Gerd Fischer, Heinrich-Heine-Universita"t, Du"sseldorf, Germany
Plane Algebraic Curves
Expected publication date is August 8, 2001
From a review for the German Edition:
"The present book provides a completely self-contained introduction to complex plane curves from the traditional algebraic-analytic viewpoint. The arrangement of the material is of outstanding instructional skill, and the text is written in a very lucid, detailed and enlightening style ... Compared to the many other textbooks on (plane) algebraic curves, the present new one comes closest in spirit and content, to the work of E. Brieskorn and H. Knoerrer ... One could say that the book under review is a beautiful, creative and justifiable abridged version of this work, which also stresses the analytic-topological point of view ... the present book is a beautiful invitation to algebraic geometry, encouraging for beginners, and a welcome source for teachers of algebraic geometry, especially for those who want to give an introduction to the subject on the undergraduate-graduate level, to cover some not too difficult topics in substantial depth, but to do so in the shortest possible time."
-- Zentralblatt fu"r Mathematik
Description
The study of the zeroes of polynomials, which for one variable is essentially algebraic, becomes a geometric theory for several variables. In this book, Fischer looks at the classic entry point to the subject: plane algebraic curves. Here one quickly sees the mix of algebra and geometry, as well as analysis and topology, that is typical of complex algebraic geometry, but without the need for advanced techniques from commutative algebra or the abstract machinery of sheaves and schemes.
In the first half of this book, Fischer introduces some elementary geometrical aspects, such as tangents, singularities, inflection points, and so on. The main technical tool is the concept of intersection multiplicity and Be'zout's theorem. This part culminates in the beautiful Plu"cker formulas, which relate the various invariants introduced earlier.
The second part of the book is essentially a detailed outline of modern methods of local analytic geometry in the context of complex curves. This provides the stronger tools needed for a good understanding of duality and an efficient means of computing intersection multiplicities introduced earlier. Thus, we meet rings of power series, germs of curves, and formal parametrizations. Finally, through the patching of the local information, a Riemann surface is associated to an algebraic curve, thus linking the algebra and the analysis.
Concrete examples and figures are given throughout the text, and when possible, procedures are given for computing by using polynomials and power series. Several appendices gather supporting material from algebra and topology and expand on interesting geometric topics.
This is an excellent introduction to algebraic geometry, which assumes only standard undergraduate mathematical topics: complex analysis, rings and fields, and topology. Reading this book will help the student establish the appropriate geometric intuition that lies behind the more advanced ideas and techniques used in the study of higher dimensional varieties.
This is the English translation of a German work originally published by Vieweg Verlag (Wiesbaden, Germany).
Contents
Introduction
Affine algebraic curves and their equations
The projective closure
Tangents and singularities
Polars and Hessian curves
The dual curve and the Plu"cker formulas
The ring of convergent power series
Parametrizing the branches of a curve by Puiseux series
Tangents and intersection multiplicities of germs of curves
The Riemann surface of an algebraic curve
The resultant
Covering maps
The implicit function theorem
The Newton polygon
A numerical invariant of singularities of curves
Harnack's inequality
Bibliography
Subject index
List of symbols
Series: Student Mathematical Library, Volume: 15
Publication Year: 2001
ISBN: 0-8218-2122-9
Paging: approximately 206 pp.
Binding: Softcover
Edward Frenkel, University of California, Berkeley, CA,
and David Ben-Zvi, University of Chicago, IL
Vertex Algebras and Algebraic Curves
Expected publication date is August 31, 2001
Description
Vertex algebras are algebraic objects that formalize the concepts of vertex operators and operator product expansion from two-dimensional conformal field theory. In the fifteen years since they were introduced by R. Borcherds, vertex algebras have turned out to be extremely useful in many areas of mathematics. They are by now ubiquitous in the representation theory of infinite-dimensional Lie algebras. They have also found applications in such fields as algebraic geometry, theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of algebraic curves.
