graduate studies in mathematics - american mathematical society

Graduate Studiesin Mathematics

Catalog of Publications

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AnAlysis

3 Algebra and Algebraic Geometry

6 Analysis

10 Applications

11 Differential Equations

11 Discrete Mathematics and Combinatorics

12 Geometry and Topology

15 logic and Foundations

15 Mathematical Physics

16 number Theory

17 Probability

18 index

19 Ordering information

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Course AdoptionFind the right textbook for your course! The AMS publishes many high-quality books for use in the classroom. To view a comprehensive list of our most widely adopted textbooks, please visit www.ams.org/bookstore/course-adoption.

To Prospective AuthorsThe AMS has been publishing books on advanced mathematics for almost 100 years. The AMS should be considered as your publisher of choice for many reasons, including:

The AMS is a nonprofit organization, and by publishing with us you will help support many activities that benefit the entire mathematical community.

The list price of your book will likely be lower than with any other publisher.

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The AMS will create and support a website for your book on which you can post additions, updates, and supplementary material at your convenience throughout the lifetime of your book.

If you would like to submit a manuscript to the AMS, please visit www.ams.org/authors.

The volumes in the GsM series are specifically designed as graduate studies texts, but are also suitable for recommended and/or supplemental course reading. With appeal to both students and professors, these texts make ideal independent study resources. The breadth and depth of the series’ coverage make it an

ideal acquisition for all academic libraries that support mathematics

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AlgebrA And AlgebrAic geometry

linear Algebra in Action Harry Dym, Weizmann Institute of Science, Rehovot, Israel

Books that address linear algebra as a mere series of definitions and theorems fail to convey how useful the subject is as a tool permeating so many disciplines, from pure and applied mathematics to physics and engineering. This book by Harry Dym is written not from the perspective of an algebraist, but from that of an analyst who has

used linear algebra and its ideas over a long, successful mathematical career.

The author explains how more than 40 years in mathematics demon-strated, to his surprise, that linear algebra is one of the most important objects of mathematical study. He begins the book with the basic mate-rial of a first course in linear algebra, then proceeds to cover applications drawn from vector calculus, numerical analysis, control theory, complex analysis, convexity and functional analysis.

Dym explains that he has taken the liberty of addressing significant areas that traditionally are not linked to linear algebra, such as fixed point theorems, extremal problems, locating the zeros of functions, and locating eigenvalues. The book includes a healthy selection of exercises and examples.

The text also includes appendices on useful facts from analysis and supplementary information from complex function theory. Throughout the text, the message that is conveyed emphasizes the rich terrain of linear algebra and that this rich subject is a tool to be applied, not simply an art to be possessed.

It is a wonderful book: very accessible and rigorous in the same time, containing basic and not-so-basic facts, discussing many (sometimes unexpected) applications... Given that and the wonderful way this book was written and organized, I think it can be used by many readers: engineering students, mathematics students, research mathematicians, and researchers in any other field where linear algebra is applied. I strongly recommend this book to anyone interested in “working” linear algebra.

—MAA Reviews

READERSHIP: Undergraduates and first-year graduates interested in linear algebra and applications.Volume 78

2007; 541 pp.; hardcover; isBn: 978-0-8218-3813-6; list Us$79; AMs members Us$63; Order code: GsM/78

Algebraic Curves and Riemann surfacesRick Miranda, Colorado State University, Ft. Collins, CO

In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader’s classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category.

Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.

One of the best introductory textbooks on the theory of algebraic curves and Riemann surfaces … very well organized … plenty of examples … strongly recommend this book as a textbook for an introduction to algebraic curves and Riemann surfaces … One of my students said that this is one of a very few books in algebraic geometry that he can read and understand. The price of the book is very affordable.

—Pawel Gajer, Johns Hopkins University

READERSHIP: Graduate students studying complex variables and alge-braic geometry.Volume 5


Graduate Algebra: Commutative View Louis Halle Rowen, Bar-Ilan University, Ramat Gan, Israel

This text focuses on the algebraic underpinnings of algebraic geometry and of number theory, featuring direct treatment of the theory of affine algebras. The author offers a thorough treatment of transcendence degree and its relation to the lengths of chains of prime ideals in an affine algebra, as well as some theory of Noetherian rings. The book is designed to prove useful to graduate students seeking a solid basis in algebra.

Topics addressed in the book include the theory of modules over a prin-cipal ideal domain and its applications to matrix theory (including the Jordan decomposition), the Galois theory of field extensions, the prime spectrum of an algebra, and localization. Later chapters involve applica-tions used in arithmetic and algebraic curves and surfaces, culminating in the algebraic aspects of elliptical curves and the underlying number theory.

This text departs from others in commutative algebra in its attention to the theorem that the Krull dimension equals the transcendence degree. It also covers several areas of recent research interest, including Grobner bases, elliptic curves, the Jacobian conjecture, and locally nilpotent deri-vations.

The text is extended through a series of appendices that present the more advanced material. Exercises provide a further extension of the text, which is suitable as either a textbook or a reference source.

READERSHIP: Graduate students interested in algebra, geometry, and number theory. Research mathematicians interested in algebra.Volume 73



Volume 78

linear Algebra in Action

Harry Dym

AlGEBRA AnD AlGEBRAiC GEOMETRy


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AnAlysisGRADUATE sTUDiEs in MAThEMATiCs

Representations of Finite and Compact GroupsBarry Simon, California Institute of Technology, Pasadena, CA

Barry Simon is the author of many well-known books, including such clas-sics as Methods of Mathematical Physics (with M. Reed) and Functional Integration and Quantum Physics. This book, based on courses given at Princeton, Caltech, ETH-Zurich, and other universities, is an intro-ductory textbook on representation theory. Two facets distinguish the approach. First, the book is relatively elementary, and second, while the bulk of the books on the subject is written from the point of view of an algebraist or a geometer, this book is written with an analytical flavor.

The exposition centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semisimple Lie groups. It culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups. Extremely well tailored both for a one-year course in representation theory and for inde-pendent study, this book is an excellent introduction to the subject which is unique in having so much innate beauty so close to the surface.

This is indeed a nice book … I would recommend it precisely for the graduate course [that] I am teaching now, “Representation Theory” … I very much like the hands-on approach and the very explicit formulae that are given … Professor Simon has done an excellent job on this beautiful material.

—Tudor Ratiu, University of California, Santa Cruz

READERSHIP: Research mathematicians and graduate students.Volume 10


A Course in Algebra E. B. Vinberg, Moscow State University, Russia

This is a comprehensive text on modern algebra written for advanced undergraduate and basic graduate algebra classes. The book is based on courses taught by the author at the Mechanics and Mathematics Department of Moscow State University and at the Mathematical College of the Independent University of Moscow.

The unique feature of the book is that it contains almost no technically difficult proofs. Following his point of view on mathematics, the author tried, whenever possible, to replace calcula-

tions and difficult deductions with conceptual proofs and to associate geometric images to algebraic objects. Another important feature is that the book presents most of the topics on several levels, allowing the student to move smoothly from initial acquaintance to thorough study and deeper understanding of the subject.

Written with extreme care and supplied with more than 200 exercises and 70 figures, the book is also an excellent text for independent study.

Vinberg has written an algebra book that is excellent, both as a classroom text or for self-study. It is plain that years of teaching abstract algebra have enabled him to say the right thing at the right time.

—Irving Kaplansky, MSRI

READERSHIP: Advanced undergraduates, graduate students, and research mathematicians interested in algebra.

Volume 56

2003; 511 pp.; hardcover; isBn: 978-0-8218-3318-6; list Us$89; AMs members Us$71; Order code: GsM/56 Also Available in Softcover 2003; 511 pp.; softcover; isBn: 978-0-8218-3413-8: list Us$59; AMs members Us$47; Order code: GsM/56.s

A scrapbook of Complex Curve Theorysecond EditionC. Herbert Clemens, Ohio State University, Columbus, OH

Volume 55


Resolution of singularitiesSteven Dale Cutkosky, University of Missouri, Columbia, MO

It has been a pleasure for the reviewer to read this beautiful book, which is a must for graduate students interested in the subject. It fills a gap in graduate texts, covering the most important results in resolution of singularities in an elegant and didactic style.

