Alexander Polishchuk

The aim of this book is to present a modern treatment of the theory of theta functions in the context of algebraic geometry. It starts with the classical theory of theta functions (in which the usual Fourier transform plays the prominent role), and then shows that in the algebraic approach to this theory, the Fourier-Mukai transform can often be used to cast new light on many important theorems. Graduate students and researchers working in algebraic geometry will find much of interest in this volume.

K. R. Goodearl, R. B. Warfield, Jr

This introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. New material includes the basic types of quantum groups, which then serve as test cases for the theory developed.

Jerzy Weyman

The central theme of this book is a detailed exposition of the geometric technique of calculating syzygies. While this is an important tool in algebraic geometry, the author has elected to write from the point of view of commutative algebra in order to avoid being tied to special cases from geometry. No prior knowledge of representation theory is assumed. Chapters on several applications are included, and numerous exercises will give the reader insight into how to apply this important method.

Vern Paulsen

In this book the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The presentation is appropriate to graduate students and experts alike. The author assumes only that the reader has a basic background in functional analysis, but also provides new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature.

Niels Lauritzen

Concrete Abstract Algebra develops the theory of abstract algebra from numbers to Gröbner bases, whilst taking in all the usual material of a traditional introductory course. Niels Lauritzen's approach to teaching is based on an extensive use of examples, applications and exercises. As the title suggests the philosophy is to develop an abstract subject on concrete and more easily comprehensible foundations. This book is built on several years of experience teaching introductory abstract algebra at Aarhus, where the emphasis on concrete and inspiring examples has improved student performance significantly. Solutions to the exercises are available to lecturers from solutions@cambridge.org.

Robert R. Colby, Kent R. Fuller

This book provides a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years. In particular, during the past dozen or so years many authors (including the authors of this book) have investigated relationships between categories of modules over a pair of rings that are induced by both covariant and contravariant representable functors, in particular by tilting and cotilting theories. By here collecting and unifying the basic results of these investigations with innovative and easily understandable proofs, the authors' aim is to provide an aid to further research in this central topic in abstract algebra, and a reference for all whose research lies in this field.

John Swallow

Combining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and assumes of its readers only a first course in abstract algebra. For readers with Maple or Mathematica, the text introduces tools for hands-on experimentation with finite extensions of the rational numbers, enabling a familiarity never before available to students of the subject. The text is appropriate for traditional lecture courses, for seminars, or for self-paced independent study by undergraduates and graduate students.

M. Aschbacher

The foundations of the theory of finite groups are developed in this book. Unifying themes include the Classification Theorem and the classical linear groups. Lie theory appears in chapters on Coxeter groups, root systems, buildings and Tits systems. There is a new proof of the Solvable Signalizer Functor Theorem and a brief outline of the proof of the Classification Theorem itself. The second edition of Finite Group Theory has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.

Edited by Leila Schneps

This book contains eight articles which focus on presenting recently developed new aspects of the theory of Galois groups and fundamental groups, avoiding classical aspects which have already been developed at length in the standard literature. Each article strives to be introductory, while containing original results. The volume grew from the special semester held at the MSRI in Berkeley in 1999 and many of the new results are due to work accomplished during that program. Among the subjects covered are elliptic surfaces, Grothendieck's anabelian conjecture, fundamental groups of curves and differential Galois theory in positive characteristic.

Christian U. Jensen, Arne Ledet, Noriko Yui

This book describes a constructive approach to the Inverse Galois problem. The main theme is an exposition of a family of 'generic' polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of 'generic dimension' to address the problem of the smallest number of parameters required by a generic polynomial.

A. A. Ivanov, S. V. Shpectorov

This is the second volume in a two-volume set, which provides a complete self-contained proof of the classification of geometries associated with sporadic simple groups: Petersen and tilde geometries. Via their systematic treatment of group amalgams, the authors establish a deep and important mathematical result. This book will be of great interest to researchers in finite group theory, finite geometries and algebraic combinatorics.

B. A. Davey, H. A. Priestley

This new edition presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.

Igor Dolgachev

Based on lectures given at University of Michigan, Harvard University and Seoul National University, the primary goal of this book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. The style is accessible and there are numerous examples and exercises to aid the reader.

Roger W. Carter, Ian G. MacDonald, Graeme B. Segal, Foreword by M. Taylor

In this excellent introduction to the theory of Lie groups and Lie algebras, three of the leading figures in this area have written up their lectures from an LMS/SERC sponsored short course in 1993. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie Groups. And in the final part, Ian Macdonald gives an introduction to special linear groups. Anybody requiring an introduction to the theory of Lie groups and their applications should look no further than this book.

