PUBLICATIONS of the American Mathematical Society up to AMS Individual 20% Member Discount Receive substantial discounts on all AMs published and co-published books! AMS Textbooks Find 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/textbooks To Prospective Contents Authors 3 Featured Selections If you would like to submit a manuscript to the AMs, please visit www.ams.org/authors 13 Algebra and Algebraic Geometry 15 Analysis 16 Applications 16 Differential Equations Applied Mathematics The AMs book publication program on 17 Discrete Mathematics and Combinatorics applied and interdisciplinary mathematics 18 Geometry and Topology strengthens the connections between white --> mathematics and other disciplines, 19 Logic and Foundations highlighting the areas where mathematics 20 Mathematical Physics is most relevant. These publications help mathematicians understand how 21 number Theory mathematical ideas may benefit other 22 Applied Mathematics sciences, while offering researchers outside of mathematics important tools to advance 24 Memoirs of the AMs their profession. to view all of our applied 25 AMs-Distributed Publications mathematics publications, go to: www.ams.org/bookstore/appliedmath 26 Index 30 ordering Information Order Online | www.ams.org/bookstore fE aturED Selections Elliptic Partial Differential Equations TEXTBOOK TEXTBOOKS 1 Second Edition FROM THE AMS Q I N G H A N Qing Han, University of Notre Dame, IN, and Fanghua Lin, Courant Institute, New York F A N G H U A L I N University, NY TEXTBOOK Elliptic Partial This volume is based on PDE courses given by the authors at the Courant Institute and at the Differential Equations University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori esti- Second Edition mates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems. This second edition has been thoroughly revised and in a new chapter the authors discuss several methods for proving the existence of solutions of primarily the Dirichlet problem for various types of elliptic equations. READERSHIP: Graduate students and research mathematicians interested in elliptic PDEs. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University. Courant Lecture Notes, Volume 1 2011; 147 pp.; softcover; ISBN: 978-0-8218-5313-9; List US$31; AMS members US$24.80; Order code: CLN/1.R Table of Contents Contents Preface ix Chapter 1. Harmonic Functions Chapter 2. Maximum Principles Chapter 3. Weak Solutions: Part I Chapter 4. Weak Solutions, Part II Chapter 5. Viscosity Solutions Chapter 6. Existence of Solutions Bibliography Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things about it—it is a wonderful book. [ —Tobias Colding ] 3 Order Online | www.ams.org/bookstore Featured Selections FeaturED Selections TEXTBOOK a Basic Course in Partial Differential Equations TEXTBOOKS FROM THE AMS A Basic Course in Qing Han, University of Notre Dame, IN Partial Differential Equations Among the many introductory texts on partial differential equations, this treatment departs from TEXTBOOK others in its awareness of the more advanced subjects in the background. In both explaining how to solve linear PDEs and teaching readers how to think about general equations, this text carries Qing Han relevance for students who will not pursue the subject further as well as for those who will go on to more advanced courses on PDEs. Based on one-semester courses taught by the author at the University of Notre Dame, the text Graduate Studies in Mathematics features the main extra ingredient of estimates for solutions of PDEs, primarily the “a priori” esti- Volume 120 mates. These estimates are crucial tools for proving the existence and uniqueness of solutions to American Mathematical Society PDEs, especially significant for nonlinear equations. Another important emphasis of this book involves its focus on simple models of PDEs, with detailed study of these equations. Chapters are devoted to the Laplace equation, the heat equa- tion and the wave equation, and several methods are used for the study of each, exploring their advantages and disadvantages. This text is designed specifically for use by first-year graduate students in mathematics, but also can be used by students in applied mathematics, physics or engineering. In deriving explicit expressions of solutions and a priori estimates, the book serves as both a concise overview and an ideal introduction to further study. READERSHIP: Advanced undergraduate and graduate students interested in PDEs. Graduate Studies in Mathematics, Volume 120 2011; 293 pp.; hardcover; ISBN: 978-0-8218-5255-2; List US$63; AMS members US$50.40; Order code: GSM/120 TEXTBOOK a Course in Minimal Surfaces TEXTBOOKS FROM THE AMS Tobias Holck Colding, Massachusetts Institute of Technology, Cambridge, MA, and A Course in William P. Minicozzi II, Johns Hopkins University, Baltimore, MD Minimal Surfaces TEXTBOOK Minimal surfaces date back to Euler and Lagrange, and many of the techniques developed have been critical to the study of geometry and partial differential equations. This book offers a thor- Tobias Holck Colding William P. Minicozzi II ough introduction to classical results and modern techniques in minimal surfaces, while also offering significant coverage of current research topics. The authors, recipients of the Oswald Veblen Prize in 2010 for their work on minimal surfaces, Graduate Studies in Mathematics open with the basic properties of minimal surfaces, including the monotonicity formula for area Volume 121 and the Bernstein theorem. Chapters 2 and 3 address generalizations of the Bernstein theorem and American Mathematical Society a proof of a generalization. The authors proceed with a solution to the classical Plateau problem for maps from surfaces, and later discuss the theory of minimal surfaces in three-manifolds. Among the various ways of approaching the subject of minimal surfaces, these experts have chosen to focus on aspects of the theory related to partial differential equations. The book also includes applications of minimal surfaces to fields such as low dimensional topology, general relativity and materials science. Only a basic knowledge of differential geometry and some familiarity with partial differential equa- tions are required of readers. The book includes many illustrations and exercises, and is ideal for an advanced undergraduate or second-year graduate course. READERSHIP: Graduate students and research mathematicians interested in the theory of minimal surfaces. Graduate Studies in Mathematics, Volume 121 2011; 313 pp.; hardcover; ISBN: 978-0-8218-5323-8; List US$63; AMS members US$50.40; Order code: GSM/121 4 Order by Phone | 1-401-455-4000 or 1-800-321-4267 Featured Selections FeaturED Selections algebraic Groups and Differential Galois theory TEXTBOOK TEXTBOOKS Teresa Crespo, Universitat de Barcelona, Spain, and Zbigniew Hajto, Jagiellonian Algebraic GroupsFROM THE AMS University, Kraków, Poland and Differential Galois Theory Differential Galois theory has been the subject of major research activity in recent years, covering TEXTBOOK areas ranging from elaboration of more general theories to applications to classical and quantum Teresa Crespo mechanics. This book introduces the topic through a self-contained presentation of Picard-Vessiot Zbigniew Hajto theory, or Galois theory of linear differential equations. The first two parts of the book, on algebraic geometry and algebraic groups, include all questions Graduate Studies in Mathematics necessary to develop differential Galois theory. Chapter 5 addresses the Picard-Vessiot extension, Volume 122 speaking of this as a field in order to keep the analogy with the splitting field in the polynomial Galois theory. The last chapter presents classical notions such as the monodromy group, Fuchsian American Mathematical Society equations and hypergeometric equations. Each chapter ends with exercise statements that involve direct application of the theory as well as working out some topics beyond it. The clarity of exposition and the completeness of proofs enhance the appeal of the text. The book’s inclusion of the results of algebraic geometry and algebraic groups necessary to develop Galois theory in its whole makes the material accessible to graduate students with a knowledge of commutative algebra and complex analysis at an undergraduate level. The book can form the basis for a graduate course of one or two semesters. READERSHIP: Graduate students and research mathematicians interested in algebraic methods in differential equations, differential Galois theory, and dynamical systems. Graduate