger Series enables students to get the most out of their MyLab (TM) Math course through an updated learning path, new review videos, and engaging new exercises that offer the support they need, when they need it. Bittinger offers superior content written by author-educators, tightly integrated with MyLab Math - the #1 choice in digital learning. Bringing the authors' voices and their approach into the MyLab course encourages student motivation and engagement, while reinforcing their understanding of the skills and concepts they need to master algebra. Also available with MyLab Math By combining trusted author content with digital tools and a flexible platform, MyLab Math personalizes the learning experience and improves results for each student. Note: You are purchasing a standalone product; MyLab Math does not come packaged with this content. Students, if interested in purchasing this title with MyLab Math, ask your instructor to confirm the correct package ISBN and Course ID. Instructors, contact your Pearson representative for more information. If you would like to purchase both the physical text and MyLab Math, search for: 0135218284 / 9780135218280 Prealgebra Plus MyLab Math with Pearson eText - Access Card Package Package consists of: 0135182565 / 9780135182567 Prealgebra 0135184371 / 9780135184370 MyLab Math with Pearson eText - Standalone Access Card - for Prealgebra

This book, an outgrowth of the author¿s lectures at the University of California at Berkeley, is intended as a textbook for a one-semester course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semisimple rings, Jacobson¿s theory of the radical, representation theory of groups and algebras, prime and semiprime rings, local and semilocal rings, perfect and semiperfect rings, etc. By aiming the level of writing at the novice rather than the connoisseur and by stressing the role of examples and motivation, the author has produced a text that is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition.

This book concentrates on the topic of evaluation of Jacobians in some specific linear as well as nonlinear matrix transformations, in the real and complex cases, which are widely applied in the statistical, physical, engineering, biological and social sciences. It aims to develop some techniques systematically so that anyone with a little exposure to multivariable calculus can easily follow the steps and understand the various methods by which the Jacobians in complicated matrix transformations are evaluated. The material is developed slowly, with lots of worked examples, aimed at self-study. Some exercises are also given, at the end of each section.The book is a valuable reference for statisticians, engineers, physicists, econometricians, applied mathematicians and people working in many other areas. It can be used for a one-semester graduate level course on Jacobians and functions of matrix argument.

The Kronecker product of matrices plays a central role in mathematics and in applications found in engineering and theoretical physics. These applications are signal processing, statistical physics, quantum groups and quantum computers. This book provides a comprehensive introduction to the Kronecker product of matrices together with its software implementation in C++ using an object-oriented design.

A completely reworked new edition of this superb textbook. This key work is geared to the needs of the graduate student. It covers, with proofs, the usual major branches of groups, rings, fields, and modules. Its inclusive approach means that all of the necessary areas are explored, while the level of detail is ideal for the intended readership. The text tries to promote the conceptual understanding of algebra as a whole, doing so with a masterful grasp of methodology. Despite the abstract subject matter, the author includes a careful selection of important examples, together with a detailed elaboration of the more sophisticated, abstract theories.

The book deals with algorithmic problems related to binary quadratic forms. It uniquely focuses on the algorithmic aspects of the theory. The book introduces the reader to important areas of number theory such as diophantine equations, reduction theory of quadratic forms, geometry of numbers and algebraic number theory. The book explains applications to cryptography and requires only basic mathematical knowledge. The author is a world leader in number theory.

In his 1974 seminal paper 'Elliptic modules', V G Drinfeld introduced objects into the arithmetic geometry of global function fields which are nowadays known as 'Drinfeld Modules'. They have many beautiful analogies with elliptic curves and abelian varieties. They study of their moduli spaces leads amongst others to explicit class field theory, Jacquet-Langlands theory, and a proof of the Shimura-Taniyama-Weil conjecture for global function fields.This book constitutes a carefully written instructional course of 12 lectures on these subjects, including many recent novel insights and examples. The instructional part is complemented by research papers centering around class field theory, modular forms and Heegner points in the theory of global function fields.The book will be indispensable for everyone who wants a clear view of Drinfeld's original work, and wants to be informed about the present state of research in the theory of arithmetic geometry over function fields.