—Mathematical ReviewsVolume 63


Enveloping AlgebrasJacques Dixmier, Paris, France

Volume 11


Classical Groups and Geometric AlgebraLarry C. Grove, University of Arizona, Tuscon, AZ

Textbook for an in-depth course … provides a nice discussion of various further topics in the study of classical groups and Chevalley groups. … the text would be great for a class or for students learning the material on their own. The topics are covered in a clean tight fashion with appropriate examples given where possible.

—Mathematical ReviewsVolume 39


introduction to Quantum Groups and Crystal BasesJin Hong and Seok-Jin Kang, Korea Institute for Advanced Study, Seoul, Korea

Volume 42


A Course in Algebra

E. B. Vinberg


Volume 56




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Representations of semisimple lie Algebras in the BGG Category James E. Humphreys, University of Massachusetts, Amherst, MA

Volume 94


Finite Group Theory I. Martin Isaacs, University of Wisconsin–Madison, WI

Volume 92


Topics in Classical Automorphic FormsHenryk Iwaniec, Rutgers University, New Brunswick, NJ

An excellent place to begin the study of the analytic approach to modular forms … a welcome addition to this growing expository of literature.

—Bulletin of the AMS

Volume 17


Twenty-Four hours of local CohomologySrikanth B. Iyengar, University of Nebraska, Lincoln, NE, Graham J. Leuschke, Syracuse University, NY, Anton Leykin, Institute for Mathematics and Its Applications, Syracuse, NY, Claudia Miller, Syracuse University, NY, Ezra Miller, University of Minnesota, Minneapolis, MN, Anurag K. Singh, University of Utah, Salt Lake City, UT, and Uli Walther, Purdue University, West Lafayette, IN

It’s all terrific stuff. I hope this book will succeed in bringing many young mathematicians to love cohomology, too, and then to go on from there.

—MAA Reviews

Volume 87


lectures on Quantum GroupsJens Carsten Jantzen, Aarhus Universitet, Denmark

The material is very well motivated … Of the various monographs available on quantum groups, this one … seems the most suitable for most mathematicians new to the subject … will also be appreciated by a lot of those with considerably more experience.

—Bulletin of the London Mathematical SocietyVolume 6


lectures on the Orbit MethodA. A. Kirillov, University of Pennsylvania, Philadelphia, PA

The book offers a nicely written, systematic and read-able description of the orbit method for various classes of Lie groups. ...should be on the shelves of mathematicians and theoretical physicists using representation theory in their work.

—EMS Newsletter

Volume 64


Growth of Algebras and Gelfand-Kirillov DimensionRevised EditionGünter R. Krause, University of Manitoba, Winnipeg, MB, Canada, and Thomas H. Lenagan, University of Edinburgh, Scotland

Volume 22


introduction to Quadratic Forms over FieldsT.Y. Lam, University of California, Berkeley, CA

Volume 67


noncommutative noetherian RingsJ. C. McConnell and J. C. Robson, University of Leeds, England

A comprehensive list of references (containing more than 500 items) is included.

—Mathematica Bohemica

Volume 30


Graduate Algebra: noncommutative ViewLouis Halle Rowen, Bar-Ilan University, Ramat Gan, Israel

Volume 91


Representation Theory of Finite Groups: Algebra and ArithmeticSteven H. Weintraub, Lehigh University, Bethlehem, PA

Volume 59















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AnAlysisGRADUATE sTUDiEs in MAThEMATiCs AnAlysis

AnAlysis

A Modern Theory of integrationRobert G. Bartle, Eastern Michigan University, Ypsilanti, MI, and University of Illinois, Urbana, IL

The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is “better” because it removes

some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with “improper” integrals.

This book is an introduction to a relatively new theory of the integral (called the “generalized Riemann integral” or the “Henstock-Kurzweil integral”) that corrects the defects in the classical Riemann theory and both simplifies and extends the Lebesgue theory of integration. Although this integral includes that of Lebesgue, its definition is very close to the Riemann integral that is familiar to students from calculus. One virtue of the new approach is that no measure theory and virtually no topology is required. Indeed, the book includes a study of measure theory as an application of the integral.

Part 1 fully develops the theory of the integral of functions defined on a compact interval. This restriction on the domain is not necessary, but it is the case of most interest and does not exhibit some of the technical problems that can impede the reader’s understanding. Part 2 shows how this theory extends to functions defined on the whole real line. The theory of Lebesgue measure from the integral is then developed, and the author makes a connection with some of the traditional approaches to the Lebesgue integral. Thus, readers are given full exposure to the main classical results.

The text is suitable for a first-year graduate course, although much of it can be readily mastered by advanced undergraduate students. Included are many examples and a very rich collection of exercises. There are partial solutions to approximately one-third of the exercises. A complete solutions manual is available separately.

READERSHIP: Advanced undergraduates, graduate students and research mathematicians, physicists, and electrical engineers interested in real functions.Volume 32


solutions Manual to A Modern Theory of integrationRobert G. Bartle, Eastern Michigan University, Ypsilanti, MI, and University of Illinois, Urbana, IL

Volume 32

2001; 72 pp.; softcover; isBn: 978-0-8218-2821-2; list Us$15; AMs members Us$12; Order code: GsM/32.M

Fourier Analysis Javier Duoandikoetxea, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Bilbao, Spain

Fourier analysis encompasses a variety of perspectives and techniques. This volume presents the real variable methods of Fourier analysis intro-duced by Calderón and Zygmund. The text was born from a graduate course taught at the Universidad Autónoma de Madrid and incorporates lecture notes from a course taught by José Luis Rubio de Francia at the same university.

Motivated by the study of Fourier series and integrals, classical topics are introduced, such as the Hardy-Littlewood maximal function and the Hilbert transform. The remaining portions of the text are devoted to the study of singular integral operators and multipliers. Both classical aspects of the theory and more recent developments, such as weighted inequali-ties, 1 , spaces, and the 1 theorem, are discussed.

Chapter 1 presents a review of Fourier series and integrals; Chapters 2 and 3 introduce two operators that are basic to the field: the Hardy-Littlewood maximal function and the Hilbert transform. Chapters 4 and 5 discuss singular integrals, including modern generalizations. Chapter 6 studies the relationship between 1 , , and singular integrals; Chapter 7 presents the elementary theory of weighted norm inequalities. Chapter 8 discusses Littlewood-Paley theory, which had developments that resulted in a number of applications. The final chapter concludes with an important result, the 1 theorem, which has been of crucial importance in the field.

This volume has been updated and translated from the Spanish edition that was published in 1995. Minor changes have been made to the core of the book; however, the sections, “Notes and Further Results” have been considerably expanded and incorporate new topics, results, and references.

READERSHIP: Graduate students and research mathematicians inter-ested in Fourier analysis.Volume 29


Functional Analysis An introductionYuli Eidelman and Vitali Milman, Tel Aviv University, Israel, and Antonis Tsolomitis, University of the Aegean, Samos, Greece

The goal of this textbook is to provide an introduction to the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators, and spectral theory of self-adjoint opera-tors. It also presents the basic theorems and methods of abstract functional analysis and a few applications of these methods to Banach algebras and the theory of unbounded self-adjoint operators.

The text corresponds to material for two semester courses (Part I and Part II, respectively), and it is as self-contained as possible. The only prerequisites for the first part are minimal amounts of linear algebra and calculus. However, for the second course (Part II), it is useful to have some knowledge of topology and measure theory. Each chapter is followed by numerous exercises, whose solutions are given at the end of the book.