The notion of a vertex algebra is introduced in the book in coordinate-independent way, allowing the authors to give global geometric meaning to vertex operators on arbitrary smooth algebraic curves, possibly equipped with some additional data. To each vertex algebra and a smooth curve, they attach an invariant called the space of conformal blocks. When the complex structure of the curve and other geometric data vary, these spaces combine into a sheaf on the relevant moduli space. From this perspective, vertex algebras appear as algebraic objects that encode the geometric structure of various moduli spaces associated with algebraic curves.
Numerous examples and applications of vertex algebras are included, such as the Wakimoto realization of affine Kac-Moody algebras, integral solutions of the Knizhnik-Zamolodchikov equations, classical and quantum Drinfeld-Sokolov reductions, and the $W$-algebras. Among other topics discussed in the book are vertex Poisson algebras, Virasoro uniformization of the moduli spaces of curves, the geometric Langlands correspondence, and the chiral de Rham complex. The authors also establish a connection between vertex algebras and chiral algebras, recently introduced by A. Beilinson and V. Drinfeld.
This book may be used by the beginners as an entry point to the modern theory of vertex algebras, and by more experienced readers as a guide to advanced studies in this dynamic field.
Contents
Introduction
Definition of vertex algebras
Vertex algebras associated to Lie algebras
Associativity and operator product expansion
Rational vertex algebras
Vertex algebra bundles
Action of internal symmetries
Vertex algebra bundles: Examples
Conformal blocks I
Conformal blocks II
Free field realization I
Free field realization II
The Knizhnik-Zamolodchikov equations
Solving the KZ equations
Quantum Drinfeld-Sokolov reduction and $\mathcal{W}$-algebras
Vertex Lie algebras and classical limits
Vertex algebras and moduli spaces I
Vertex algebras and moduli spaces II
Chiral algebras
Appendix A
Bibliography
Index
List of frequently used notation
Series: Mathematical Surveys and Monographs,Volume: 88
Publication Year: 2001
ISBN: 0-8218-2894-0
Paging: 348 pp.
Binding: Hardcover
Martin Schechter, University of California, Irvine, CA
Principles of Functional Analysis: Second Edition
Expected publication date is September 22, 2001
From a review for the First Edition:
"`Charming' is a word that seldom comes to the mind of a science reviewer, but if he is charmed by a treatise, why not say so? I am charmed by this book.
"Professor Schechter has written an elegant introduction to functional analysis including related parts of the theory of integral equations. It is easy to read and is full of important applications. He presupposes very little background beyond advanced calculus; in particular, the treatment is not burdened by topological `refinements' which nowadays have a tendency of dominating the picture.
"The book can be warmly recommended to any reader who wants to learn about this subject without being deterred by less relevant introductory matter or scared away by heavy prerequisites."
-- The American Scientist
Description
Functional analysis plays a crucial role in the applied sciences as well as in mathematics. It is a beautiful subject that can be motivated and studied for its own sake. In keeping with this basic philosophy, the author has made this introductory text accessible to a wide spectrum of students, including beginning-level graduates and advanced undergraduates.
The exposition is inviting, following threads of ideas, describing each as fully as possible, before moving on to a new topic. Supporting material is introduced as appropriate, and only to the degree needed. Some topics are treated more than once, according to the different contexts in which they arise.
The prerequisites are minimal, requiring little more than advanced calculus and no measure theory. The text focuses on normed vector spaces and their important examples, Banach spaces and Hilbert spaces. The author also includes topics not usually found in texts on the subject.
This Second Edition incorporates many new developments while not overshadowing the book's original flavor. Areas in the book that demonstrate its unique character have been strengthened. In particular, new material concerning Fredholm and semi-Fredholm operators is introduced, requiring minimal effort as the necessary machinery was already in place. Several new topics are presented, but relate to only those concepts and methods emanating from other parts of the book. These topics include perturbation classes, measures of noncompactness, strictly singular operators, and operator constants.
Overall, the presentation has been refined, clarified, and simplified, and many new problems have been added.
Contents
Basic notions
Duality
Linear operators
The Riesz theory for compact operators
Fredholm operators
Spectral theory
Unbounded operators
Reflexive Banach spaces
Banach algebras
Semigroups
Hilbert space
Bilinear forms
Selfadjoint operators
Measures of operators
Examples and applications
Bibliography
Index
Series: Graduate Studies in Mathematics, Volume: 36
Publication Year: 2001
ISBN: 0-8218-2895-9
Paging: 393 pp.