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. It includes proofs of the main basic results (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an introduction to height functions and Lehmer's problem, several proofs of Baker's theorem as well as explicit measures of linear independence of logarithms. An original feature is the systematic use, in proofs, of Laurent's interpolation determinants. The most general result is the so-called Theorem of the Linear Subgroup, an effective version of which is also included. It yields new results of simultaneous approximation and of algebraic independence. Two chapters written by D. Roy provide complete and at the same time simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

This book, an abridgment of Volumes I and II of the highly respected Group Theory in Physics, presents a carefully constructed introduction to group theory and its applications in physics. The book provides anintroduction to and description of the most important basic ideas and the role that they play in physical problems. The clearly written text contains many pertinent examples that illustrate the topics, even for those with no background in group theory.
This work presents important mathematical developments to theoretical physicists in a form that is easy to comprehend and appreciate. Finite groups, Lie groups, Lie algebras, semi-simple Lie algebras, crystallographic point groups and crystallographic space groups, electronic energy bands in solids, atomic physics, symmetry schemes for fundamental particles, and quantum mechanics are all covered in this compact new edition.
Key Features
* Covers both group theory and the theory of Lie algebras
* Includes studies of solid state physics, atomic physics, and fundamental particle physics
* Contains a comprehensive index
* Provides extensive examples

The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications. This book describes the Schur complement as a rich and basic tool in mathematical research and applications and discusses many significant results that illustrate its power and fertility. The eight chapters of the book cover themes and variations on the Schur complement, including its historical development, basic properties, eigenvalue and singular value inequalities, matrix inequalities in both finite and infinite dimensional settings, closure properties, and applications in statistics, probability, and numerical analysis. Although the book is primarily intended to serve as a research reference, it will also be useful for graduate and advanced undergraduate courses in mathematics, applied mathematics, and statistics. The contributing authors' exposition makes most of the material accessible to readers with a sound foundation in linear algebra.

Spectral theoryis an important part of functional analysis.It has numerousapp- cations in many parts of mathematics and physics including matrix theory, fu- tion theory, complex analysis, di?erential and integral equations, control theory and quantum physics. In recent years, spectral theory has witnessed an explosive development. There are many types of spectra, both for one or several commuting operators, with important applications, for example the approximate point spectrum, Taylor spectrum, local spectrum, essential spectrum, etc. The present monograph is an attempt to organize the available material most of which exists only in the form of research papers scattered throughout the literature. The aim is to present a survey of results concerning various types of spectra in a uni?ed, axiomatic way. The central unifying notion is that of a regularity, which in a Banach algebra isasubsetofelementsthatareconsideredtobe nice .AregularityRinaBanach algebraA de?nes the corresponding spectrum ? (a)={ C: a / ? R} in R the same wayas the ordinaryspectrum is de?ned by means of invertible elements, ?(a)={ C: a / ? Inv(A)}. Axioms of a regularity are chosen in such a way that there are many natural interesting classes satisfying them. At the same time they are strong enough for non-trivial consequences, for example the spectral mapping theorem. Spectra ofn-tuples ofcommuting elements ofa Banachalgebraaredescribed similarly by means of a notion of joint regularity. This notion is closely related to ? the axiomatic spectral theory of Zelazko and S lodkowski."

From Newton's Law of Gravity to the Black-Scholes model used by bankers to predict the markets, equations, are everywhere -- and they are fundamental to everyday life.Seventeen Equations that Changed the World examines seventeen ground-breaking equations that have altered the course of human history. He explores how Pythagoras's Theorem led to GPS and Satnav; how logarithms are applied in architecture; why imaginary numbers were important in the development of the digital camera, and what is really going on with Schroedinger's cat. Entertaining, surprising and vastly informative, Seventeen Equations that Changed the World is a highly original exploration -- and explanation -- of life on earth.

A systematic survey of all the basic results on the theory of discrete subgroups of Lie groups, presented in a convenient form for users. The book makes the theory accessible to a wide audience, and will be a standard reference for many years to come.

In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system - harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincare symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.