READERSHIP: Graduate students and research mathematicians inter-ested in operator theory and functional analysis.Volume 66


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Function Theory of One Complex VariableThird EditionRobert E. Greene, University of California, Los Angeles, CA, and Steven G. Krantz, Washington University, St. Louis, MO

Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point.

The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat spaces and Painlevé’s theorem on smoothness to the boundary for conformal maps.

This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors’ expertise both as mathematicians and as expositors.

READERSHIP: Graduate students interested in complex analysis.Volume 40

2006; 504 pp.; hardcover; isBn: 978-0-8218-3962-1; list Us$79; AMs members Us$63; Order code: GsM/40.R

Analysis second EditionElliott H. Lieb, Princeton University, NJ, and Michael Loss, Georgia Institute of Technology, Atlanta, GA

Significantly revised and expanded, this new Second Edition provides readers at all levels—from beginning students to practicing analysts—with the basic concepts and standard tools necessary to solve problems of analysis, and

how to apply these concepts to research in a variety of areas.

Authors Elliott Lieb and Michael Loss take you quickly from basic topics to methods that work successfully in mathematics and its applications. While omitting many usual typical textbook topics, Analysis includes all necessary definitions, proofs, explanations, examples, and exercises to bring the reader to an advanced level of understanding with a minimum of fuss, and, at the same time, doing so in a rigorous and pedagogical way. Many topics that are useful and important, but usually left to advanced monographs, are presented in Analysis, and these give the beginner a sense that the subject is alive and growing.

This new Second Edition incorporates numerous changes since the publication of the original 1997 edition and includes:

Features:• a new chapter on eigenvalues that covers the min-max principle, semi-

classical approximation, coherent states, Lieb-Thirring inequalities, and more

• extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities

• new material on Measure and Integration

• many new exercises• and much more …

The Second Edition continues its no-nonsense approach to the topic that has made it one of the best selling books on the subject. It is an authori-tative, straight-forward volume that readers—from the graduate student, to the professional mathematician, to the physicist or engineer using analytical methods—will find useful both as a reference and as a guide to real problem solving.

READERSHIP: Introductory level graduate students in mathematics; research mathematicians, natural scientists, and engineers interested in learning some of the important tools of modern analysis.Volume 14

2001; 346 pp.; hardcover; isBn: 978-0-8218-2783-3; list Us$41; AMs members Us$33; Order code: GsM/14.R

Principles of Functional Analysis second EditionMartin Schechter, University of California, Irvine, CA

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.

READERSHIP: Advanced undergraduates, graduate students, and pure and applied research mathematicians interested in functional analysis and operator theory.Volume 36


Elliott H. LiebMichael Loss

AnAlysis


Volume 14

American Mathematical SocietyFOUNDED 1888

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AnAlysisAnAlysisGRADUATE sTUDiEs in MAThEMATiCs AnAlysis

Measure Theory and integration Michael E. Taylor, University of North Carolina, Chapel Hill, NC

This self-contained treatment of measure and integration begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. From there the reader is led to the general notion of measure, to the construction of the Lebesgue integral on a

measure space, and to the major limit theorems, such as the Monotone and Dominated Convergence Theorems. The treatment proceeds to spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to 2 spaces as Hilbert spaces, with a useful geometrical structure.

Having gotten quickly to the heart of the matter, the text proceeds to broaden its scope. There are further constructions of measures, including Lebesgue measure on -dimensional Euclidean space. There are also discussions of surface measure, and more generally of Riemannian mani-folds and the measures they inherit, and an appendix on the integration of differential forms. Further geometric aspects are explored in a chapter on Hausdorff measure. The text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales.

READERSHIP: This text will prepare graduate students for more advanced studies in functional analysis, harmonic analysis, stochastic analysis, and geometric measure theory.Volume 76


Complex Made simple David C. Ullrich, Oklahoma State University, Stillwater, OK

Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from?

In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work.

Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem.

READERSHIP: This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis..Volume 97


An invitation to Operator TheoryY. A. Abramovich, Indiana University–Purdue University, Indianapolis, IN, and C. D. Aliprantis, Purdue University, West Lafayette, IN

Volume 50


Problems in Operator TheoryY. A. Abramovich, Indiana University–Purdue University, Indianapolis, IN, and C. D. Aliprantis, Purdue University, West Lafayette, IN

Volume 51


Pick interpolation and hilbert Function spacesJim Agler, University of California at San Diego, CA, and John E. McCarthy, Washington University, St. Louis, MO

Material is wonderfully presented, and the book serves as a lovely introduction to the subject. It is written by two authorities in the field, and helps grad students get entry into an exciting, modern, and very active research area.

—Palle JorgensenVolume 44


Pseudo-differential Operators and the nash–Moser TheoremSerge Alinhac and Patrick Gérard, Université Paris-Sud, Orsay, France

Translated by Stephen S. Wilson

Volume 82


Cones and DualityCharalambos D. Aliprantis, Purdue University, West Lafayette, IN, and Rabee Tourky, The University of Queensland, Brisbane, Queensland, Australia

Volume 84


C -Algebras and Finite-Dimensional ApproximationsNathanial P. Brown, Pennsylvania State University, State College, PA, and Narutaka Ozawa, University of California, Los Angeles, CA, and University of Tokyo, Japan

Volume 88



Volume 76

Measure Theory and integration

Michael E. Taylor










AnAlysis


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A Course in Operator TheoryJohn B. Conway, University of Tennessee, Knoxville, TN

Volume 21


Theta Constants, Riemann surfaces and the Modular GroupAn introduction with Applications to Uniformization Theorems, Partition identities and Combinatorial number TheoryHershel M. Farkas, The Hebrew University, Jerusalem, Israel, and Irwin Kra, State University of New York, Stony Brook, NY

Volume 37


The integrals of lebesgue, Denjoy, Perron, and henstockRussell A. Gordon, Whitman College, Walla Walla, WA

Volume 4


Fundamentals of the Theory of Operator Algebras. Volume iElementary TheoryRichard V. Kadison, University of Pennsylvania, Philadelphia, and John R. Ringrose, University of Newcastle, Newcastle upon Tyne, England

Drawing on their rich experience, the authors have succeeded in presenting a very attractive and well-written book that conveys the flavor and the beauty of classical operator algebra theory and that should be ideally suited as a text for a graduate course on the subject.

-- Zentralblatt MATHVolume 15


Fundamentals of the Theory of Operator Algebras. Volume iiAdvanced TheoryRichard V. Kadison, University of Pennsylvania, Philadelphia, and John R. Ringrose, University of Newcastle, Newcastle upon Tyne, England

Volume 16


A Companion to AnalysisA second First and First second Course in AnalysisT. W. Körner, University of Cambridge, England

This book not only provides a lot of solid information about real analysis, it also answers those questions which students want to ask but cannot figure how to formulate. To read this book is to spend time with one of the modern masters in the subject.

—Steven G. Krantz, Washington University, St. LouisVolume 62


Applied Asymptotic Analysis Peter D. Miller, University of Michigan, Ann Arbor, MI

The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is known as the Courant point of view!!

--Percy Deift, Courant Institute, New YorkVolume 75


An introduction to Measure and integrationsecond EditionInder K. Rana, Indian Institute of Technology, Powai, Mumbai, IndiaThe AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka.

Volume 45


Topics in Optimal TransportationCédric Villani, École Normale Supérieure de Lyon, France

Cedric Villani’s book is a lucid and very readable documentation of the tremendous recent analytic progress in “optimal mass transportation” theory and of its diverse and unexpected applications in optimization, nonlinear PDE, geometry, and mathematical physics.

—Lawrence C. Evans, University of California at BerkeleyVolume 58


GSM Series books have played a profound role in the teaching and research activities in our department. We sincerely hope that this important series of books continues to develop and grow.