Binding: Hardcover
Balkrishnan, N., McMaster University, Hamilton, Canada, Ibragimov, I.A., St. Petersburg State University, Russia, Nevzorov, V.B., St. Petersburg State University, Russia
Asymptotic Methods in Probability and Statistics with Applications
2001. Approx. 568 pages. Hardcover
ISBN 3-7643-4214-5
This volume 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. Its main features include:
A broad coverage of recent material on wavelet analysis, and time-frequency signal analysis and other applications that are not usually covered in other recent reference books.
The material presented in this volume brings together a rich variety of ideas that blend most aspects of the subject mentioned above.
This volume brings together a detailed account of major recent developments in wavelets, wavelet transforms and time-frequency signal analysis.
This volume provides the reader with a thorough mathematical background and a wide variety of applications that are sufficient to do interdisciplinary collaborative research in applied mathematics.
The book provides information that puts the reader at the forefront of the current resarch. An up-to-date bibliography is included at the end of each chapter to stimulate new interest in future study and research.
Enns, R.H., Simon Fraser University, Burnaby, CDN,
McGuire, G.C., University College of the Fraser Valley, Abbotsford, CDN
Nonlinear Physics with MATHEMATICA for Scientists and Engineers
2001. Approx. 656 pages. Hardcover. CD-ROM included
ISBN 3-7643-4223-4
Due in August 2001
Nonlinear physics continues to be an area of dynamic modern research, with applications to physics, engineering, chemistry, mathematics, computer science, biology, medicine and economics. This textbook, together with its supplementary CD-ROM, is designed and centered around the Mathematica computer algebra system and its use in investigating a wide variety nonlinear models.
No prior knowledge of Mathematica or programming is assumed. The authors have included a CD-ROM that contains over 130 annotated Mathematica files. These files may be used to solve and explore the text's 400 problems. This book includes 33 experimental activities that are designed to deepen and broaden the reader's understanding of nonlinear physics. These activities are correlated with Part I, the theoretical framework of the text.
Additional features:
* User-friendly, accessible presentation integrating theory, experiments, and the provided Mathematica notebooks; as the concepts of nonlinear science are developed, readers are gently introduced to Mathematica as an auxiliary tool
* CD-ROM includes a wide variety of illustrative nonlinear examples solved with Mathematica-command structures introduced on a need-to-know basis
* Notebooks designed to make use of Mathematica's sound capability
* Mathematica notebook using the EulerEquation command incorporated into the text
This work is a valuable tool for undergraduate and graduate students as well as a useful resource for working scientists.
From the Table of Contents:
Preface
Part I: THEORY
1. Introduction
2. Nonlinear Systems, Part I
3. Nonlinear Systems, Part II
4. Topological Analysis
5. Analytic Methods
6. The Numerical Approach
7. Limit Cycles
8. Forced Oscillators
9. Nonlinear Maps
10. Nonlinear PDE Phenomena
11. Numerical Simulation
12. Inverse Scattering Method
Part II: EXPERIMENTAL ACTIVITIES
Introduction to Nonlinear Experiments
1. Spin Toy Pendulum
2. Driven Eardrum
3. Nonlinear Damping
4. Anaharmonic Potential
5. Iron Core Conductor
6. Nonlinear LRC Circuit
7. Tunnel Diode Negative Resistance Curve
8. Tunnel Diode Self-Excited Oscillator
9. Forced Duffing Equation
10. Focal Point Instability
11. Compound Pendulum
12. Stable Limit Cycle
13. Van der Pol Limit Cycle
14. Relaxation Oscillations: Neon Bulb
15. Relaxation Oscillations: Drinking Bird
16. Relaxation Oscillations: Tunnel Diode
17. Hard Spring
18. Nonlinear Resonance Curve: Mechanical
19. Nonlinear Resonance Curve: Electrical
20. Nonlinear Resonance Curve: Magnetic
21. Subharmonic Response: Period Doubling
22. Diode: Period Doubling
23. Five-Well