Pure and Applied Mathematics ➤ ALGEBRa / COMBINATORICS ➤ LOGIc / SET THEORY ➤ NUMBER THEORY ➤ GEOMETRY / TOPOLOGY ➤ GROUPS AND REPRESENTATION THEORY ➤ ANALYSIS / OPERATOR THEORY ➤ PROBABILITY / STATISTICS / NUMERIcaL METHODS ➤ MaTHEMATIcaL PHYSICS / PHYSICS ➤ DIFFERENTIAL EQUATIONS / DYNAMIcaL SYSTEMS ➤ APPLIED MaTHEMATICS ➤ GENERAL INTEREST / REFERENCE / MaTH EDUcaTION www.birkhauser.com ALGEBRA / COMBINATORICS Cornerstones is a new Birkhäuser series that is comprised of textbooks focusing on “what students need to know” and Advanced Algebra “what faculty should teach” in various selected topics in pure ANTHONY W. KNAPP, State University of New York, Stony Brook, NY mathematics and related subjects. Basic Algebra and Advanced Algebra systematically develop Series Editors: concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together the two CHARLES L. EPSTEIN, University of Pennsylvania, Philadelphia, PA books give the reader a global view of algebra and its role in math- STEVEN G. KRANTZ, Washington University, St. Louis, MO ematics as a whole. Basic Algebra Key Topics and Features of Advanced Algebra: ANTHONY W. KNAPP, State University of • Topics build upon the linear algebra, group theory, factoriza- New York at Stony Brook, NY tions of ideals, structure of fields, Galois theory, and elementary theory of modules as developed in Basic Algebra Basic Algebra and Advanced Algebra systematically develop concepts and tools • Chapters on modern algebra treat various topics in commuta- in algebra that are vital to every mathema- tive and noncommutative algebra and povide introductions to tician, whether pure or applied,, aspiring the theory of associative algebras, homological algebras, alge- or established. Together, the two books braic number theory, and algebraic geometry give the reader a global view of algebra • Continues to carry on the three prominent themes recurring in and its role in mathematics as a whole. Basic Algebra: the anaology between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory Key Topics and Features of Basic Algebra: and geometry • Linear algebra and group theory build on each other continually • Many examples and hundreds of problems are included, along • Chapters on modern algebra treat groups, rings, fields, modules, with hints or complete solutions for most of the problems and Galois groups with emphasis on methods of computation • The exposition proceeds from the particular to the general, throughout often providing examples well before a theory that incorporates • Three prominent themes recur and blend together at times: the them; it includes blocks of problems that introduce additional analogy between integers and polynomials in one variable over topics and applications for futher study a field, the interplay between linear algebra and group theory, Advanced Algebra presents its subject matter in a forward-look- and the relationship between number theory and geometry ing way that takes into accoutn the historical development of the • Many examples and hundreds of problems are included, along subject. It is suitable as a text in a two-semester advanced under- with a separate 90-page section giving hints or complete solu- graduated or first-year graduate sequence in algebra, possibly sup- tions for most of the problems plemented for a graduate sequence by some material from Basic • The exposition proceeds from the particular to the general, of- Algebra. It requires of the reader only familiarity with matrix algebra, ten providing examples well before a theory that incorporates an understanding of the geometry and reduction of linear equa- them; includes blocks of problems that introduce additional tions, and an acquantance with proofs. topics and applications for future study Contents: Preface. • Guide for the Reader. • Transition to Modern • Applications to science and engineering (e.g., the fast Fourier Number Theory. • Wedderburn-Artin Ring Theory. • Brauer Group. transform, the theory of error-correcting codes, the use of the • Homological Algebra. • Three Theorems in Algebraic Number Theory. Jordan canonical form in solving linear systems of ordinary dif- • Reinterpretation with Adeles and Ideles. • Infinite Field Extensions. ferential equations, and constructions of interest in mathemati- • Background for Algebraic Geometry. • The Number Theory of cal physics) appear in sequences of problems Algebraic Curves. • Methods of Algebraic Geometry. • Index 2007 / APPROX. 