— Glen R. Himmelberg University of Missouri, Columbia

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ApplicAtions

Classical and Quantum ComputationA. Yu. Kitaev, California Institute of Technology, Pasadena, CA, and A. H. Shen and M. N. Vyalyi, Independent University of Moscow, Russia

This book presents a concise introduction to an emerging and increasingly important topic, the theory of quantum computing. The development of quantum computing exploded in 1994 with the

discovery of its use in factoring large numbers—an extremely difficult and time-consuming problem when using a conventional computer. In less than 300 pages, the authors set forth a solid foundation to the theory, including results that have not appeared elsewhere and improvements on existing works.

The book starts with the basics of classical theory of computation, including NP-complete problems and the idea of complexity of an algorithm. Then the authors introduce general principles of quantum computing and pass to the study of main quantum computation algo-rithms: Grover’s algorithm, Shor’s factoring algorithm, and the Abelian hidden subgroup problem. In concluding sections, several related topics are discussed (parallel quantum computation, a quantum analog of NP-completeness, and quantum error-correcting codes).

This is a suitable textbook for a graduate course in quantum computing. Prerequisites are very modest and include linear algebra, elements of group theory and probability, and the notion of an algorithm (on a formal or an intuitive level). The book is complete with problems, solutions, and an appendix summarizing the necessary results from number theory.

The aim of the book is to teach the wonders of the qubit-algorithms. While other books, such as Nielsen-Chuang, serve as (more or less) comprehensive references, the present book is focused on complexity. Mathematical prerequisites are minimal, but a reader with some understanding of basic ideas from CS, and quantum theory will get more out of Kitaev, et al ... Really well done, and nicely updated; a handy appendix was added, covering elementary math terms that are used ... The book does a great job in explaining the fundamentals ... The big question is why some qubit-algorithms are a lot better than classical counterparts ... a reader comes away with a good understanding of this in the end.

-- Palle Jorgensen

READERSHIP: Advanced undergraduates, graduate students, research mathematicians, physicists, and computer scientists interested in computer science and quantum theory.Volume 47

2002; 257 pp.; softcover; isBn: 978-0-8218-3229-5; list Us$38; AMs members Us$30; Order code: GsM/47.s

introduction to the Mathematics of FinanceR. J. Williams, University of California, San Diego, La Jolla, CA

This introduction to topics in the evolving area of mathematical finance progresses from the development of the basic ideas of hedging and pricing of derivatives to the application of the Black-Scholes model. Readers of this monograph will be able to go on to read more advanced texts and research papers in mathematical finance, a field that has under-gone significant development over the past two decades.

The text begins by outlining the basic ideas of hedging and pricing of derivatives in the discrete time, discrete state setting of binomial

tree models. A general discrete finite market model is defined, under which the fundamental theorems of asset pricing are proved. In this discrete framework, key probabilistic concepts of conditional expecta-tion, martingale, change of measure and martingale representation are first used.

The author then moves to the continuous time, continuous state setting, where the concepts of Brownian motion and stochastic calculus come into play. The Black-Scholes model offers the simplest model in the continuous setting. The book concludes with a description of the funda-mental theorems of asset pricing for a continuous time, continuous state market model that generalizes the Black-Scholes model in several directions.

The text is clearly written and well-arranged and most of the results are proved in detail. Each chapter is completed with exercises, which makes the textbook very comprehensive.

-- EMS Newsletter

READERSHIP: Graduate students interested in financial mathematics.Volume 72


Option Pricing and Portfolio OptimizationModern Methods of Financial MathematicsRalf Korn and Elke Korn, University of Kaiserslautern, Germany

Especially useful for students seeking a lively introduction to Itô calculus.

— Short Book Reviews, International Statistical Institute

Volume 31


Concise numerical Mathematics Robert Plato, Technical University of Berlin, Germany

From a review of the German edition:

Appealing result of [the author's] endeavours ... The presentation is concise ... avoiding unnecessary redundancies, but nevertheless is self-contained ... even instructors are offered new views and insights ... the author offers many well-chosen exercises ... The book really is a valuable contribution to the literature on its subject.

— Zentralblatt MATHVolume 57

2003; 453 pp.; hardcover; isBn: 978-0-8218-2953-0; list Us$85; AMs members Us$68; Order code: GsM/57 Also Available in Softcover 2003; 453 pp.; softcover; isBn: 978-0-8218-3414-5; list Us$55; AMs members Us$44; Order code: GsM/57.s

APPliCATiOns







differentiAl equAtions

Partial Differential Equations Lawrence C. Evans, University of California, Berkeley, CA

This text presents a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDE), with particular emphasis on nonlinear equations. The exposition is divided into three parts: 1) representation formulas for solutions, 2) theory for linear partial differential equations, and 3) theory for nonlinear partial differential equations.

Included are complete treatments of the method of characteristics, energy methods, regularity for second-order elliptic, parabolic, and hyperbolic equations, maximum principles, the multidimensional calculus of varia-tions, viscosity solutions of Hamilton–Jacobi equations, shock waves and entropy criteria for conservation laws, and much more. The author also summarizes in appendices the relevant mathematical background required.

While he has reworked and simplified much of the classical theory, the author primarily emphasizes the modern interplay between func-tional analytic insights and calculus-type estimates within the context of Sobolev spaces. The treatment of all topics is complete and self-contained. The book’s wide scope and clear exposition make it a suitable text for a graduate course in PDE.

READERSHIP: Graduate students and research mathematicians inter-ested in PDEs.Volume 19


lectures on Elliptic and Parabolic Equations in sobolev spacesN. V. Krylov, University of Minnesota, Minneapolis, MN

Volume 96


A Geometric Approach to Free Boundary ProblemsLuis Caffarelli, University of Texas, Austin, TX, and Sandro Salsa, Politecnico di Milano, Italy

Volume 68


lectures on Analytic Differential EquationsYulij Ilyashenko, Cornell University, Ithaca, NY, and Independent University of Moscow, Russia, and Sergei Yakovenko, Weizmann Institute of Science, Rehovot, Israel

Volume 86


lectures on Elliptic and Parabolic Equations in hölder spacesN. V. Krylov, University of Minnesota, Minneapolis, MN

Volume 12


introduction to the Theory of Differential inclusionsGeorgi V. Smirnov, University of Porto, Portugal

Volume 41


discrete mAthemAtics And combinAtorics

A Course on the Web GraphAnthony Bonato, Wilfrid Laurier University, Waterloo, ON, Canada

This textbook offers a comprehensive examina-tion of Internet mathematics, an active field of study with many recent research developments. It is the first mathematically rigorous textbook that discusses both models of the web graph and algorithms for searching the web. The text encourages interest in the fascinating area of

graph theory and theoretical computer science. This book is jointly published by the AMS and the Atlantic Association for Research in the Mathematical Sciences (AARMS).

Volume 89


Combinatorial RigidityJack Graver, Syracuse University, NY, and Brigitte Servatius and Herman Servatius, Worcester Polytechnic Institute, MA

Volume 2


An introduction to Gröbner Bases William W. Adams, University of Maryland, College Park, MD, and Philippe Loustaunau, George Mason University, Fairfax, VA

Volume 3


A Course in ConvexityAlexander Barvinok, University of Michigan, Ann Arbor, MI

Volume 54



Volume 89

A Course onthe Web Graph

Anthony Bonato

American Mathematical Society

Atlantic Association for Researchin the Mathematical Sciences

DiFFEREnTiAl EQUATiOns













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geometry And topology

A Course in Differential GeometryThierry Aubin, University of Paris, France

This textbook for graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the impor-tant theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III addresses integration of vector fields and -plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds by deducing global prop-erties from local properties of curvature, the final goal being to determine the manifold completely. Chapter VI explores some problems in PDEs suggested by the geometry of manifolds.

The author is well known for his significant contributions to the field of geometry and PDEs—particularly for his work on the Yamabe problem—and for his expository accounts on the subject.

The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.

