Anthony W. Knapp

Representation Theory of Semisimple Groups:

An Overview Based on Examples. (PMS-36)
Paper | December 2001
794 pp. | 6 x 9

Reviews

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Anthony W. Knapp is Emeritus Professor of Mathematics, State University of New York at Stony Brook. The author of numerous books, he is the former editor of the Notices of the American Mathematical Society.

Reviews:
''Anthony Knapp has written a marvelous book. . . . Written with accuracy, style, and a genuine desire to communicate the materials. . . . This is one of the finest books I have ever had the pleasure to read, and I recommend it in the strongest possible terms to anyone wishing to appreciate the intricate beauty and technical difficulty of representation theory of semisimple Lie groups.''--R. J. Plymen, Bulletin of the London Mathematical Society

''Each [theme] is developed carefully and thoroughly, with beautifully worked examples and proofs that reflect long experience in teaching and research. . . . This result is delightful: a readable text that loses almost none of its value as a reference work.''--David A. Vogan Jr., Bulletin of the American Mathematical Society

Nicholas M. Katz

Twisted L-Functions and Monodromy

237 pp. | 6 x 9

For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves?

Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves.

The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

Nicholas M. Katz is Professor of Mathematics at Princeton University. He is the author of four other books in this series: Arithmetic Moduli of Elliptic Curves (with Barry Mazur); Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; and Rigid Local Systems.

Series: Annals of Mathematics Studies

ISBN: 0-691-09151-X Paper:
ISBN: 0-691-09150-1 Cloth:

Michael Harris and Richard Taylor

The Geometry and Cohomology of Some Simple Shimura Varieties

291 pp. | 6 x 9

This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the ''simple'' Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory.

The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GL n(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary.

Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.

Michael Harris is Professor of Mathematics at Universite' Paris 7--Denis Diderot. His research focuses on automorphic forms and representation theory, with a view to applications in number theory and arithmetical algebraic geometry. Richard Taylor is Professor of Mathematics at Harvard University. His main research interest is number theory, particularly the theory of modular forms, automorphic forms, and Galois representations.

Series: Annals of Mathematics Studies

ISBN: 0-691-09092-0 Paper:
ISBN: 0-691-09090-4 Cloth:

Edited by Michel Janssen, Robert Schulmann, Jo'zsef Illy, and Christoph Lehner

The Collected Papers of Albert Einstein:
Volume 7. The Berlin Years: Writings, 1918-1921

Cloth | January 2002
728 pp. | 7 x 10

In the spring of 1919, two British solar eclipse expeditions confirmed the correctness of general relativity theory and propelled Albert Einstein to instant celebrity. Before this major turning point, the majority of Einstein's writings published in this volume dealt with the clarification of general relativistic problems, such as the status of the metric field, the character of gravitational waves, the problem of energy-momentum conservation, and questions of cosmology, such as the nature and size of the universe and the distribution of matter within it. After his rise to international fame, Einstein's publications changed markedly. He faced an increasing demand for popular articles and lectures on relativity, its development and meaning. He also felt compelled to respond to a host of commentators, ranging from skeptical physicists to philosophers trying to reconcile his revolutionary theory with their views. For the first time, he also responded in print to outspoken anti-relativists, some of them fueled by cultural conservatism and, frequently, anti-Semitism.

Einstein used his newly won fame to lend prestige to political causes, especially to the reconciliation among European nations and to Zionism. In the early years of Weimar Germany, Einstein spoke out vigorously for the young republic, emphasizing the rights of the individual. He agonized over the misery of the Central Europeans in the grip of starvation and economic collapse, praised the support of individuals and groups such as the Quakers, and championed the cause of Eastern European Jews. His rejection of assimilation, combined with a fierce defense of the right of Jews to higher education, led Einstein to campaign for the establishment of a university in Palestine, the land which he conceived of as a cultural center for all Jews.

Michel Janssen is Assistant Professor of the History of Science at the University of Minnesota. Robert Schulmann has been a longstanding editor of the Einstein Papers and Associate Professor of History at Boston University. Jo'zsef Illy was Senior Research Associate at the Hungarian Academy of Sciences. Christoph Lehner is Senior Research Fellow in Philosophy at the California Institute of Technology. Diana Kormos Buchwald is Associate Professor of History at the California Institute of Technology.

ISBN: 0-691-05717-6 Cloth:
ISBN: 0-691-05718-4 Paper (English translation of selected texts)

Emil Simiu

Chaotic Transitions in Deterministic and Stochastic Dynamical Systems:
Applications of Melnikov Processes in Engineering, Physics, and Neuroscience

January 2002
176 pp. | 6 x 9 | 94 line illus.

Endorsements

The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This book develops a unified treatment of deterministic and stochastic systems that extends the applicability of the Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, white, colored, or dichotomous noise. The extended Melnikov method yields the novel result that motions with transitions are chaotic regardless of whether the excitation is deterministic or stochastic. It explains the role in the occurrence of transitions of the characteristics of the system and its deterministic or stochastic excitation, and is a powerful modeling and identification tool.

The book is designed primarily for readers interested in applications. The level of preparation required corresponds to the equivalent of a first-year graduate course in applied mathematics. No previous exposure to dynamical systems theory or the theory of stochastic processes is required. The theoretical prerequisites and developments are presented in the first part of the book. The second part of the book is devoted to applications, ranging from physics to mechanical engineering, naval architecture, oceanography, nonlinear control, stochastic resonance, and neurophysiology.

Emil Simiu is a NIST Fellow, National Institute of Standards and Technology, and Research Professor, Whiting School of Engineering, The Johns Hopkins University. A specialist in flow-structure interaction, he is the coauthor of Wind Effects on Structures and was the 1984 recipient of the Federal Engineer of the Year award.

Endorsement:
''The author has chosen an excellent subject, which will probably become a main direction of research in the field of stochastic differential equations. This book is addressed to a wide readership: specialists in dynamical systems, in probability and stochastics, mathematicians, physicists, and neuroscientists. The author manages to present the material such that it makes it interesting to all these groups of researchers.''--Florin Diacu, PIMS University of Victoria

ISBN: 0-691-05094-5 Cloth