650 PP., 10 ILLUS. / HARDCOVER / $69.95 Basic Algebra presents the subject matter in a forward-looking way ISBN_13 978-0-8176-4533-5 that takes into account its historical development. It is suitable as a ISBN_10 0-8176-4533-0 text in a two-semester advanced undergraduate or first-year graduate CORNERSTONES sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs. Set of Basic Algebra Contents: Preface. • Guide for the Reader • Preliminaries about the and Advanced Algebra Integers, Polynomials, and Matrices. • Vector Spaces over Q R and C. 2007/$89.95 (TENT.) ISBN_13 978-0-8176-4533-5 • Inner-Product Spaces. • Group and Group Actions. • Theory of a Single ISBN_10 0-8176-4533-0 Linear Transformation. • Multilinear Algebra. • Advanced Group Theory. CORNERSTONES • Commutative Rings and Their Modules. • Fields and Galois Theory. • Modules over Noncommutative Rings. • Appendix. • Index See pages 28 and 30 for additional titles in the Cornerstones Series 2006 / 718 PP., 42 ILLUS. / HARDCOVER / $69.95 ISBN_13 978-0-8176-3248-9 ISBN_10 0-8176-3248-4 CORNERSTONES Order By Phone: 1-800-777-4643 • Fax: 01-348-4505 • Or Online: www.birkhauser.com ALGEBRA / COMBINATORICS Representation Theory and Artinian Modules Over Group Automorphic Forms Rings TOSHIYUKI KOBAYASHI, Kyoto University, RIMS, Kyoto, Japan; LEONID KURDacHENKO, Dnipropetrovsk National University, Ukraine; WILFRIED ScHMID, Harvard University, Cambridge, MA; JAVIER OTAL, University of Zaragoza, Spain; JAE-HYUN YaNG, Inha University, Incheon, Korea (Eds.) IGOR Ya. SUBBOTIN, National University, Los Angeles, CA This volume uses a unified approach to representation theory and This book highlights important developments on artinian modules automorphic forms. The invited papers, written by leading mathema- over group rings of generalized nilpotent groups. Along with tradi- ticians in related fields, track recent progress in the ever expanding tional topics such as direct decompositions of artinian modules, fields of representation theory and automorphic forms, and their as- criteria of complementability for some important modules, and cri- sociation with number theory and differential geometry. teria of semisimplicity of artinian modules, it also focuses on recent Contributors: T. Ikeda, T. Kobayashi, S. Miller, D. Ramakrishnan, advanced results. W. Schmid, F. Shahidi, K. Yoshikawa The theory of modules over groups has its own specific character 2007 / APPROX. 310 PP., 10 ILLUS. / HARDCOVER / $89.95 (TENT.) that plays an imperative role here and, for example, allows for a ISBN_13 978-0-8176-4505-2 significant generalization of the classical Maschke Theorem on ISBN_10 0-8176-4505-5 PROGRESS IN MATHEMATICS, VOL. 55 some classes of infinite groups. Conversely, it leads to establishing direct decompositions of artinian modules related to important natural formations, which, in turn, find very efficient applications in infinite groups. Schubert Varieties As self-contained as possible, this book will be useful for students as V. LakSHMIBAI, Northeastern University, Boston, MA; well as for experts in group theory, ring theory, and module theory. PETER LITTELMANN, Bergische Universität, Wuppertal, Germany; 2007 / APPROX. 264 PP. / SOFTCOVER / $49.95 (TENT.) C.S. SESHADRI, Chennai Mathematical Institute, Chennai, India ISBN_13 978-3-7643-7764-9 ISBN_10 3-7643-7764-X Over the past two decades, and more intensely in recent years, the FRONTIERS IN MATHEMATICS algebro-geometric study of Schubert Varieties has had considerable impact on the theory of algebraic groups. One of the most interest- ing developments in the theory has been the construction of natural bases of representations of the full linear group, the orthogonal The Geometry of the group, and the symplectic group. This construction gives character Word Problem for Finitely formulas of these representations which are quite different in spirit from the famous character formulas of H. Weyl. In fact, they connect Generated Groups to monomial theory and the work of Hodge which was done more NOEL BRADY, University of Oklahoma, Norman, OK; than fifty years ago, and to the very recent developments in path HaMISH SHORT, Université de Provence, Marseille, France; models, Frobenius splittings, and quantum groups. TIM RILEY, Cornell University, Ithaca, NY Written by three of the world’s leading mathematicians in algebraic The origins of the word problem in group theory are decidability geometry, group theory, and combinatorics, this excellent self- con- and complexity, but through the vision of M. Gromov and the lan- tained exposition on Schubert Varieties unfolds systematically, from guage of filling functions, the topic now impacts the world of large- relevant introductory