READERSHIP: Graduate students, research mathematicians, and math-ematics educators interested in differential geometry.Volume 27


Differential Geometry, lie Groups, and symmetric spacesSigurdur Helgason, Massachusetts Institute of Technology, Cambridge, MA

The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason’s classic Differential Geometry, Lie Groups, and Symmetric Spaces has been—and continues to be—the standard source for this material.

Helgason begins with a concise, self-contained introduction to differen-tial geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduc-tion and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over C and Cartan’s classification of simple Lie algebras over R.

The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book.

Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.

READERSHIP: Graduate students and research mathematicians inter-ested in differential geometry, Lie groups, and symmetric spaces.Volume 34


Topics in Differential Geometry Peter W. Michor, Universität Wien, Austria, and Erwin Schrödinger Institut für Mathematische Physik, Wien, Austria

This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections,

Riemann manifolds, isometric actions, and symplectic and Poisson geometry.

The layout of the material stresses naturality and functoriality from the beginning and is as coordinate-free as possible. Coordinate formulas are always derived as extra information. Some attractive unusual aspects of this book are as follows:• Initial submanifolds and the Frobenius theorem for distributions of

nonconstant rank (the Stefan-Sussman theory) are discussed.• Lie groups and their actions are treated early on, including the slice

theorem and invariant theory.• De Rham cohomology includes that of compact Lie groups, leading to

the study of (nonabelian) extensions of Lie algebras and Lie groups.• The Frölicher-Nijenhuis bracket for tangent bundle valued differential

forms is used to express any kind of curvature and second Bianchi identity, even for fiber bundles (without structure groups). Riemann geometry starts with a careful treatment of connections to geodesic structures to sprays to connectors and back to connections, going via the second and third tangent bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point of view.

• Symplectic and Poisson geometry emphasizes group actions, momentum mappings, and reductions.

This book gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good knowledge of under-graduate analysis and linear algebra.

READERSHIP: Graduate students, research mathematicians and physi-cists interested in differential geometry, mechanics, and relativity.Volume 93


Elements of Combinatorial and Differential TopologyV. V. Prasolov, Independent University of Moscow, Russia

This book focuses on the methods of combinatorial topology and those of differential topology, whose use can be applied to solving many topo-logical problems. The text departs from most other books in topology by advancing the study of the properties of topological spaces without the use of complicated techniques.

The book starts with a consideration of graphs, the simplest topological objects. Chapter 1 concludes with a detailed discussion of the polyno-mial invariants of graphs, which have been the subject of keen interest since the discovery of their relationship to knot invariants. Subsequent chapters discuss Euclidean space with standard topology and elements of general topology, where three properties that facilitate the study of topological spaces are considered (Hausdorffness, normality, and para-compactness).

Chapter 5 turns to an examination of differential topology, considering smooth manifolds and the application of smooth maps to topology. This includes introducing basic tools and proving existence theorems for

GEOMETRy AnD TOPOlOGy


Volume 93

Topics in DifferentialGeometry

Peter W. Michor








embeddings and immersions, which play an important role in the study of smooth manifolds. The section on differential topology also includes explicit calculations of fundamental groups for some spaces and to appli-cations of fundamental groups.

The book contains useful figures and many problems of varying degrees of difficulty. Simpler exercises do not come with solutions, but problems and challenging problems (the latter making up the content of a whole scientific paper) have solutions given at the end of the book.

READERSHIP: Advanced undergraduates and graduate students inter-ested in combinatorial and differential topology.Volume 74


Elements of homology Theory V. V. Prasolov, Independent University of Moscow, Russia

Offering a natural continuation of Prasolov’s earlier book Elements of Combinatorial and Differential Topology, this text begins with an exami-nation of simplicial homology and cohomology and proceeds to address singular homology and cohomology. The book’s many applications, exercises and examples will appeal to graduate students with an interest in algebraic topology.

The author departs from other texts on the subject by defining simplicial homology and cohomology at the outset, as simplicial homology has a simpler and more natural definition than singular homology. Chapter 2 considers an important algebraic structure on cohomology: the cup product of Kolmogorov and Alexander, which is particularly useful in the case of manifolds. Once applications of simplicial homology are discussed, the author moves to defining singular homology and cohomology and describing some of their applications.

The book’s final chapter is devoted largely to homology theory’s applica-tions to the topology of manifolds. This includes a detailed account of the Alexander polynomials and a proof of the strong Whitney embedding theorem.

Numerous problems and their solutions are offered, as are many exer-cises. Although the text is geared toward graduate students at a variety of levels, many of the book’s applications will be of interest even to experts in the field.

READERSHIP: Graduate students interested in algebraic topology.Volume 81


Global AnalysisDifferential Forms in Analysis, Geometry and PhysicsIlka Agricola and Thomas Friedrich, Humboldt University, Berlin, Germany

Volume 52


The General Topology of Dynamical systemsEthan Akin, City College (CUNY), New York, NY

Volume 1


Recurrence and TopologyJohn M. Alongi, Northwestern University, Evanston, IL, and Gail S. Nelson, Carleton College, Northfield, MN

Volume 85


An introduction to symplectic GeometryRolf Berndt, University of Hamburg, Germany

Volume 26


Discrete Differential Geometryintegrable structureAlexander I. Bobenko, Technische Universität Berlin, Germany, and Yuri B. Suris, Technische Universität München, Garching bei München, GermanyVolume 98 2008; approximately 401 pp.; hardcover; isBn: 978-0-8218-4700-8; list Us$69; AMs members Us$55; Order code: GsM/98

A Course in Metric GeometryDmitri Burago, Pennsylvania State University, University Park, PA, and Yuri Burago and Sergei Ivanov, Steklov Institute of Mathematics, St. Petersburg, Russia

Volume 33


Foliations iAlberto Candel, California Institute of Technology, Pasadena, CA, and Lawrence Conlon, Washington University, St. Louis, MO

Volume 23


GEOMETRy AnD TOPOlOGy

A first-rate series of monographs and texts. Whether on classical material or on more advanced, modern topics, there is always an interesting angle, a fresh point of view, both for the beginning graduate student and for the working mathematician. Besides, one can count on impeccable layout and typesetting, and on an affordable price. One of the best things to have happened in mathematics publishing in recent memory; may it stay that way.

— Ioannis Karatzas Columbia University

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AnAlysisGRADUATE sTUDiEs in MAThEMATiCs GEOMETRy AnD TOPOlOGy

Where you can …

Get the best deals• special Member Pricing —

up to 20% off

• Online sales every month — discounts up to 75%

Browse new and forthcoming books

• View the Table of Contents

• Read a sample chapter

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Find the right textbook for your course• specifically designed for

undergraduate or graduate courses

www.ams.org/bookstore

Foliations iiAlberto Candel, California State University, Northridge, CA, and Lawrence Conlon, Washington University, St. Louis, MO

Volume 60


hamilton’s Ricci FlowBennett Chow, University of California, San Diego, La Jolla, CA, Peng Lu, University of Oregon, Eugene, OR, and Lei Ni, University of California, San Diego, La Jolla, CAThis book is co-published with Science Press.

Volume 77


lecture notes in Algebraic TopologyJames F. Davis and Paul Kirk, Indiana University, Bloomington, IN

Volume 35


Functions of several Complex Variables and Their singularitiesWolfgang Ebeling, Leibniz Universität Hannover, Germany

Translated by Philip G. Spain

Volume 83


introduction to the -PrincipleY. Eliashberg, Stanford University, CA, and N. Mishachev, Lipetsk Technical University, Russia

Volume 48


Dirac Operators in Riemannian GeometryThomas Friedrich, Humboldt-Universität, Berlin, Germany

Volume 25


4-Manifolds and Kirby CalculusRobert E. Gompf, University of Texas, Austin, TX, and András I. Stipsicz, ELTE, TTK, Budapest, Hungary

Volume 20


secondary Cohomology OperationsJohn R. Harper, University of Rochester, NY

Volume 49

2002; 268 pp.; hardcover; isBn: 978-0-8218-3198-4; list Us$67; AMs members Us$54; Order code: GsM/49 Also Available in Softcover 2002; 268 pp.; softcover; isBn: 978-0-8218-3270-7; list Us$51; AMs members Us$41; Order code: GsM/49.s

Cartan for BeginnersDifferential Geometry via Moving Frames and Exterior Differential systemsThomas A. Ivey, College of Charleston, SC, and J.M. Landsberg, Georgia Institute of Technology, Atlanta, GA

Volume 61


Curves and surfaces Sebastián Montiel and Antonio Ros, Universidad de Granada, SpainThis book is jointly published by the AMS and the Real Sociedad Matemática Española (RSME).