Roger A. Horn, Charles R. Johnson

Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research. In this book the authors present classical and recent results of matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematical science, but the necessary material has appeared only sporadically in the literature and in university curricula. As interest in applied mathematics has grown, the need for a text and reference offering a broad selection of topics in matrix theory has become apparent, and this book meets that need. This volume reflects two concurrent views of matrix analysis. First, it encompasses topics in linear algebra that have arisen out of the needs of mathematical analysis. Second, it is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis. Both views are reflected in its choice and treatment of topics.

Ákos Seress

Permutation group algorithms played an indispensable role in the proof of many deep results, including the construction and study of sporadic finite simple groups. This book describes the theory behind permutation group algorithms, up to the most recent developments. Rigorous complexity estimates, implementation hints, and advanced exercises are included throughout. The book fills a significant gap in the symbolic computation literature. It is recommended for everyone interested in using computers in group theory, and is suitable for advanced graduate courses.

W. K. Nicholson, M. F. Yousif

A ring is called quasi-Frobenius if it is right or left selfinjective, and right or left artinian (all four combinations are equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field, and the present extent of the theory is vast. This book makes no attempt to be encyclopedic; instead it provides an elementary, self-contained account at a level allowing researchers and graduate students to gain entry to the field.

Marc Cabanes, Michel Enguehard

At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesising the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Throughout the text is illustrated by many examples and background is provided by several introductory chapters on basic results and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

Gordon James, Martin Liebeck

This is the second edition of the popular textbook on representation theory of finite groups. The authors have greatly revised the text and added new sections. Each chapter is accompanied by a variety of exercises, and full solutions to all the exercises are provided at the end of the book. This will be suitable as a text for those teaching a course in representation theory, and in view of the applications of the subject, will be of interest to chemists and physicists as well as mathematicians.

Roe Goodman, Nolan R. Wallach

This book presents an updated version of Weyl's invariant theory of the classical groups, together with many of the important recent developments. Requiring only an abstract algebra course as a prerequisite, it will introduce students of mathematics to the structure and finite-dimensional representation theory of the complex classical groups and will serve as a reference for researchers in mathematics, statistics, physics and chemistry whose work involves symmetry groups, representation theory, invariant theory and algebraic group theory.

Teo Mora

This is the first in a three-volume handbook presenting classical and modern work on polynomial equations, from the viewpoint of solution. Here, Mora covers the classical theory of finding roots of a univariate polynomial, emphasising computational aspects, especially the representation and manipulation of algebraic numbers, enlarged by more recent representations like the Duval Model and the Thom Codification. He shows that solving a polynomial equation really means finding algorithms that help one manipulate roots rather than simply computing them; to that end he also surveys algorithms for factorizing univariate polynomials.

George E. Andrews, Richard Askey, Ranjan Roy

Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series. In addition to relatively new work, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematical physics, differential equations, mathematical computing, number theory, and combinatorics.\

Roger A. Horn, Charles R. Johnson

Building on the foundations of its predecessor volume, Matrix Analysis, this book treats in detail several topics with important applications and of special mathematical interest in matrix theory not included in the previous text. These topics include the field of values, stable matrices and inertia, singular values, matrix equations and Kronecker products, Hadamard products, and matrices and functions. The authors assume a background in elementary linear algebra and knowledge of rudimentary analytical concepts. The book should be welcomed by graduate students and researchers in a variety of mathematical fields both as an advanced text and as a modern reference work.

Edited by Luchezar L. Avramov, Mark Green, Craig Huneke, Karen E. Smith, Bernd Sturmfels

This book is based on lectures by six internationally known experts presented at the 2002 MSRI introductory workshop on commutative algebra. These focus on the interaction of commutative algebra with other areas of mathematics, including algebraic geometry, group cohomology and representation theory, and combinatorics, with all necessary background provided. Six short complementary papers describing work at the research frontier are also included. The unusual scope and format should make the book invaluable reading for graduate students and researchers interested in commutative algebra and its various uses.

William Fulton

This book describes combinatorics involving Young tableaux and some of their uses in representation theory and algebraic geometry. The emphasis is on the relations among these three fields, rather than a self-contained development of any one of them. The applications to representation theory of the symmetric groups and general linear groups, and to geometry of Grassmannians and flag varieties, are basic to many areas of mathematics. Much of this material has never appeared in book form. Researchers in representation theory and algebraic geometry as well as in combinatorics will find Young Tableaux interesting and useful; beginning graduate students will welcome the examples and exercises and find the intuitive presentation easy to follow.