Volume 69


notes on seiberg-Witten TheoryLiviu I. Nicolaescu, University of Notre Dame, IN

Volume 28


Modern Geometric structures and FieldsS. P. Novikov, University of Maryland, College Park, MD, and I. A. Taimanov, Russian Academy of Sciences, Novosibirsk, Russia

Volume 71


Global CalculusS. Ramanan, Chennai Mathematics Institute, India

Volume 65


several Complex Variables with Connections to Algebraic Geometry and lie GroupsJoseph L. Taylor, University of Utah, Salt Lake City, UT

Volume 46

















logic And foundAtions

Discovering Modern set Theory. i: The BasicsWinfried Just, Ohio University, Athens, OH, and Martin Weese, Humboldt University, Berlin, Germany

This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of math-ematics. Topics covered include formal languages

and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn’s Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.

READERSHIP: Graduate students and mathematicians interested in set-theoretical techniques.Volume 8


Discovering Modern set Theory. ii: set-Theoretic Tools for Every MathematicianWinfried Just, Ohio University, Athens, and Martin Weese, University of Potsdam, Germany

Volume 18


mAthemAticAl physics

Quantum Mechanics for MathematiciansLeon A. Takhtajan, Stony Brook University, NY

Quantum physics has been a driving force behind many dramatic achievements in mathematics, yet its study has never been a significant part of the graduate mathematics curriculum. This book, based on graduate courses taught by the author, offers a comprehensive and self-contained introduction to quantum mechanics that is accessible for the general mathematics audience.

Topics covered in the book include mathematical foundations, quantiza-tion, the Schrödinger equation, the Feynman path integral and functional methods, and supersymmetry. The book adopts the approach that classical mechanics and quantum mechanics are simply two different realizations of the fundamental mathematical structure of a physical theory that uses the notions of observables, states, measurement and the time evolution — dynamics.

The author is a well-known expert in mathematical problems of quantum mechanics, quantum field theory and statistical physics. In this book he pays careful attention to precise mathematics formulation of methods and results. Numerous problems are included to help the reader master the subject.

Development of many disciplines of mathematics has been stimulated by quantum mechanics. This book provides the reader with fundamental knowledge of quantum mechanics while also potentially serving as a bridge to study more advanced topics, such as quantum field theory.

READERSHIP: Graduate students and research mathematicians inter-ested in mathematical aspects of quantum mechanics.Volume 95


lOGiC AnD FOUnDATiOns | MAThEMATiCAl PhysiCs


Volume 8


Winfried JustMartin Weese

Discovering Modern set Theory. i: The Basics

Where you can …

Get the best deals• special Member Pricing —

up to 20% off

• Online sales every month — discounts up to 75%

Browse new and forthcoming books

• View the Table of Contents

• Read a sample chapter

• Browse the “What’s new” section for upcoming titles

Find the right textbook for your course• specifically designed for

undergraduate or graduate courses

www.ams.org/bookstore

The AMS Bookstore






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number theory

Basic Quadratic Forms Larry J. Gerstein, University of California, Santa Barbara, CA

The arithmetic theory of quadratic forms is a rich branch of number theory that has had important applications to several areas of pure mathematics—particularly group theory and topology—as well as to cryptography and coding theory. This book is a self-contained introduc-tion to quadratic forms that is based on graduate courses the author has taught many times. It leads the reader from foundation material up to topics of current research interest—with special attention to the theory over the integers and over polynomial rings in one variable over a field—and requires only a basic background in linear and abstract algebra as a prerequisite. Whenever possible, concrete constructions are chosen over more abstract arguments. The book includes many exercises and explicit examples, and it is appropriate as a textbook for graduate courses or for independent study. To facilitate further study, a guide to the extensive literature on quadratic forms is provided.

READERSHIP: Graduate students interested in number theory and algebra. Mathematicians seeking an introduction to the study of quadratic forms on lattices over the integers and related rings.Volume 90


Algebraic number Fieldssecond EditionGerald J. Janusz, University of Illinois, Urbana, IL

The book is directed toward students with a minimal background who want to learn class field theory for number fields. The only prerequisite for reading it is some elementary Galois theory. The first three chapters lay out the necessary back-ground in number fields, such as the arithmetic of

fields, Dedekind domains, and valuations. The next two chapters discuss class field theory for number fields. The concluding chapter serves as an illustration of the concepts introduced in previous chapters. In particular, some interesting calculations with quadratic fields show the use of the norm residue symbol.

For the second edition the author added some new material, expanded many proofs, and corrected errors found in the first edition. The main objective, however, remains the same as it was for the first edition: to give an exposition of the introductory material and the main theorems about class fields of algebraic number fields that would require as little background preparation as possible. Janusz’s book can be an excellent textbook for a year-long course in algebraic number theory; the first three chapters would be suitable for a one-semester course. It is also very suit-able for independent study.

READERSHIP: Mathematics graduate students and faculty.Volume 7


spectral Methods of Automorphic Formssecond EditionHenryk Iwaniec, Rutgers University, Piscataway, NJ

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style.

The first edition of this volume was an underground classic, both as a textbook and as a respected source for results, ideas, and references. The book’s reputation sparked a growing interest in the mathematical community to bring it back into print. The AMS has answered that call with the publication of this second edition.

In the book, Iwaniec treats the spectral theory of automorphic forms as the study of the space 2 , where is the upper half-plane and is a discrete subgroup of volume-preserving transformations of . He combines various techniques from analytic number theory. Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues.

Henryk Iwaniec was awarded the 2002 Cole Prize for his fundamental contributions to number theory.

READERSHIP: Graduate students and researchers working in analytic number theory.This book is copublished by the AMS and Revista Matemática Iberoamericana (RMI), Madrid, Spain.

Volume 53


An invitation to Arithmetic GeometryDino Lorenzini, University of Georgia, Athens, GA

Volume 9


number TheoryAlgebraic numbers and FunctionsHelmut Koch, Humboldt-University, Berlin, Germany

Volume 24


Modular Forms, a Computational ApproachWilliam Stein, University of Washington, Seattle, WA

with an appendix by Paul E. Gunnells

This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments.

—John E. Cremona, University of NottinghamVolume 79


nUMBER ThEORy


Volume 7


Gerald J. Janusz

Algebraic number Fieldssecond Edition








probAbility

Probability Theory in Finance A Mathematical Guide to the Black-scholes FormulaSeán Dineen, University College Dublin, Ireland

The use of the Black-Scholes model and formula is pervasive in financial markets. There are very few undergraduate textbooks available on the subject and, until now, almost none written by mathematicians. Based on a course given by the author, the goal of this book is to introduce advanced undergraduates and beginning graduate students studying the mathematics of finance to the Black-Scholes formula. The author uses a first-principles approach, developing only the minimum background necessary to justify mathematical concepts and placing mathematical developments in context.

The book skillfully draws the reader toward the art of thinking mathemat-ically and then proceeds to lay the foundations in analysis and probability theory underlying modern financial mathematics. It rigorously reveals the mathematical secrets of topics such as abstract measure theory, conditional expectations, martingales, Wiener processes, the Itô calculus, and other ingredients of the Black-Scholes formula. In explaining these topics, the author uses examples drawn from the universe of finance. The book also contains many exercises, some included to clarify simple points of exposition, others to introduce new ideas and techniques, and a few containing relatively deep mathematical results.

With the modest prerequisite of a first course in calculus, the book is suitable for undergraduates and graduate students in mathematics, finance, and economics and can be read, using appropriate selections, at a number of levels.

The book by Seán Dineen is recommended to students in the program, “Master of Science in Financial Engineering” at Kent State University. The book is an especially appreciated reference since it is very self-contained and so requires little or no background material that is not already within its pages.

—Richard M. Aron, Kent State University

For budding financial engineers, this is an outstanding introduction to the mathematics that underlies derivatives pricing theory....It also offers plenty of exercises that will help you build mastery.

—Glyn Holton, founder of Contingency Analysis and Riskbook.com

READERSHIP: Undergraduate and graduate students in mathematics, economics, or finance interested in studying probability theory and other mathematics used in finance.Volume 70


Probability Davar Khoshnevisan, University of Utah, Salt Lake City, UT

Probability has a profound role in mathematics, science and engineering and is taught in an increasing number of graduate courses. This text departs from others designed for graduate-level work by presenting a cohesive graduate course in measure-theoretic probability that has the one-semester student in mind.

Khoshnevisan, one of this generation’s leading probabilists, uses a lucid writing style to develop ideas central to modern probability theory and showcase them by presenting their applications. Early chapters of the book focus on a proof of the De Moivre-Laplace central limit theorem and on basic results in analysis. Later chapters discuss Brownian motion and stochastic integration, including the Itô integral, an important ingre-dient in the applications of probability to mathematical finance.

The author has included numerous exercises of varying levels of diffi-culty in the text. Each chapter also concludes with historical notes. Throughout the text, Khoshnevisan strikes a balance between explaining to the reader and challenging the reader to think.

While this book has ample material to cover a typical year-long course at a more leisurely pace, it is ideal for a one-semester course in that it focuses on the themes most central to probability. In this way it reflects the author’s distinct ability to achieve brevity and depth at the same time in examining this important topic.

READERSHIP: Graduate students interested in probability theory.Volume 80


stochastic Analysis on ManifoldsElton P. Hsu, Northwestern University, Evanston, IL

Volume 38


introduction to the Theory of Random ProcessesN. V. Krylov, University of Minnesota, Minneapolis, MN

Volume 43


The Ergodic Theory of Discrete sample PathsPaul C. Shields, University of Toledo, OH

Volume 13


PROBABiliTy

The Graduate Studies in Mathematics is a great series of mathematical textbooks, covering a wide variety of topics on different levels. The amount of exercises in the books as well as the style makes them an ideal choice for graduate courses, reading classes and individual study. I personally have used books from the series for classes as well as pointing my graduate students towards the series for further readings.

— Gestur Olafsson Louisiana State University

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Probability


Volume 80

Davar Khoshnevisan








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AnAlysisGRADUATE sTUDiEs in MAThEMATiCs inDEx

AAbramovich, Y. A.. . . . . . . . 8Adams, William W. . . . . . . 11Agler, Jim . . . . . . . . . . . . . . 8Agricola, Ilka . . . . . . . . . . 13Akin, Ethan . . . . . . . . . . . 13Algebraic Curves and

Riemann Surfaces . . . . . . 3Algebraic Number Fields . . 16Alinhac, Serge . . . . . . . . . . . 8Aliprantis, C. D. . . . . . . . . . 8Alongi, John M. . . . . . . . . 13Analysis . . . . . . . . . . . . . . . 7Applied Asymptotic Analysis . 9Aubin, Thierry . . . . . . . . . 12

BBartle, Robert G. . . . . . . . . 6Barvinok, Alexander . . . . . 11Basic Quadratic Forms . . . 16Berndt, Rolf . . . . . . . . . . . 13Bobenko, Alexander I. . . . 13Bonato, Anthony . . . . . . . 11Brown, Nathanial P. . . . . . . 8Burago, Dmitri . . . . . . . . . 13Burago, Yuri . . . . . . . . . . . 13

cC*-Algebras and

Finite-Dimensional Approximations . . . . . . . . 8

Caffarelli, Luis . . . . . . . . . 11Candel, Alberto . . . . . 13, 14Cartan for Beginners. . . . . 14Chow, Bennett . . . . . . . . . 14Classical and Quantum

Computation . . . . . . . . . 10Classical Groups and

Geometric Algebra . . . . . . 4Clemens, C. Herbert . . . . . 4Combinatorial Rigidity . . . 11A Companion to Analysis . . . 9Complex Made Simple . . . . . 8Concise Numerical

Mathematics . . . . . . . . . 10Cones and Duality . . . . . . . 8Conlon, Lawrence . . . 13, 14Conway, John B. . . . . . . . . 9A Course in Algebra . . . . . . 4A Course in Convexity . . . . 11A Course in Differential

Geometry . . . . . . . . . . . . 12A Course in Metric Geometry

. . . . . . . . . . . . . . . . . . . 13A Course in Operator Theory 9A Course on the Web Graph 11Curves and Surfaces . . . . . 14Cutkosky, Steven Dale . . . . 4

dDavis, James F. . . . . . . . . . 14Differential Geometry, Lie

Groups, and Symmetric Spaces . . . . . . . . . . . . . . 12

Dineen, Seán . . . . . . . . . . 17Dirac Operators in

Riemannian Geometry . 14Discovering Modern Set

Theory. I: The Basics . . . . 15

Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician . . . . . . . 15

Discrete Differential Geometry . . . . . . . . . . . . . . . . . . . 13

Dixmier, Jacques . . . . . . . . . 4Duoandikoetxea, Javier . . . . 6Dym, Harry . . . . . . . . . . . . 3

EEbeling, Wolfgang . . . . . . 14Eidelman, Yuli . . . . . . . . . . 6Elements of Combinatorial

and Differential Topology 12Elements of Homology Theory

. . . . . . . . . . . . . . . . . . . 13Eliashberg, Y. . . . . . . . . . . 14Enveloping Algebras . . . . . . 4Evans, Lawrence C. . . . . . 11

FFarkas, Hershel M. . . . . . . . 9Finite Group Theory . . . . . . 5Foliations I . . . . . . . . . . . . 13Foliations II . . . . . . . . . . . 144-Manifolds and Kirby

Calculus . . . . . . . . . . . . 14Fourier Analysis . . . . . . . . . 6Friedrich, Thomas . . . 13, 14Function Theory of One

Complex Variable . . . . . . 7Functional Analysis . . . . . . . 6Functions of Several Complex

Variables and Their Singularities . . . . . . . . . 14

Fundamentals of the Theory of Operator Algebras. Volume I . . . . . . . . . . . . . . . . . . . . 9

Fundamentals of the Theory of Operator Algebras. Volume II . . . . . . . . . . . . . . . . . . 9

GA Geometric Approach to Free

Boundary Problems . . . . 11Gérard, Patrick . . . . . . . . . . 8Gerstein, Larry J. . . . . . . . 16Global Analysis . . . . . . . . . 13Global Calculus . . . . . . . . . 14Gompf, Robert E. . . . . . . 14Gordon, Russell A. . . . . . . . 9Graduate Algebra:

Commutative View . . . . . 3Graduate Algebra:

Noncommutative View . . . 5Graver, Jack . . . . . . . . . . . 11Greene, Robert E. . . . . . . . 7Grove, Larry C. . . . . . . . . . 4Growth of Algebras and

Gelfand-Kirillov Dimension . . . . . . . . . . . . . . . . . . . . 5

HHamilton’s Ricci Flow . . . . 14Harper, John R. . . . . . . . . 14Helgason, Sigurdur . . . . . . 12Hong, Jin . . . . . . . . . . . . . . 4Hsu, Elton P. . . . . . . . . . . 17Humphreys, James E. . . . . . 5

IIlyashenko, Yulij . . . . . . . . 11

An Introduction to Gröbner Bases . . . . . . . . . . . . . . . 11

An Introduction to Measure and Integration . . . . . . . . 9

Introduction to Quadratic Forms over Fields . . . . . . . 5

Introduction to Quantum Groups and Crystal Bases . 4

An Introduction to Symplectic Geometry . . . . . . . . . . . . 13

Introduction to the -Principle . . . . . . . . . . . . . . . . . . . 14

Introduction to the Mathematics of Finance . . . . . . . . . . . 10

Introduction to the Theory of Differential Inclusions . . . 11

Introduction to the Theory of Random Processes . . . . . 17

An Invitation to Arithmetic Geometry . . . . . . . . . . . . 16

An Invitation to Operator Theory . . . . . . . . . . . . . . . 8

Isaacs, I. Martin . . . . . . . . . 5Ivanov, Sergei . . . . . . . . . . 13Ivey, Thomas A. . . . . . . . . 14Iwaniec, Henryk . . . . . . 5, 16Iyengar, Srikanth B. . . . . . . 5

JJantzen, Jens Carsten . . . . . 5Janusz, Gerald J. . . . . . . . . 16Just, Winfried . . . . . . . . . . 15

KKadison, Richard V. . . . . . . 9Kang, Seok-Jin . . . . . . . . . . 4Khoshnevisan, Davar . . . . . 17Kirillov, A. A. . . . . . . . . . . . 5Kirk, Paul . . . . . . . . . . . . . 14Kitaev, A. Yu. . . . . . . . . . . 10Koch, Helmut . . . . . . . . . 16Korn, Elke . . . . . . . . . . . . 10Korn, Ralf. . . . . . . . . . . . . 10Körner, T. W. . . . . . . . . . . . 9Kra, Irwin . . . . . . . . . . . . . . 9Krantz, Steven G. . . . . . . . . 7Krause, Günter R. . . . . . . . . 5Krylov, N. V. . . . . . . . 11, 17

LLam, T.Y. . . . . . . . . . . . . . . 5Landsberg, J.M. . . . . . . . . 14Lecture Notes in Algebraic

Topology . . . . . . . . . . . . . 14Lectures on Analytic

Differential Equations . . 11Lectures on Elliptic and

Parabolic Equations in Hölder Spaces . . . . . . . . 11

Lectures on Elliptic and Parabolic Equations in Sobolev Spaces . . . . . . . . 11

Lectures on Quantum Groups . . . . . . . . . . . . . . 5Lectures on the Orbit Method 5Lenagan, Thomas H. . . . . . 5Leuschke, Graham J. . . . . . 5Leykin, Anton. . . . . . . . . . . 5Lieb, Elliott H. . . . . . . . . . . 7Linear Algebra in Action . . 3Lorenzini, Dino . . . . . . . . 16Loss, Michael . . . . . . . . . . . 7

Loustaunau, Philippe . . . . 11Lu, Peng . . . . . . . . . . . . . 14

MMcCarthy, John E. . . . . . . . 8McConnell, J. C. . . . . . . . . 5Measure Theory and

Integration . . . . . . . . . . . 8Michor, Peter W. . . . . . . . 12Miller, Claudia . . . . . . . . . . 5Miller, Ezra . . . . . . . . . . . . 5Miller, Peter D. . . . . . . . . . 9Milman, Vitali . . . . . . . . . . 6Miranda, Rick . . . . . . . . . . . 3Mishachev, N. . . . . . . . . . . 14Modern Geometric Structures

and Fields . . . . . . . . . . . 14A Modern Theory of

Integration . . . . . . . . . . . 6Modular Forms, a

Computational Approach 16Montiel, Sebastián . . . . . . 14

NNelson, Gail S. . . . . . . . . . 13Ni, Lei . . . . . . . . . . . . . . . 14Nicolaescu, Liviu I. . . . . . . 14Noncommutative Noetherian

Rings . . . . . . . . . . . . . . . 5Notes on Seiberg-Witten Theory

. . . . . . . . . . . . . . . . . . . 14Novikov, S. P. . . . . . . . . . . 14Number Theory . . . . . . . . . 16

OOption Pricing and Portfolio

Optimization . . . . . . . . . 10Ozawa, Narutaka . . . . . . . . 8

PPartial Differential Equations

. . . . . . . . . . . . . . . . . . . 11Pick Interpolation and Hilbert

Function Spaces . . . . . . . . 8Plato, Robert . . . . . . . . . . 10Prasolov, V. V. . . . . . . 12, 13Principles of Functional

Analysis . . . . . . . . . . . . . . 7Probability . . . . . . . . . . . . 17Probability Theory in Finance

. . . . . . . . . . . . . . . . . . . 17Problems in Operator Theory 8Pseudo-differential Operators

and the Nash–Moser Theorem . . . . . . . . . . . . . 8

QQuantum Mechanics for

Mathematicians . . . . . . . 15

RRamanan, S. . . . . . . . . . . . 14Rana, Inder K. . . . . . . . . . . 9Recurrence and Topology . . . 13Representation Theory of

Finite Groups: Algebra and Arithmetic . . . . . . . . . . . . 5

Representations of Finite and Compact Groups . . . . . . . 4

Representations of Semisimple Lie Algebras in the BGG Category . . . . . . . . . . . 5

Resolution of Singularities . . . 4Ringrose, John R. . . . . . . . . 9

Robson, J. C. . . . . . . . . . . . 5Ros, Antonio . . . . . . . . . . 14Rowen, Louis Halle . . . . 3, 5

SSalsa, Sandro . . . . . . . . . . 11Schechter, Martin . . . . . . . . 7A Scrapbook of Complex Curve

Theory . . . . . . . . . . . . . . . 4Secondary Cohomology

Operations . . . . . . . . . . . 14Servatius, Brigitte . . . . . . . 11Servatius, Herman . . . . . . 11Several Complex Variables with

Connections to Algebraic Geometry and Lie Groups 14

Shen, A. H. . . . . . . . . . . . 10Shields, Paul C. . . . . . . . . 17Simon, Barry . . . . . . . . . . . 4Singh, Anurag K. . . . . . . . . 5Smirnov, Georgi V. . . . . . . 11Solutions Manual to A Modern

Theory of Integration . . . . 6Spectral Methods of

Automorphic Forms . . . . 16Stein, William . . . . . . . . . . 16Stipsicz, András I. . . . . . . . 14Stochastic Analysis on

Manifolds . . . . . . . . . . . 17Suris, Yuri B. . . . . . . . . . . 13

TTaimanov, I. A. . . . . . . . . . 14Takhtajan, Leon A. . . . . . . 15Taylor, Joseph L. . . . . . . . 14Taylor, Michael E. . . . . . . . 8The Ergodic Theory of Discrete

Sample Paths . . . . . . . . . 17The General Topology of

Dynamical Systems . . . . . 13The Integrals of Lebesgue,

Denjoy, Perron, and Henstock . . . . . . . . . . . . . 9

Theta Constants, Riemann Surfaces and the Modular Group . . . . . . . . . . . . . . . 9

Topics in Classical Automorphic Forms . . . . . 5

Topics in Differential Geometry . . . . . . . . . . . . 12

Topics in Optimal Transportation . . . . . . . . 9

Tourky, Rabee . . . . . . . . . . 8Tsolomitis, Antonis . . . . . . . 6Twenty-Four Hours of Local

Cohomology . . . . . . . . . . . 5

UUllrich, David C. . . . . . . . . 8

VVillani, Cédric. . . . . . . . . . . 9Vinberg, E. B. . . . . . . . . . . 4Vyalyi, M. N. . . . . . . . . . . 10

WWalther, Uli . . . . . . . . . . . . 5Weese, Martin. . . . . . . . . . 15Weintraub, Steven H. . . . . . 5Williams, R. J. . . . . . . . . . 10

YYakovenko, Sergei . . . . . . . 11


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